Research ArticleNumerical Solution of First-Order Linear DifferentialEquations in Fuzzy Environment by Runge-Kutta-FehlbergMethod and Its Application

Sankar Prasad Mondal1 Susmita Roy1 and Biswajit Das2

1Department of Mathematics National Institute of Technology Agartala Jirania Tripura 799046 India2Department of Mechanical Engineering National Institute of Technology Agartala Jirania Tripura 799046 India

Correspondence should be addressed to Sankar Prasad Mondal sankarres07gmailcom

Received 31 July 2015 Accepted 16 February 2016

Academic Editor Yucheng Liu

Copyright copy 2016 Sankar Prasad Mondal et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The numerical algorithm for solving ldquofirst-order linear differential equation in fuzzy environmentrdquo is discussed A scheme namelyldquoRunge-Kutta-Fehlberg methodrdquo is described in detail for solving the said differential equation The numerical solutions arecompared with (i)-gH and (ii)-gH differential (exact solutions concepts) system The method is also followed by complete erroranalysis The method is illustrated by solving an example and an application

1 Introduction

Fuzzy Differential Equation In modeling of real naturalphenomena differential equations play an important role inmany areas of discipline exemplary in economics biomathe-matics science and engineering Many experts in such areaswidely use differential equations in order to make someproblems under study more comprehensible In many casesinformation about the physical phenomena related is alwaysimmanent with uncertainty

Today the study of differential equationswith uncertaintyis instantaneously growing as a new area in fuzzy analysisThe terms such as ldquofuzzy differential equationrdquo and ldquofuzzydifferential inclusionrdquo are used interchangeably in mentionto differential equations with fuzzy initial values or fuzzyboundary values or even differential equations dealing withfunctions on the space of fuzzy numbers In the year 1987the term ldquofuzzy differential equation (FDE)rdquo was introducedby Kandel and Byatt [1] There are different approaches todiscuss the FDEs (i) the Hukuhara derivative of a fuzzynumber valued function is used (ii) Hullermeier [2] andDiamond andWatson [3ndash5] suggested a different formulationfor the fuzzy initial value problems (FIVP) based on a familyof differential inclusions (iii) in [6 7] Bede et al defined

generalized differentiability of the fuzzy number valuedfunctions and studied FDE and (iv) applying a parametricrepresentation of fuzzy numbers Chen et al [8] establisheda new definition for the differentiation of a fuzzy valuedfunction and used it in FDE

Solution of Fuzzy Differential Equation by Numerical Tech-niques Numerical methods are the methods by which wecan find the solution of differential equation where the exactsolution is critical to find There exist various numericalmethods for solving differential equation such as Setia etal [9] Liu [10] and Setia et al [11] Our aim is to findthe numerical techniques by which the solution of a linearor nonlinear first-order fuzzy differential equation comeseasily and the solution is very close to the exact solutionThere exist many techniques of numerical methods forfinding the solution of fuzzy differential equation Authorsapplied the method in certain types of fuzzy differentialequation which shows that their techniques are best fit forthat particular problem The first paper on fuzzy differentialequation and numerical analysis was published in 1999 byMa et al [12] Allahviranloo et al [13] apply the two-stepmethod on fuzzy differential equations Allahviranloo et al[14] find the numerical solution by using predictor-corrector

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2016 Article ID 8150497 14 pageshttpdxdoiorg10115520168150497

2 International Journal of Differential Equations

method Allahviranloo et al [15] find an algorithm for findingthe solution 119873th-order fuzzy linear differential equationsusing numerical techniques Pederson and Sambandhamin [16] use characterization theorem on hybrid fuzzy ini-tial value problem A soft computing technique namelyartificial neural network is implicated for solving FDE byEffati and Pakdaman [17] Duraisamy and Usha [18] usedmodified Eulerrsquos method The extension principle methodwas compared by Eulerrsquos method in Saberi Najafi et al [19]article Rostami et al [20] find a numerical algorithm forsolving nonlinear fuzzy differential equations Moghadamand Dahaghin [21] apply two-step methods for numericalsolution of FDE Batiha in [22] finds an iterative solutionof multispecies predator-prey model by variational iterationmethod Ahmad and Hasan [23] proposed a new fuzzyversion of Eulerrsquos method for solving differential equationswith fuzzy initial value Nirmala and Chenthur Pandian[24] give an idea for improving the numerical result onFDE Shafiee et al [25] use predictor-corrector method fornonlinear fuzzy Volterra integral equations Comparisonresults on some numerical techniques on first-order fuzzydifferential equation are illustrated by Ghanbari [26]The useof variational iteration method for solving 119873th-order fuzzydifferential equations is shown by Jafari et al [27] Tapaswiniand Chakraverty [28] discuss a new approach to fuzzy initialvalue problem by Improved Euler method The solution ofFIVP is compared by Least Square method and AdomianDecomposition method by Ahmed and Fadhel [29] Solutionof differential equation by Eulerrsquosmethod using fuzzy conceptis developed by Saikia [30] Ezzati et al [31] find the numericalsolution of Volterra-Fredholm integral equations with thehelp of inverse and direct discrete fuzzy transforms andcollocation technique The Adomian method is applied onsecond-order FDE by Wang and Guo [32] Fard [33] usesiterative scheme to find the solution of generalized system oflinear FDE whereas Blockmethod is used byMehrkanoon etal [34] Asady and Alavi [35] apply a numerical method forsolving119873th-order linear fuzzy differential equation

Solution of Fuzzy Differential Equation by Runge-KuttaMethod Runge-Kutta method is well known for finding theapproximate or numerical solution In the last decade Runge-Kutta method is applied in fuzzy differential equation forfinding the numerical solution The researchers are givingvarious types of view to apply these methods Someonechanges the order and someone applies different types onFDE a comparison of another method to Runge-Kuttamethod The details of published work done in Runge-Kuttamethod are summarized below

Numerical Solution of Fuzzy Differential Equationsby Runge-Kutta method of order three is developed byDuraisamy and Usha [36] Solution techniques for fourth-order Runge-Kutta method with higher order derivativeapproximations are developed by Nirmala and ChenthurPandian [37] Runge-Kutta method of order five is devel-oped by Jayakumar et al [38] The techniques extendedRunge-Kutta-like formulae of order four are developed byGhazanfari and Shakerami [39] Third-order Runge-Kuttamethod is developed by Kanagarajan and Sambath [40]

Runge-Kutta-Fehlberg method for hybrid fuzzy differentialequation is solved by Jayakumar and Kanagarajan [41]A different approach followed by Runge-Kutta method isapplied by Akbarzadeh Ghanaie and Mohseni Moghadam[42] ldquoNumerical Solution of Fuzzy IVP with Trapezoidaland Triangular Fuzzy Numbers by Using Fifth-Order Runge-Kutta Methodrdquo is solved by Ghanbari [43] ldquoNew Multi-Step Runge-Kutta Method for Solving Fuzzy DifferentialEquationsrdquo is solved by Nirmala and Chenthur Pandian [44]ldquoNumerical Solution of Fuzzy Hybrid Differential Equationby Third Order Runge-Kutta Nystrom Methodrdquo is solved bySaveetha and Chenthur Pandian [45] A new approach tosolve fuzzy differential equation by using third-order Runge-Kutta method is developed by Deshmukh [46] Runge-Kutta method of order four is developed by Duraisamy andUsha [47] and order five is developed by Jayakumar andKanagarajan [48]

Application of Fuzzy Differential Equation Fuzzy differentialequations play a significant role in the fields of biologyengineering and physics as well as among other fieldsof science for example in population models [49] civilengineering [50] bioinformatics and computational biology[51] quantum optics and gravity [52] modeling hydraulic[53] HIVmodel [54] decaymodel [55] predator-preymodel[56] population dynamics model [57] friction model [58]growthmodel [59] bacteria culturemodel [60] bank accountand drug concentration problem [61] barometric pressureproblem [62] concentration problem [63] weight loss andoil production model [64] arm race model [65] vibration ofmass [66] and fractional predator-prey equation [67]

Novelties Although some developments are done some newinterest and newwork have been done by ourselves which arementioned below

(i) Differential equation is solved in fuzzy environmentby numerical techniques that is coefficients andinitial condition both are taken as fuzzy numberof a differential equation and solved by numericaltechniques

(ii) The numerical solution is compared with the exactsolution ((i)-gH and (ii)-gH both cases)

(iii) Runge-Kutta-Fehlberg method for solving fuzzy dif-ferential equation is used

(iv) For application purpose a mixture problem is consid-ered

(v) The solutions are found using different step length forbetter accuracy of the result

(vi) The necessary algorithm for numerical solution isgiven

Structure of the Paper The paper is organized as follows inPreliminary Concepts the preliminary concepts and basicconcepts on fuzzy number and fuzzy derivative are givenThe method for finding the exact solution is discussed inExact Solution of Fuzzy Differential Equation In Numer-ical Solution of Fuzzy Differential Equation we proposed

International Journal of Differential Equations 3

Runge-Kutta-Fehlberg method in fuzzy environment Theconvergence of the said method and algorithm for findingthe numerical results are also discussed in this sectionNumerical Example shows a numerical example In Appli-cation an important application namely mixture problemis illustrated in fuzzy environment Finally conclusions andfuture research scope of this paper are drawn in last sectionConclusion

2 Preliminary Concepts

Definition 1 (fuzzy set) A fuzzy set is defined by =

(119909 120583

119860

(119909)) 119909 isin 119860 120583

119860

(119909) isin [0 1] In the pair (119909 120583

119860

(119909))

the first element 119909 belongs to the classical set 119860 and thesecond element 120583

119860

(119909) belongs to the interval [0 1] calledmembership function

Definition 2 (120572-cut of a fuzzy set) The 120572-level set (or intervalof confidence at level 120572 or 120572-cut) of the fuzzy set of 119883 isa crisp set 119860

120572

that contains all the elements of 119883 that havemembership values in 119860 greater than or equal to 120572 that is = 119909 120583

119860

(119909) ge 120572 119909 isin 119883 0 lt 120572 le 1

Definition 3 (fuzzy number) The basic definition of fuzzynumber is as follows [30] if we denote the set of all realnumbers by R and the set of all fuzzy numbers on R isindicated by RF then a fuzzy number is mapping such that119906 R rarr [0 1] which satisfies the following four properties

(i) 119906 is upper semicontinuous(ii) 119906 is a fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for all 119909 119910 isin R 120582 isin [0 1](iii) 119906 is normal that is exist119909

0

isin R for which 119906(1199090

) = 1(iv) supp 119906 = 119909 isin R | 119906(119909) gt 0 is support of 119906 and the

closure of (supp 119906) is compact

Definition 4 (parametric form of fuzzy number [31]) Afuzzy number is represented by an ordered pair of functions(1199061

(120572) 1199062

(120572)) 0 le 120572 le 1 that satisfy the following condition(1) 1199061

(120572) is a bounded left continuous nondecreasingfunction for any 120572 isin [0 1]

(2) 1199062

(120572) is a bounded left continuous nonincreasingfunction for any 120572 isin [0 1]

(3) 1199061

(120572) le 1199062

(120572) for any 120572 isin [0 1]

Note If 1199061

(120572) = 1199062

(120572) = 120572 then 120572 is a crisp number

Definition 5 (generalized Hukuhara difference [20]) Thegeneralized Hukuhara difference of two fuzzy numbers 119906 V isin

RF is defined as follows

119906⊖gHV = 119908 lArrrArr

(i) 119906 = V oplus 119908

or (ii) V = 119906 oplus (minus1)119908

(1)

Consider [119908]120572

= [1199081

(120572) 1199081

(120572)] then 1199081

(120572) = min1199061

(120572) minus

V1

(120572) 1199062

(120572) minus V2

(120572) and1199082

(120572) = max1199061

(120572) minus V1

(120572) 1199062

(120572) minus

V2

(120572)

Here the parametric representation of a fuzzy valuedfunction 119891 [119886 119887] rarr RF is expressed by [119891(119905)]

120572

=

[1198911

(119905 120572) 1198912

(119905 120572)] 119905 isin [119886 119887] 120572 isin [0 1]

Definition 6 (generalized Hukuhara derivative for first order[20]) The generalized Hukuhara derivative of a fuzzy valuedfunction 119891 (119886 119887) rarr RF at 119905

0

is defined as

1198911015840

(1199050

) = limℎrarr0

119891 (1199050

+ ℎ) ⊖gH119891 (1199050

)

ℎ (2)

If 1198911015840

(1199050

) isin RF satisfying (2) exists we say that 119891 isgeneralized Hukuhara differentiable at 119905

0

Also we say that 119891(119905) is (i)-gH differentiable at 1199050

if

[1198911015840

(1199050

)]120572

= [1198911015840

1

(1199050

120572) 1198911015840

2

(1199050

120572)] (3)

and 119891(119905) is (ii)-gH differentiable at 1199050

if

[1198911015840

(1199050

)]120572

= [1198911015840

2

(1199050

120572) 1198911015840

1

(1199050

120572)] (4)

Definition 7 (see [6]) For arbitrary 119906 = (1199061

1199062

) and V =

(V1

V2

) isin 1198641 the quantity

119863 (119906 V) = (int

1

0

(1199061

minus V1

)2

+ int

1

0

(1199062

minus V2

)2

)

12

(5)

is the distance between fuzzy numbers 119906 and V

Definition 8 (triangular fuzzy number) A triangular fuzzynumber (TFN) denoted by is defined as (119886 119887 119888) where themembership function

120583

119860

(119909) =

0 119909 le 119886

119909 minus 119886

119887 minus 119886 119886 le 119909 le 119887

1 119909 = 119887

119888 minus 119909

119888 minus 119887 119887 le 119909 le 119888

0 119909 ge 119888

(6)

Definition 9 (120572-cut of a fuzzy set ) The 120572-cut of = (119886 119887 119888)

is given by

119860120572

= [119886 + 120572 (119887 minus 119886) 119888 minus 120572 (119888 minus 119887)] forall120572 isin [0 1] (7)

Definition 10 (fuzzy ordinary differential equation (FODE))Consider a simple 1st-order linear ordinary differential equa-tion as follows

119889119909

119889119905= 119896119909 + 119909

0

with initial condition 119909 (1199050

) = 120574 (8)

The above ordinary differential equation is called fuzzyordinary differential equation if any one of the following threecases holds

(i) Only 120574 is a fuzzy number (Type-I)(ii) Only 119896 is a fuzzy number (Type-II)(iii) Both 119896 and 120574 are fuzzy numbers (Type-III)

4 International Journal of Differential Equations

3 Exact Solution ofFuzzy Differential Equation

Consider the fuzzy initial value problem

1199101015840

(119905) = 119891 (119905 119910 (119905)) 119905 isin 119868 = [0 119879] with 119910 (0) = 1199100

(9)

where 119891 is a continuous mapping from 119877+

times 119877 into 119877 and1199100

isin 119864 with 119903-level sets

[1199100

]119903

= [1199101

(0 120572) 1199102

(0 120572)] 120572 isin (0 1] (10)

We write 119891(119905 119910) = [1198911

(119905 119910) 1198912

(119905 119910)] and 1198911

(119905 119910) =

119865[119905 1199101

1199102

] 1198912

(119905 119910) = 119866[119905 1199101

1199102

]Because of 1199101015840(119905) = 119891(119905 119910) we have the followingWhen 119910(119905 119910) is (i)-gH differentiable

1199101015840

1

(119905 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

1199101015840

2

(119905 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

(11)

When 119910(119905 119910) is (ii)-gH differentiable

1199101015840

2

(119905 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

1199101015840

1

(119905 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

(12)

where by using extension principle we have themembershipfunction

119891 (119905 119910 (119905)) (119904) = Sup 119910 (119905) (120591) 119904 = 119891 (119905 120591) 119904 isin 119877 (13)

So fuzzy number is 119891(119905 119910(119905)) From this it follows that

[119891 (119905 119910 (119905))]120572

= [1198911

(119905 119910 (119905) 120572) 1198912

(119905 119910 (119905) 120572)]

120572 isin [0 1]

(14)

where

1198911

(119905 119910 (119905) 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

= min 119891 (119905 119906) 119906 isin [1199101

(119905 120572) 1199102

(119905 120572)]

1198912

(119905 119910 (119905) 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

= max 119891 (119905 119906) 119906 isin [1199101

(119905 120572) 1199102

(119905 120572)]

(15)

Note (1) Both cases ((i)-gH and (ii)-gH) can be applied to aFDE for finding exact solution

(2) After taking 120572-cut of the given FDE it transforms tosystem of ordinary differential equation

4 Numerical Solution ofFuzzy Differential Equation

41 Runge-Kutta-FehlbergMethod for Ordinary (Crisp) Differ-ential Equation Consider the initial value problem 119910

1015840

(119905) =

119891(119905 119910(119905)) 119910(1199050

) = 1199100

The Runge-Kutta-Fehlberg method (denoted as RKF45)

is one way to try to resolve this problem

The problem is to solve the initial value problem in aboveequation by means of Runge-Kutta methods of order 4 andorder 5

First we need some definitions

1198961

= ℎ119891 (119905119894

119910119894

)

1198962

= ℎ119891(119905119894

+1

4ℎ 119910119894

+1

41198961

)

1198963

= ℎ119891(119905119894

+3

8ℎ 119910119894

+3

321198961

+9

321198962

)

1198964

= ℎ119891(119905119894

+12

13ℎ 119910119894

+1932

21971198961

minus7200

21971198962

+7296

21971198963

)

1198965

= ℎ119891(119905119894

+ ℎ 119910119894

+439

2161198961

minus 81198962

+3680

5131198963

minus845

41041198964

)

1198966

= ℎ119891(119905119894

+ ℎ 119910119894

minus8

271198961

+ 21198962

minus3544

25651198963

+1859

41041198964

minus11

401198965

)

(16)

Then an approximation to the solution of initial valueproblem is made using Runge-Kutta method of order 4

119910119894+1

= 119910119894

+25

2161198961

+1408

25651198963

+2197

41011198964

minus1

51198965

(17)

A better value for the solution is determined using a Runge-Kutta method of order 5

119911119894+1

= 119910119894

+16

1351198961

+6656

128251198963

+2856156430

1198964

minus9

501198965

+2

551198966

(18)

The optimal step size 119904ℎ can be determined bymultiplying thescalar 119904 times the step size ℎ The scalar 119904 is

119904 = (120598ℎ

21003816100381610038161003816119911119894+1 minus 119910

119894+1

1003816100381610038161003816

)

14

= 00840896(120598ℎ

1003816100381610038161003816119911119894+1 minus 119910119894+1

1003816100381610038161003816

)

14

(19)

where 120598 is the specified error control toleranceNote that RK4 requires 4 function evaluations and RK5

requires 6 evaluations that is 10 for RK4 and RK5 Fehlbergdevised a method to get RK4 and RK5 results using only6 function evaluations by using some of 119870 values in bothmethods

42 Runge-Kutta-Fehlberg Method for Solving Fuzzy Differen-tial Equations Let 119884 = [119884

1

1198842

] be the exact solution and let

International Journal of Differential Equations 5

119910 = [1199101

1199102

] be the approximated solution of the fuzzy initialvalue problem

Let [119884(119905)]120572

= [1198841

(119905 120572) 1198842

(119905 120572)] [119910(119905)]119903

= [1199101

(119905 120572) 1199102

(119905

120572)]Throughout this argument the value of 119903 is fixed Then

the exact and approximated solution at 119905119899

are respectivelydenoted by

[119884 (119905119899

)]120572

= [1198841

(119905119899

120572) 1198842

(119905119899

120572)]

[119910 (119905119899

)]120572

= [1199101

(119905119899

120572) 1199102

(119905119899

120572)]

(20)

The grid points at which the solution is calculated are ℎ =

(119879 minus 1199050

)119873 119905119894

= 1199050

+ 119894ℎ 0 le 119894 le 119873Then we obtained

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) +16

1351198701

+6656

128251198703

+2856156430

1198704

minus9

501198705

+2

551198706

(21)

where

1198701

= ℎ119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198702

= ℎ119865(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198701

1199102

(119905119899

120572)

+1

41198701

)

1198703

= ℎ119865(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198701

+9

321198702

1199102

(119905119899

120572) +3

321198701

+9

321198702

)

1198704

= ℎ119865(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

1199102

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

)

1198705

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

1199102

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

)

1198706

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

1199102

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

)

(22)

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) +16

1351198711

+6656

128251198713

+2856156430

1198714

minus9

501198715

+2

551198716

(23)

where

1198711

= ℎ119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198712

= ℎ119866(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198711

1199102

(119905119899

120572)

+1

41198711

)

1198713

= ℎ119866(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198711

+9

321198712

1199102

(119905119899

120572) +3

321198711

+9

321198712

)

1198714

= ℎ119866(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

1199102

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

)

1198715

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

1199102

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

)

1198716

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

1199102

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

)

(24)

43 Convergence of Fuzzy Runge-Kutta-FehlbergMethod Thesolution is calculated by grid points at 119886 = 119905

0

le 1199051

le sdot sdot sdot le

119905119873

= 119887 and ℎ = (119887 minus 119886)119873 = 119905119899+1

minus 119905119899

Therefore we have

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572) + 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) + 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) + 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

(25)

6 International Journal of Differential Equations

Clearly 1199101

(119905 120572) and 1199102

(119905 120572) converge to 1198841

(119905 120572) and 1198842

(119905 120572)respectively whenever ℎ rarr 0 that is

limℎrarr0

1199101

(119905 120572) = 1198841

(119905 120572)

limℎrarr0

1199102

(119905 120572) = 1198842

(119905 120572)

(26)

Proof Before we go to the main proof we need to know someresults

Lemma 11 Let the sequence of numbers 119882119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le 1198601003816100381610038161003816119882119899

1003816100381610038161003816 + 119861 0 le 119899 le 119873 minus 1 (27)

for some given positive constants 119860 and 119861 Then

10038161003816100381610038161198821198991003816100381610038161003816 le 119860119899 10038161003816100381610038161198820

1003816100381610038161003816 + 119861119860119899

minus 1

119860 minus 1 0 le 119899 le 119873 (28)

Lemma 12 Let the sequence of numbers 119882119873

119899=0

and 119881119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 119860max 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 + 119861

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 119860max

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 + 119861

(29)

for some given positive constants 119860 and 119861 and denote

119880119899

=1003816100381610038161003816119882119899

1003816100381610038161003816 +1003816100381610038161003816119881119899

1003816100381610038161003816 0 le 119899 le 119873 (30)

Then

119880119899

le 119860119899

1198800

+ 119861119860119899

minus 1

119860 minus 1

0 le 119899 le 119873 (31)

where 119860 = 1 + 2119860 and 119861 = 2119861

Let 119865(119905 119906 V) and 119866(119905 119906 V) be obtained by substituting[1199101

(119905 120572) 1199102

(119905 120572)] = [119906 V] in (21) and (23) that is

119865 (119905 119906 V) =16

1351198701

(119905 119906 V) +6656

128251198703

(119905 119906 V)

+2856156430

1198704

(119905 119906 V) minus9

501198705

(119905 119906 V)

+2

551198706

(119905 119906 V)

119866 (119905 119906 V) =16

1351198711

(119905 119906 V) +6656

128251198713

(119905 119906 V)

+2856156430

1198714

(119905 119906 V) minus9

501198715

(119905 119906 V)

+2

551198716

(119905 119906 V)

(32)

The domain where 119865 and 119866 are defined is as

119867 = (119905 119906 V) | 0 le 119905 le 119879 minusinfin lt V lt infin minusinfin lt 119906

le V (33)

Theorem 13 Let 119865(119905 119906 V) and 119866(119905 119906 V) belong to 119862119901minus1

(119870)

and let the partial derivative of 119865 and 119866 be bounded over 119870Then for arbitrary fixed 0 le 120572 le 1 the numerical solu-tion of (9) [119910

1

(119905 120572) 1199102

(119905 120572)] converges to the exact solution[1198841

(119905 120572) 1198842

(119905 120572)]

Proof (see [46]) By using Taylorrsquos theorem we get

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572)

+ ℎ119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

1

(1205851198991

)

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ ℎ119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

2

(1205851198992

)

(34)

where 1205851198991

1205851198992

isin (119905119899

119905119899+1

)Now if we denote

119882119899

= 1198841

(119905119899

120572) minus 1199101

(119905119899

120572)

119881119899

= 1198842

(119905119899

120572) minus 1199102

(119905119899

120572)

(35)

then the above two expressions converted to

119882119899+1

= 119882119899

+ ℎ 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

1

(1205851198991

)

119881119899+1

= 119881119899

+ ℎ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

2

(1205851198992

)

(36)

Hence we can write1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 2119871ℎmax 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 +ℎ119901+1

(119901 + 1)119872

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 2119871ℎmax

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 +

ℎ119901+1

(119901 + 1)119872

(37)

where 119872 = maxmax |119884(119901+1)

1

(119905 120572)|max |119884(119901+1)

2

(119905 120572)| for 119905 isin

[0 119879] and 119871 gt 0 is a bound for the partial derivative of 119865 and119866

Therefore we can write1003816100381610038161003816119882119899

1003816100381610038161003816 le (1 + 4119871ℎ)119899 10038161003816100381610038161198800

1003816100381610038161003816 +2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

10038161003816100381610038161198811198991003816100381610038161003816 le (1 + 4119871ℎ)

119899 100381610038161003816100381611988001003816100381610038161003816 +

2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

(38)

where |1198800

| = |1198820

| + |1198810

|

International Journal of Differential Equations 7

In particular

10038161003816100381610038161198821198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

10038161003816100381610038161198811198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

(39)

Since1198820

= 1198810

= 0 we have

10038161003816100381610038161198821198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

10038161003816100381610038161198811198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

(40)

Thus if ℎ rarr 0 we get119882119873

rarr 0 and119881119873

rarr 0 which completesthe proof

44 Algorithm for Finding the Numerical Solution

Step 1 119865(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

119866(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

Step 2 Read 119905(0) 1199101

(0) 1199102

(0) ℎ limit

Step 3 For 119894 = 0(1) limit

1198701

larr ℎ119865(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198702

larr ℎ sdot 119865(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198701

1199102

(119905119894

119903) +

(14)1198701

)

1198703

larr ℎsdot119865(119905119894

+(38)ℎ 1199101

(119905119894

119903)+(332)1198701

+(932)1198702

1199102

(119905119894

119903) + (332)1198701

+ (932)1198702

)

1198704

larr ℎ sdot 119865(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198701

minus

(72002197)1198702

+ (72962197)1198703

1199102

(119905119894

119903) + (1932

2197)1198701

minus (72002197)1198702

+ (72962197)1198703

)

1198705

larr ℎ119865(119905119894

+ℎ 1199101

(119905119894

119903)+(439216)1198701

minus81198702

+(3680

513)1198703

minus (8454104)1198704

1199102

(119905119894

119903) + (439216)1198701

minus

81198702

+ (3680513)1198703

minus (8454104)1198704

)

1198706

larr ℎ119865(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198701

+ 21198702

minus (3544

2565)1198703

+ (18594104)1198704

minus (1140)1198705

1199102

(119905119894

119903) minus

(827)1198701

+ 21198702

minus (35442565)1198703

+ (18594104)1198704

minus

(1140)1198705

)

1198711

larr ℎ sdot 119866(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198712

larr ℎ119866(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198711

1199102

(119905119894

119903) +

(14)1198711

)

1198713

larr ℎ119866(119905119894

+ (38)ℎ 1199101

(119905119894

119903) + (332)1198711

+ (932)1198712

1199102

(119905119894

119903) + (332)1198711

+ (932)1198712

)

1198714

larr ℎ119866(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198711

minus

(72002197)1198712

+ (72962197)1198713

1199102

(119905119894

119903) + (1932

2197)1198711

minus (72002197)1198712

+ (72962197)1198713

)

1198715

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) + (439216)1198711

minus 81198712

+

(3680513)1198713

minus (8454104)1198714

1199102

(119905119894

119903) + (439

216)1198711

minus 81198712

+ (3680513)1198713

minus (8454104)1198714

)

1198716

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198711

+ 21198712

minus

(35442565)1198713

+ (18594104)1198714

minus (1140)1198715

1199102

(119905119894

119903) minus (827)1198711

+ 21198712

minus (35442565)1198713

+ (1859

4104)1198714

minus (1140)1198715

)

Step 4

1199101

(119905119894+1

119903) = 1199101

(119905119894

119903)+(16135)1198701

+(665612825)1198703

+

(2856156430)1198704

minus (950)1198705

+ (255)1198706

Step 5

1199102

(119905119894+1

119903) = 1199102

(119905119894

119903)+(16135)1198711

+(665612825)1198713

+

(2856156430)1198714

minus (950)1198715

+ (255)1198716

Step 6 119905119894+1

= 119905119894

+ ℎ Write 1199101

(119905119894+1

119903) 1199102

(119905119894+1

119903) 119905119894+1

Step 7 Next 119894

Step 8 End

5 Numerical Example

Example Solve 1199101015840

= minus119910 + 119905 + 1 with initial condition 119910(0) =

(096 1 101) Then find the solution at 119905 = 01

Solution For (i)-gH differentiable case the exact solution is

1199101

(119903 119905) = 119905 + (096 + 004119903) 119890minus119905

1199102

(119903 119905) = 119905 + (101 minus 001119903) 119890minus119905

(41)

and for (ii)-gH differentiable case the exact solution is

1199101

(119903 119905) = 1 + 119905 + (minus004 + 004119903) 119890119905

1199102

(119903 119905) = 1 + 119905 + (001 minus 001119903) 119890119905

(42)

Remark 14 From Figure 1 and Table 1 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

001 (for ℎ = 0001 is nearly equal) whereas the lowerexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

Remark 15 From Figure 2 and Table 1 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the upperexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

6 Application

Problem A tank initially contains 300 gals of brine which hasdissolved in 119888 lbs of salt Coming into the tank at 3 galsmin

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

International Journal of Differential Equations 3

Runge-Kutta-Fehlberg method in fuzzy environment Theconvergence of the said method and algorithm for findingthe numerical results are also discussed in this sectionNumerical Example shows a numerical example In Appli-cation an important application namely mixture problemis illustrated in fuzzy environment Finally conclusions andfuture research scope of this paper are drawn in last sectionConclusion

2 Preliminary Concepts

Definition 1 (fuzzy set) A fuzzy set is defined by =

(119909 120583

119860

(119909)) 119909 isin 119860 120583

119860

(119909) isin [0 1] In the pair (119909 120583

119860

(119909))

the first element 119909 belongs to the classical set 119860 and thesecond element 120583

119860

(119909) belongs to the interval [0 1] calledmembership function

Definition 2 (120572-cut of a fuzzy set) The 120572-level set (or intervalof confidence at level 120572 or 120572-cut) of the fuzzy set of 119883 isa crisp set 119860

120572

that contains all the elements of 119883 that havemembership values in 119860 greater than or equal to 120572 that is = 119909 120583

119860

(119909) ge 120572 119909 isin 119883 0 lt 120572 le 1

Definition 3 (fuzzy number) The basic definition of fuzzynumber is as follows [30] if we denote the set of all realnumbers by R and the set of all fuzzy numbers on R isindicated by RF then a fuzzy number is mapping such that119906 R rarr [0 1] which satisfies the following four properties

(i) 119906 is upper semicontinuous(ii) 119906 is a fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for all 119909 119910 isin R 120582 isin [0 1](iii) 119906 is normal that is exist119909

0

isin R for which 119906(1199090

) = 1(iv) supp 119906 = 119909 isin R | 119906(119909) gt 0 is support of 119906 and the

closure of (supp 119906) is compact

Definition 4 (parametric form of fuzzy number [31]) Afuzzy number is represented by an ordered pair of functions(1199061

(120572) 1199062

(120572)) 0 le 120572 le 1 that satisfy the following condition(1) 1199061

(120572) is a bounded left continuous nondecreasingfunction for any 120572 isin [0 1]

(2) 1199062

(120572) is a bounded left continuous nonincreasingfunction for any 120572 isin [0 1]

(3) 1199061

(120572) le 1199062

(120572) for any 120572 isin [0 1]

Note If 1199061

(120572) = 1199062

(120572) = 120572 then 120572 is a crisp number

Definition 5 (generalized Hukuhara difference [20]) Thegeneralized Hukuhara difference of two fuzzy numbers 119906 V isin

RF is defined as follows

119906⊖gHV = 119908 lArrrArr

(i) 119906 = V oplus 119908

or (ii) V = 119906 oplus (minus1)119908

(1)

Consider [119908]120572

= [1199081

(120572) 1199081

(120572)] then 1199081

(120572) = min1199061

(120572) minus

V1

(120572) 1199062

(120572) minus V2

(120572) and1199082

(120572) = max1199061

(120572) minus V1

(120572) 1199062

(120572) minus

V2

(120572)

Here the parametric representation of a fuzzy valuedfunction 119891 [119886 119887] rarr RF is expressed by [119891(119905)]

120572

=

[1198911

(119905 120572) 1198912

(119905 120572)] 119905 isin [119886 119887] 120572 isin [0 1]

Definition 6 (generalized Hukuhara derivative for first order[20]) The generalized Hukuhara derivative of a fuzzy valuedfunction 119891 (119886 119887) rarr RF at 119905

0

is defined as

1198911015840

(1199050

) = limℎrarr0

119891 (1199050

+ ℎ) ⊖gH119891 (1199050

)

ℎ (2)

If 1198911015840

(1199050

) isin RF satisfying (2) exists we say that 119891 isgeneralized Hukuhara differentiable at 119905

0

Also we say that 119891(119905) is (i)-gH differentiable at 1199050

if

[1198911015840

(1199050

)]120572

= [1198911015840

1

(1199050

120572) 1198911015840

2

(1199050

120572)] (3)

and 119891(119905) is (ii)-gH differentiable at 1199050

if

[1198911015840

(1199050

)]120572

= [1198911015840

2

(1199050

120572) 1198911015840

1

(1199050

120572)] (4)

Definition 7 (see [6]) For arbitrary 119906 = (1199061

1199062

) and V =

(V1

V2

) isin 1198641 the quantity

119863 (119906 V) = (int

1

0

(1199061

minus V1

)2

+ int

1

0

(1199062

minus V2

)2

)

12

(5)

is the distance between fuzzy numbers 119906 and V

Definition 8 (triangular fuzzy number) A triangular fuzzynumber (TFN) denoted by is defined as (119886 119887 119888) where themembership function

120583

119860

(119909) =

0 119909 le 119886

119909 minus 119886

119887 minus 119886 119886 le 119909 le 119887

1 119909 = 119887

119888 minus 119909

119888 minus 119887 119887 le 119909 le 119888

0 119909 ge 119888

(6)

Definition 9 (120572-cut of a fuzzy set ) The 120572-cut of = (119886 119887 119888)

is given by

119860120572

= [119886 + 120572 (119887 minus 119886) 119888 minus 120572 (119888 minus 119887)] forall120572 isin [0 1] (7)

Definition 10 (fuzzy ordinary differential equation (FODE))Consider a simple 1st-order linear ordinary differential equa-tion as follows

119889119909

119889119905= 119896119909 + 119909

0

with initial condition 119909 (1199050

) = 120574 (8)

The above ordinary differential equation is called fuzzyordinary differential equation if any one of the following threecases holds

(i) Only 120574 is a fuzzy number (Type-I)(ii) Only 119896 is a fuzzy number (Type-II)(iii) Both 119896 and 120574 are fuzzy numbers (Type-III)

4 International Journal of Differential Equations

3 Exact Solution ofFuzzy Differential Equation

Consider the fuzzy initial value problem

1199101015840

(119905) = 119891 (119905 119910 (119905)) 119905 isin 119868 = [0 119879] with 119910 (0) = 1199100

(9)

where 119891 is a continuous mapping from 119877+

times 119877 into 119877 and1199100

isin 119864 with 119903-level sets

[1199100

]119903

= [1199101

(0 120572) 1199102

(0 120572)] 120572 isin (0 1] (10)

We write 119891(119905 119910) = [1198911

(119905 119910) 1198912

(119905 119910)] and 1198911

(119905 119910) =

119865[119905 1199101

1199102

] 1198912

(119905 119910) = 119866[119905 1199101

1199102

]Because of 1199101015840(119905) = 119891(119905 119910) we have the followingWhen 119910(119905 119910) is (i)-gH differentiable

1199101015840

1

(119905 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

1199101015840

2

(119905 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

(11)

When 119910(119905 119910) is (ii)-gH differentiable

1199101015840

2

(119905 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

1199101015840

1

(119905 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

(12)

where by using extension principle we have themembershipfunction

119891 (119905 119910 (119905)) (119904) = Sup 119910 (119905) (120591) 119904 = 119891 (119905 120591) 119904 isin 119877 (13)

So fuzzy number is 119891(119905 119910(119905)) From this it follows that

[119891 (119905 119910 (119905))]120572

= [1198911

(119905 119910 (119905) 120572) 1198912

(119905 119910 (119905) 120572)]

120572 isin [0 1]

(14)

where

1198911

(119905 119910 (119905) 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

= min 119891 (119905 119906) 119906 isin [1199101

(119905 120572) 1199102

(119905 120572)]

1198912

(119905 119910 (119905) 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

= max 119891 (119905 119906) 119906 isin [1199101

(119905 120572) 1199102

(119905 120572)]

(15)

Note (1) Both cases ((i)-gH and (ii)-gH) can be applied to aFDE for finding exact solution

(2) After taking 120572-cut of the given FDE it transforms tosystem of ordinary differential equation

4 Numerical Solution ofFuzzy Differential Equation

41 Runge-Kutta-FehlbergMethod for Ordinary (Crisp) Differ-ential Equation Consider the initial value problem 119910

1015840

(119905) =

119891(119905 119910(119905)) 119910(1199050

) = 1199100

The Runge-Kutta-Fehlberg method (denoted as RKF45)

is one way to try to resolve this problem

The problem is to solve the initial value problem in aboveequation by means of Runge-Kutta methods of order 4 andorder 5

First we need some definitions

1198961

= ℎ119891 (119905119894

119910119894

)

1198962

= ℎ119891(119905119894

+1

4ℎ 119910119894

+1

41198961

)

1198963

= ℎ119891(119905119894

+3

8ℎ 119910119894

+3

321198961

+9

321198962

)

1198964

= ℎ119891(119905119894

+12

13ℎ 119910119894

+1932

21971198961

minus7200

21971198962

+7296

21971198963

)

1198965

= ℎ119891(119905119894

+ ℎ 119910119894

+439

2161198961

minus 81198962

+3680

5131198963

minus845

41041198964

)

1198966

= ℎ119891(119905119894

+ ℎ 119910119894

minus8

271198961

+ 21198962

minus3544

25651198963

+1859

41041198964

minus11

401198965

)

(16)

Then an approximation to the solution of initial valueproblem is made using Runge-Kutta method of order 4

119910119894+1

= 119910119894

+25

2161198961

+1408

25651198963

+2197

41011198964

minus1

51198965

(17)

A better value for the solution is determined using a Runge-Kutta method of order 5

119911119894+1

= 119910119894

+16

1351198961

+6656

128251198963

+2856156430

1198964

minus9

501198965

+2

551198966

(18)

The optimal step size 119904ℎ can be determined bymultiplying thescalar 119904 times the step size ℎ The scalar 119904 is

119904 = (120598ℎ

21003816100381610038161003816119911119894+1 minus 119910

119894+1

1003816100381610038161003816

)

14

= 00840896(120598ℎ

1003816100381610038161003816119911119894+1 minus 119910119894+1

1003816100381610038161003816

)

14

(19)

where 120598 is the specified error control toleranceNote that RK4 requires 4 function evaluations and RK5

requires 6 evaluations that is 10 for RK4 and RK5 Fehlbergdevised a method to get RK4 and RK5 results using only6 function evaluations by using some of 119870 values in bothmethods

42 Runge-Kutta-Fehlberg Method for Solving Fuzzy Differen-tial Equations Let 119884 = [119884

1

1198842

] be the exact solution and let

International Journal of Differential Equations 5

119910 = [1199101

1199102

] be the approximated solution of the fuzzy initialvalue problem

Let [119884(119905)]120572

= [1198841

(119905 120572) 1198842

(119905 120572)] [119910(119905)]119903

= [1199101

(119905 120572) 1199102

(119905

120572)]Throughout this argument the value of 119903 is fixed Then

the exact and approximated solution at 119905119899

are respectivelydenoted by

[119884 (119905119899

)]120572

= [1198841

(119905119899

120572) 1198842

(119905119899

120572)]

[119910 (119905119899

)]120572

= [1199101

(119905119899

120572) 1199102

(119905119899

120572)]

(20)

The grid points at which the solution is calculated are ℎ =

(119879 minus 1199050

)119873 119905119894

= 1199050

+ 119894ℎ 0 le 119894 le 119873Then we obtained

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) +16

1351198701

+6656

128251198703

+2856156430

1198704

minus9

501198705

+2

551198706

(21)

where

1198701

= ℎ119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198702

= ℎ119865(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198701

1199102

(119905119899

120572)

+1

41198701

)

1198703

= ℎ119865(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198701

+9

321198702

1199102

(119905119899

120572) +3

321198701

+9

321198702

)

1198704

= ℎ119865(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

1199102

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

)

1198705

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

1199102

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

)

1198706

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

1199102

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

)

(22)

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) +16

1351198711

+6656

128251198713

+2856156430

1198714

minus9

501198715

+2

551198716

(23)

where

1198711

= ℎ119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198712

= ℎ119866(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198711

1199102

(119905119899

120572)

+1

41198711

)

1198713

= ℎ119866(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198711

+9

321198712

1199102

(119905119899

120572) +3

321198711

+9

321198712

)

1198714

= ℎ119866(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

1199102

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

)

1198715

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

1199102

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

)

1198716

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

1199102

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

)

(24)

43 Convergence of Fuzzy Runge-Kutta-FehlbergMethod Thesolution is calculated by grid points at 119886 = 119905

0

le 1199051

le sdot sdot sdot le

119905119873

= 119887 and ℎ = (119887 minus 119886)119873 = 119905119899+1

minus 119905119899

Therefore we have

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572) + 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) + 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) + 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

(25)

6 International Journal of Differential Equations

Clearly 1199101

(119905 120572) and 1199102

(119905 120572) converge to 1198841

(119905 120572) and 1198842

(119905 120572)respectively whenever ℎ rarr 0 that is

limℎrarr0

1199101

(119905 120572) = 1198841

(119905 120572)

limℎrarr0

1199102

(119905 120572) = 1198842

(119905 120572)

(26)

Proof Before we go to the main proof we need to know someresults

Lemma 11 Let the sequence of numbers 119882119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le 1198601003816100381610038161003816119882119899

1003816100381610038161003816 + 119861 0 le 119899 le 119873 minus 1 (27)

for some given positive constants 119860 and 119861 Then

10038161003816100381610038161198821198991003816100381610038161003816 le 119860119899 10038161003816100381610038161198820

1003816100381610038161003816 + 119861119860119899

minus 1

119860 minus 1 0 le 119899 le 119873 (28)

Lemma 12 Let the sequence of numbers 119882119873

119899=0

and 119881119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 119860max 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 + 119861

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 119860max

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 + 119861

(29)

for some given positive constants 119860 and 119861 and denote

119880119899

=1003816100381610038161003816119882119899

1003816100381610038161003816 +1003816100381610038161003816119881119899

1003816100381610038161003816 0 le 119899 le 119873 (30)

Then

119880119899

le 119860119899

1198800

+ 119861119860119899

minus 1

119860 minus 1

0 le 119899 le 119873 (31)

where 119860 = 1 + 2119860 and 119861 = 2119861

Let 119865(119905 119906 V) and 119866(119905 119906 V) be obtained by substituting[1199101

(119905 120572) 1199102

(119905 120572)] = [119906 V] in (21) and (23) that is

119865 (119905 119906 V) =16

1351198701

(119905 119906 V) +6656

128251198703

(119905 119906 V)

+2856156430

1198704

(119905 119906 V) minus9

501198705

(119905 119906 V)

+2

551198706

(119905 119906 V)

119866 (119905 119906 V) =16

1351198711

(119905 119906 V) +6656

128251198713

(119905 119906 V)

+2856156430

1198714

(119905 119906 V) minus9

501198715

(119905 119906 V)

+2

551198716

(119905 119906 V)

(32)

The domain where 119865 and 119866 are defined is as

119867 = (119905 119906 V) | 0 le 119905 le 119879 minusinfin lt V lt infin minusinfin lt 119906

le V (33)

Theorem 13 Let 119865(119905 119906 V) and 119866(119905 119906 V) belong to 119862119901minus1

(119870)

and let the partial derivative of 119865 and 119866 be bounded over 119870Then for arbitrary fixed 0 le 120572 le 1 the numerical solu-tion of (9) [119910

1

(119905 120572) 1199102

(119905 120572)] converges to the exact solution[1198841

(119905 120572) 1198842

(119905 120572)]

Proof (see [46]) By using Taylorrsquos theorem we get

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572)

+ ℎ119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

1

(1205851198991

)

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ ℎ119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

2

(1205851198992

)

(34)

where 1205851198991

1205851198992

isin (119905119899

119905119899+1

)Now if we denote

119882119899

= 1198841

(119905119899

120572) minus 1199101

(119905119899

120572)

119881119899

= 1198842

(119905119899

120572) minus 1199102

(119905119899

120572)

(35)

then the above two expressions converted to

119882119899+1

= 119882119899

+ ℎ 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

1

(1205851198991

)

119881119899+1

= 119881119899

+ ℎ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

2

(1205851198992

)

(36)

Hence we can write1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 2119871ℎmax 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 +ℎ119901+1

(119901 + 1)119872

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 2119871ℎmax

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 +

ℎ119901+1

(119901 + 1)119872

(37)

where 119872 = maxmax |119884(119901+1)

1

(119905 120572)|max |119884(119901+1)

2

(119905 120572)| for 119905 isin

[0 119879] and 119871 gt 0 is a bound for the partial derivative of 119865 and119866

Therefore we can write1003816100381610038161003816119882119899

1003816100381610038161003816 le (1 + 4119871ℎ)119899 10038161003816100381610038161198800

1003816100381610038161003816 +2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

10038161003816100381610038161198811198991003816100381610038161003816 le (1 + 4119871ℎ)

119899 100381610038161003816100381611988001003816100381610038161003816 +

2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

(38)

where |1198800

| = |1198820

| + |1198810

|

International Journal of Differential Equations 7

In particular

10038161003816100381610038161198821198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

10038161003816100381610038161198811198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

(39)

Since1198820

= 1198810

= 0 we have

10038161003816100381610038161198821198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

10038161003816100381610038161198811198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

(40)

Thus if ℎ rarr 0 we get119882119873

rarr 0 and119881119873

rarr 0 which completesthe proof

44 Algorithm for Finding the Numerical Solution

Step 1 119865(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

119866(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

Step 2 Read 119905(0) 1199101

(0) 1199102

(0) ℎ limit

Step 3 For 119894 = 0(1) limit

1198701

larr ℎ119865(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198702

larr ℎ sdot 119865(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198701

1199102

(119905119894

119903) +

(14)1198701

)

1198703

larr ℎsdot119865(119905119894

+(38)ℎ 1199101

(119905119894

119903)+(332)1198701

+(932)1198702

1199102

(119905119894

119903) + (332)1198701

+ (932)1198702

)

1198704

larr ℎ sdot 119865(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198701

minus

(72002197)1198702

+ (72962197)1198703

1199102

(119905119894

119903) + (1932

2197)1198701

minus (72002197)1198702

+ (72962197)1198703

)

1198705

larr ℎ119865(119905119894

+ℎ 1199101

(119905119894

119903)+(439216)1198701

minus81198702

+(3680

513)1198703

minus (8454104)1198704

1199102

(119905119894

119903) + (439216)1198701

minus

81198702

+ (3680513)1198703

minus (8454104)1198704

)

1198706

larr ℎ119865(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198701

+ 21198702

minus (3544

2565)1198703

+ (18594104)1198704

minus (1140)1198705

1199102

(119905119894

119903) minus

(827)1198701

+ 21198702

minus (35442565)1198703

+ (18594104)1198704

minus

(1140)1198705

)

1198711

larr ℎ sdot 119866(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198712

larr ℎ119866(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198711

1199102

(119905119894

119903) +

(14)1198711

)

1198713

larr ℎ119866(119905119894

+ (38)ℎ 1199101

(119905119894

119903) + (332)1198711

+ (932)1198712

1199102

(119905119894

119903) + (332)1198711

+ (932)1198712

)

1198714

larr ℎ119866(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198711

minus

(72002197)1198712

+ (72962197)1198713

1199102

(119905119894

119903) + (1932

2197)1198711

minus (72002197)1198712

+ (72962197)1198713

)

1198715

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) + (439216)1198711

minus 81198712

+

(3680513)1198713

minus (8454104)1198714

1199102

(119905119894

119903) + (439

216)1198711

minus 81198712

+ (3680513)1198713

minus (8454104)1198714

)

1198716

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198711

+ 21198712

minus

(35442565)1198713

+ (18594104)1198714

minus (1140)1198715

1199102

(119905119894

119903) minus (827)1198711

+ 21198712

minus (35442565)1198713

+ (1859

4104)1198714

minus (1140)1198715

)

Step 4

1199101

(119905119894+1

119903) = 1199101

(119905119894

119903)+(16135)1198701

+(665612825)1198703

+

(2856156430)1198704

minus (950)1198705

+ (255)1198706

Step 5

1199102

(119905119894+1

119903) = 1199102

(119905119894

119903)+(16135)1198711

+(665612825)1198713

+

(2856156430)1198714

minus (950)1198715

+ (255)1198716

Step 6 119905119894+1

= 119905119894

+ ℎ Write 1199101

(119905119894+1

119903) 1199102

(119905119894+1

119903) 119905119894+1

Step 7 Next 119894

Step 8 End

5 Numerical Example

Example Solve 1199101015840

= minus119910 + 119905 + 1 with initial condition 119910(0) =

(096 1 101) Then find the solution at 119905 = 01

Solution For (i)-gH differentiable case the exact solution is

1199101

(119903 119905) = 119905 + (096 + 004119903) 119890minus119905

1199102

(119903 119905) = 119905 + (101 minus 001119903) 119890minus119905

(41)

and for (ii)-gH differentiable case the exact solution is

1199101

(119903 119905) = 1 + 119905 + (minus004 + 004119903) 119890119905

1199102

(119903 119905) = 1 + 119905 + (001 minus 001119903) 119890119905

(42)

Remark 14 From Figure 1 and Table 1 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

001 (for ℎ = 0001 is nearly equal) whereas the lowerexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

Remark 15 From Figure 2 and Table 1 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the upperexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

6 Application

Problem A tank initially contains 300 gals of brine which hasdissolved in 119888 lbs of salt Coming into the tank at 3 galsmin

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

4 International Journal of Differential Equations

3 Exact Solution ofFuzzy Differential Equation

Consider the fuzzy initial value problem

1199101015840

(119905) = 119891 (119905 119910 (119905)) 119905 isin 119868 = [0 119879] with 119910 (0) = 1199100

(9)

where 119891 is a continuous mapping from 119877+

times 119877 into 119877 and1199100

isin 119864 with 119903-level sets

[1199100

]119903

= [1199101

(0 120572) 1199102

(0 120572)] 120572 isin (0 1] (10)

We write 119891(119905 119910) = [1198911

(119905 119910) 1198912

(119905 119910)] and 1198911

(119905 119910) =

119865[119905 1199101

1199102

] 1198912

(119905 119910) = 119866[119905 1199101

1199102

]Because of 1199101015840(119905) = 119891(119905 119910) we have the followingWhen 119910(119905 119910) is (i)-gH differentiable

1199101015840

1

(119905 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

1199101015840

2

(119905 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

(11)

When 119910(119905 119910) is (ii)-gH differentiable

1199101015840

2

(119905 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

1199101015840

1

(119905 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

(12)

where by using extension principle we have themembershipfunction

119891 (119905 119910 (119905)) (119904) = Sup 119910 (119905) (120591) 119904 = 119891 (119905 120591) 119904 isin 119877 (13)

So fuzzy number is 119891(119905 119910(119905)) From this it follows that

[119891 (119905 119910 (119905))]120572

= [1198911

(119905 119910 (119905) 120572) 1198912

(119905 119910 (119905) 120572)]

120572 isin [0 1]

(14)

where

1198911

(119905 119910 (119905) 120572) = 119865 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

= min 119891 (119905 119906) 119906 isin [1199101

(119905 120572) 1199102

(119905 120572)]

1198912

(119905 119910 (119905) 120572) = 119866 [119905 1199101

(119905 120572) 1199102

(119905 120572)]

= max 119891 (119905 119906) 119906 isin [1199101

(119905 120572) 1199102

(119905 120572)]

(15)

Note (1) Both cases ((i)-gH and (ii)-gH) can be applied to aFDE for finding exact solution

(2) After taking 120572-cut of the given FDE it transforms tosystem of ordinary differential equation

4 Numerical Solution ofFuzzy Differential Equation

41 Runge-Kutta-FehlbergMethod for Ordinary (Crisp) Differ-ential Equation Consider the initial value problem 119910

1015840

(119905) =

119891(119905 119910(119905)) 119910(1199050

) = 1199100

The Runge-Kutta-Fehlberg method (denoted as RKF45)

is one way to try to resolve this problem

The problem is to solve the initial value problem in aboveequation by means of Runge-Kutta methods of order 4 andorder 5

First we need some definitions

1198961

= ℎ119891 (119905119894

119910119894

)

1198962

= ℎ119891(119905119894

+1

4ℎ 119910119894

+1

41198961

)

1198963

= ℎ119891(119905119894

+3

8ℎ 119910119894

+3

321198961

+9

321198962

)

1198964

= ℎ119891(119905119894

+12

13ℎ 119910119894

+1932

21971198961

minus7200

21971198962

+7296

21971198963

)

1198965

= ℎ119891(119905119894

+ ℎ 119910119894

+439

2161198961

minus 81198962

+3680

5131198963

minus845

41041198964

)

1198966

= ℎ119891(119905119894

+ ℎ 119910119894

minus8

271198961

+ 21198962

minus3544

25651198963

+1859

41041198964

minus11

401198965

)

(16)

Then an approximation to the solution of initial valueproblem is made using Runge-Kutta method of order 4

119910119894+1

= 119910119894

+25

2161198961

+1408

25651198963

+2197

41011198964

minus1

51198965

(17)

A better value for the solution is determined using a Runge-Kutta method of order 5

119911119894+1

= 119910119894

+16

1351198961

+6656

128251198963

+2856156430

1198964

minus9

501198965

+2

551198966

(18)

The optimal step size 119904ℎ can be determined bymultiplying thescalar 119904 times the step size ℎ The scalar 119904 is

119904 = (120598ℎ

21003816100381610038161003816119911119894+1 minus 119910

119894+1

1003816100381610038161003816

)

14

= 00840896(120598ℎ

1003816100381610038161003816119911119894+1 minus 119910119894+1

1003816100381610038161003816

)

14

(19)

where 120598 is the specified error control toleranceNote that RK4 requires 4 function evaluations and RK5

requires 6 evaluations that is 10 for RK4 and RK5 Fehlbergdevised a method to get RK4 and RK5 results using only6 function evaluations by using some of 119870 values in bothmethods

42 Runge-Kutta-Fehlberg Method for Solving Fuzzy Differen-tial Equations Let 119884 = [119884

1

1198842

] be the exact solution and let

International Journal of Differential Equations 5

119910 = [1199101

1199102

] be the approximated solution of the fuzzy initialvalue problem

Let [119884(119905)]120572

= [1198841

(119905 120572) 1198842

(119905 120572)] [119910(119905)]119903

= [1199101

(119905 120572) 1199102

(119905

120572)]Throughout this argument the value of 119903 is fixed Then

the exact and approximated solution at 119905119899

are respectivelydenoted by

[119884 (119905119899

)]120572

= [1198841

(119905119899

120572) 1198842

(119905119899

120572)]

[119910 (119905119899

)]120572

= [1199101

(119905119899

120572) 1199102

(119905119899

120572)]

(20)

The grid points at which the solution is calculated are ℎ =

(119879 minus 1199050

)119873 119905119894

= 1199050

+ 119894ℎ 0 le 119894 le 119873Then we obtained

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) +16

1351198701

+6656

128251198703

+2856156430

1198704

minus9

501198705

+2

551198706

(21)

where

1198701

= ℎ119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198702

= ℎ119865(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198701

1199102

(119905119899

120572)

+1

41198701

)

1198703

= ℎ119865(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198701

+9

321198702

1199102

(119905119899

120572) +3

321198701

+9

321198702

)

1198704

= ℎ119865(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

1199102

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

)

1198705

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

1199102

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

)

1198706

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

1199102

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

)

(22)

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) +16

1351198711

+6656

128251198713

+2856156430

1198714

minus9

501198715

+2

551198716

(23)

where

1198711

= ℎ119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198712

= ℎ119866(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198711

1199102

(119905119899

120572)

+1

41198711

)

1198713

= ℎ119866(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198711

+9

321198712

1199102

(119905119899

120572) +3

321198711

+9

321198712

)

1198714

= ℎ119866(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

1199102

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

)

1198715

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

1199102

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

)

1198716

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

1199102

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

)

(24)

43 Convergence of Fuzzy Runge-Kutta-FehlbergMethod Thesolution is calculated by grid points at 119886 = 119905

0

le 1199051

le sdot sdot sdot le

119905119873

= 119887 and ℎ = (119887 minus 119886)119873 = 119905119899+1

minus 119905119899

Therefore we have

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572) + 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) + 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) + 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

(25)

6 International Journal of Differential Equations

Clearly 1199101

(119905 120572) and 1199102

(119905 120572) converge to 1198841

(119905 120572) and 1198842

(119905 120572)respectively whenever ℎ rarr 0 that is

limℎrarr0

1199101

(119905 120572) = 1198841

(119905 120572)

limℎrarr0

1199102

(119905 120572) = 1198842

(119905 120572)

(26)

Proof Before we go to the main proof we need to know someresults

Lemma 11 Let the sequence of numbers 119882119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le 1198601003816100381610038161003816119882119899

1003816100381610038161003816 + 119861 0 le 119899 le 119873 minus 1 (27)

for some given positive constants 119860 and 119861 Then

10038161003816100381610038161198821198991003816100381610038161003816 le 119860119899 10038161003816100381610038161198820

1003816100381610038161003816 + 119861119860119899

minus 1

119860 minus 1 0 le 119899 le 119873 (28)

Lemma 12 Let the sequence of numbers 119882119873

119899=0

and 119881119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 119860max 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 + 119861

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 119860max

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 + 119861

(29)

for some given positive constants 119860 and 119861 and denote

119880119899

=1003816100381610038161003816119882119899

1003816100381610038161003816 +1003816100381610038161003816119881119899

1003816100381610038161003816 0 le 119899 le 119873 (30)

Then

119880119899

le 119860119899

1198800

+ 119861119860119899

minus 1

119860 minus 1

0 le 119899 le 119873 (31)

where 119860 = 1 + 2119860 and 119861 = 2119861

Let 119865(119905 119906 V) and 119866(119905 119906 V) be obtained by substituting[1199101

(119905 120572) 1199102

(119905 120572)] = [119906 V] in (21) and (23) that is

119865 (119905 119906 V) =16

1351198701

(119905 119906 V) +6656

128251198703

(119905 119906 V)

+2856156430

1198704

(119905 119906 V) minus9

501198705

(119905 119906 V)

+2

551198706

(119905 119906 V)

119866 (119905 119906 V) =16

1351198711

(119905 119906 V) +6656

128251198713

(119905 119906 V)

+2856156430

1198714

(119905 119906 V) minus9

501198715

(119905 119906 V)

+2

551198716

(119905 119906 V)

(32)

The domain where 119865 and 119866 are defined is as

119867 = (119905 119906 V) | 0 le 119905 le 119879 minusinfin lt V lt infin minusinfin lt 119906

le V (33)

Theorem 13 Let 119865(119905 119906 V) and 119866(119905 119906 V) belong to 119862119901minus1

(119870)

and let the partial derivative of 119865 and 119866 be bounded over 119870Then for arbitrary fixed 0 le 120572 le 1 the numerical solu-tion of (9) [119910

1

(119905 120572) 1199102

(119905 120572)] converges to the exact solution[1198841

(119905 120572) 1198842

(119905 120572)]

Proof (see [46]) By using Taylorrsquos theorem we get

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572)

+ ℎ119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

1

(1205851198991

)

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ ℎ119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

2

(1205851198992

)

(34)

where 1205851198991

1205851198992

isin (119905119899

119905119899+1

)Now if we denote

119882119899

= 1198841

(119905119899

120572) minus 1199101

(119905119899

120572)

119881119899

= 1198842

(119905119899

120572) minus 1199102

(119905119899

120572)

(35)

then the above two expressions converted to

119882119899+1

= 119882119899

+ ℎ 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

1

(1205851198991

)

119881119899+1

= 119881119899

+ ℎ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

2

(1205851198992

)

(36)

Hence we can write1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 2119871ℎmax 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 +ℎ119901+1

(119901 + 1)119872

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 2119871ℎmax

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 +

ℎ119901+1

(119901 + 1)119872

(37)

where 119872 = maxmax |119884(119901+1)

1

(119905 120572)|max |119884(119901+1)

2

(119905 120572)| for 119905 isin

[0 119879] and 119871 gt 0 is a bound for the partial derivative of 119865 and119866

Therefore we can write1003816100381610038161003816119882119899

1003816100381610038161003816 le (1 + 4119871ℎ)119899 10038161003816100381610038161198800

1003816100381610038161003816 +2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

10038161003816100381610038161198811198991003816100381610038161003816 le (1 + 4119871ℎ)

119899 100381610038161003816100381611988001003816100381610038161003816 +

2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

(38)

where |1198800

| = |1198820

| + |1198810

|

International Journal of Differential Equations 7

In particular

10038161003816100381610038161198821198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

10038161003816100381610038161198811198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

(39)

Since1198820

= 1198810

= 0 we have

10038161003816100381610038161198821198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

10038161003816100381610038161198811198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

(40)

Thus if ℎ rarr 0 we get119882119873

rarr 0 and119881119873

rarr 0 which completesthe proof

44 Algorithm for Finding the Numerical Solution

Step 1 119865(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

119866(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

Step 2 Read 119905(0) 1199101

(0) 1199102

(0) ℎ limit

Step 3 For 119894 = 0(1) limit

1198701

larr ℎ119865(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198702

larr ℎ sdot 119865(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198701

1199102

(119905119894

119903) +

(14)1198701

)

1198703

larr ℎsdot119865(119905119894

+(38)ℎ 1199101

(119905119894

119903)+(332)1198701

+(932)1198702

1199102

(119905119894

119903) + (332)1198701

+ (932)1198702

)

1198704

larr ℎ sdot 119865(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198701

minus

(72002197)1198702

+ (72962197)1198703

1199102

(119905119894

119903) + (1932

2197)1198701

minus (72002197)1198702

+ (72962197)1198703

)

1198705

larr ℎ119865(119905119894

+ℎ 1199101

(119905119894

119903)+(439216)1198701

minus81198702

+(3680

513)1198703

minus (8454104)1198704

1199102

(119905119894

119903) + (439216)1198701

minus

81198702

+ (3680513)1198703

minus (8454104)1198704

)

1198706

larr ℎ119865(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198701

+ 21198702

minus (3544

2565)1198703

+ (18594104)1198704

minus (1140)1198705

1199102

(119905119894

119903) minus

(827)1198701

+ 21198702

minus (35442565)1198703

+ (18594104)1198704

minus

(1140)1198705

)

1198711

larr ℎ sdot 119866(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198712

larr ℎ119866(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198711

1199102

(119905119894

119903) +

(14)1198711

)

1198713

larr ℎ119866(119905119894

+ (38)ℎ 1199101

(119905119894

119903) + (332)1198711

+ (932)1198712

1199102

(119905119894

119903) + (332)1198711

+ (932)1198712

)

1198714

larr ℎ119866(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198711

minus

(72002197)1198712

+ (72962197)1198713

1199102

(119905119894

119903) + (1932

2197)1198711

minus (72002197)1198712

+ (72962197)1198713

)

1198715

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) + (439216)1198711

minus 81198712

+

(3680513)1198713

minus (8454104)1198714

1199102

(119905119894

119903) + (439

216)1198711

minus 81198712

+ (3680513)1198713

minus (8454104)1198714

)

1198716

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198711

+ 21198712

minus

(35442565)1198713

+ (18594104)1198714

minus (1140)1198715

1199102

(119905119894

119903) minus (827)1198711

+ 21198712

minus (35442565)1198713

+ (1859

4104)1198714

minus (1140)1198715

)

Step 4

1199101

(119905119894+1

119903) = 1199101

(119905119894

119903)+(16135)1198701

+(665612825)1198703

+

(2856156430)1198704

minus (950)1198705

+ (255)1198706

Step 5

1199102

(119905119894+1

119903) = 1199102

(119905119894

119903)+(16135)1198711

+(665612825)1198713

+

(2856156430)1198714

minus (950)1198715

+ (255)1198716

Step 6 119905119894+1

= 119905119894

+ ℎ Write 1199101

(119905119894+1

119903) 1199102

(119905119894+1

119903) 119905119894+1

Step 7 Next 119894

Step 8 End

5 Numerical Example

Example Solve 1199101015840

= minus119910 + 119905 + 1 with initial condition 119910(0) =

(096 1 101) Then find the solution at 119905 = 01

Solution For (i)-gH differentiable case the exact solution is

1199101

(119903 119905) = 119905 + (096 + 004119903) 119890minus119905

1199102

(119903 119905) = 119905 + (101 minus 001119903) 119890minus119905

(41)

and for (ii)-gH differentiable case the exact solution is

1199101

(119903 119905) = 1 + 119905 + (minus004 + 004119903) 119890119905

1199102

(119903 119905) = 1 + 119905 + (001 minus 001119903) 119890119905

(42)

Remark 14 From Figure 1 and Table 1 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

001 (for ℎ = 0001 is nearly equal) whereas the lowerexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

Remark 15 From Figure 2 and Table 1 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the upperexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

6 Application

Problem A tank initially contains 300 gals of brine which hasdissolved in 119888 lbs of salt Coming into the tank at 3 galsmin

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

International Journal of Differential Equations 5

119910 = [1199101

1199102

] be the approximated solution of the fuzzy initialvalue problem

Let [119884(119905)]120572

= [1198841

(119905 120572) 1198842

(119905 120572)] [119910(119905)]119903

= [1199101

(119905 120572) 1199102

(119905

120572)]Throughout this argument the value of 119903 is fixed Then

the exact and approximated solution at 119905119899

are respectivelydenoted by

[119884 (119905119899

)]120572

= [1198841

(119905119899

120572) 1198842

(119905119899

120572)]

[119910 (119905119899

)]120572

= [1199101

(119905119899

120572) 1199102

(119905119899

120572)]

(20)

The grid points at which the solution is calculated are ℎ =

(119879 minus 1199050

)119873 119905119894

= 1199050

+ 119894ℎ 0 le 119894 le 119873Then we obtained

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) +16

1351198701

+6656

128251198703

+2856156430

1198704

minus9

501198705

+2

551198706

(21)

where

1198701

= ℎ119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198702

= ℎ119865(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198701

1199102

(119905119899

120572)

+1

41198701

)

1198703

= ℎ119865(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198701

+9

321198702

1199102

(119905119899

120572) +3

321198701

+9

321198702

)

1198704

= ℎ119865(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

1199102

(119905119899

120572) +1932

21971198701

minus7200

21971198702

+7296

21971198703

)

1198705

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

1199102

(119905119899

120572) +439

2161198701

minus 81198702

+3680

5131198703

minus845

41041198704

)

1198706

= ℎ119865(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

1199102

(119905119899

120572) minus8

271198701

+ 21198702

minus3544

25651198703

+1859

41041198704

minus11

401198705

)

(22)

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) +16

1351198711

+6656

128251198713

+2856156430

1198714

minus9

501198715

+2

551198716

(23)

where

1198711

= ℎ119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1198712

= ℎ119866(119905119899

+1

4ℎ 1199101

(119905119899

120572) +1

41198711

1199102

(119905119899

120572)

+1

41198711

)

1198713

= ℎ119866(119905119899

+3

8ℎ 1199101

(119905119899

120572) +3

321198711

+9

321198712

1199102

(119905119899

120572) +3

321198711

+9

321198712

)

1198714

= ℎ119866(119905119899

+12

13ℎ 1199101

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

1199102

(119905119899

120572) +1932

21971198711

minus7200

21971198712

+7296

21971198713

)

1198715

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

1199102

(119905119899

120572) +439

2161198711

minus 81198712

+3680

5131198713

minus845

41041198714

)

1198716

= ℎ119866(119905119899

+ ℎ 1199101

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

1199102

(119905119899

120572) minus8

271198711

+ 21198712

minus3544

25651198713

+1859

41041198714

minus11

401198715

)

(24)

43 Convergence of Fuzzy Runge-Kutta-FehlbergMethod Thesolution is calculated by grid points at 119886 = 119905

0

le 1199051

le sdot sdot sdot le

119905119873

= 119887 and ℎ = (119887 minus 119886)119873 = 119905119899+1

minus 119905119899

Therefore we have

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572) + 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

1199101

(119905119899+1

120572) = 1199101

(119905119899

120572) + 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

1199102

(119905119899+1

120572) = 1199102

(119905119899

120572) + 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572))

(25)

6 International Journal of Differential Equations

Clearly 1199101

(119905 120572) and 1199102

(119905 120572) converge to 1198841

(119905 120572) and 1198842

(119905 120572)respectively whenever ℎ rarr 0 that is

limℎrarr0

1199101

(119905 120572) = 1198841

(119905 120572)

limℎrarr0

1199102

(119905 120572) = 1198842

(119905 120572)

(26)

Proof Before we go to the main proof we need to know someresults

Lemma 11 Let the sequence of numbers 119882119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le 1198601003816100381610038161003816119882119899

1003816100381610038161003816 + 119861 0 le 119899 le 119873 minus 1 (27)

for some given positive constants 119860 and 119861 Then

10038161003816100381610038161198821198991003816100381610038161003816 le 119860119899 10038161003816100381610038161198820

1003816100381610038161003816 + 119861119860119899

minus 1

119860 minus 1 0 le 119899 le 119873 (28)

Lemma 12 Let the sequence of numbers 119882119873

119899=0

and 119881119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 119860max 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 + 119861

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 119860max

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 + 119861

(29)

for some given positive constants 119860 and 119861 and denote

119880119899

=1003816100381610038161003816119882119899

1003816100381610038161003816 +1003816100381610038161003816119881119899

1003816100381610038161003816 0 le 119899 le 119873 (30)

Then

119880119899

le 119860119899

1198800

+ 119861119860119899

minus 1

119860 minus 1

0 le 119899 le 119873 (31)

where 119860 = 1 + 2119860 and 119861 = 2119861

Let 119865(119905 119906 V) and 119866(119905 119906 V) be obtained by substituting[1199101

(119905 120572) 1199102

(119905 120572)] = [119906 V] in (21) and (23) that is

119865 (119905 119906 V) =16

1351198701

(119905 119906 V) +6656

128251198703

(119905 119906 V)

+2856156430

1198704

(119905 119906 V) minus9

501198705

(119905 119906 V)

+2

551198706

(119905 119906 V)

119866 (119905 119906 V) =16

1351198711

(119905 119906 V) +6656

128251198713

(119905 119906 V)

+2856156430

1198714

(119905 119906 V) minus9

501198715

(119905 119906 V)

+2

551198716

(119905 119906 V)

(32)

The domain where 119865 and 119866 are defined is as

119867 = (119905 119906 V) | 0 le 119905 le 119879 minusinfin lt V lt infin minusinfin lt 119906

le V (33)

Theorem 13 Let 119865(119905 119906 V) and 119866(119905 119906 V) belong to 119862119901minus1

(119870)

and let the partial derivative of 119865 and 119866 be bounded over 119870Then for arbitrary fixed 0 le 120572 le 1 the numerical solu-tion of (9) [119910

1

(119905 120572) 1199102

(119905 120572)] converges to the exact solution[1198841

(119905 120572) 1198842

(119905 120572)]

Proof (see [46]) By using Taylorrsquos theorem we get

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572)

+ ℎ119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

1

(1205851198991

)

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ ℎ119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

2

(1205851198992

)

(34)

where 1205851198991

1205851198992

isin (119905119899

119905119899+1

)Now if we denote

119882119899

= 1198841

(119905119899

120572) minus 1199101

(119905119899

120572)

119881119899

= 1198842

(119905119899

120572) minus 1199102

(119905119899

120572)

(35)

then the above two expressions converted to

119882119899+1

= 119882119899

+ ℎ 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

1

(1205851198991

)

119881119899+1

= 119881119899

+ ℎ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

2

(1205851198992

)

(36)

Hence we can write1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 2119871ℎmax 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 +ℎ119901+1

(119901 + 1)119872

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 2119871ℎmax

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 +

ℎ119901+1

(119901 + 1)119872

(37)

where 119872 = maxmax |119884(119901+1)

1

(119905 120572)|max |119884(119901+1)

2

(119905 120572)| for 119905 isin

[0 119879] and 119871 gt 0 is a bound for the partial derivative of 119865 and119866

Therefore we can write1003816100381610038161003816119882119899

1003816100381610038161003816 le (1 + 4119871ℎ)119899 10038161003816100381610038161198800

1003816100381610038161003816 +2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

10038161003816100381610038161198811198991003816100381610038161003816 le (1 + 4119871ℎ)

119899 100381610038161003816100381611988001003816100381610038161003816 +

2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

(38)

where |1198800

| = |1198820

| + |1198810

|

International Journal of Differential Equations 7

In particular

10038161003816100381610038161198821198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

10038161003816100381610038161198811198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

(39)

Since1198820

= 1198810

= 0 we have

10038161003816100381610038161198821198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

10038161003816100381610038161198811198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

(40)

Thus if ℎ rarr 0 we get119882119873

rarr 0 and119881119873

rarr 0 which completesthe proof

44 Algorithm for Finding the Numerical Solution

Step 1 119865(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

119866(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

Step 2 Read 119905(0) 1199101

(0) 1199102

(0) ℎ limit

Step 3 For 119894 = 0(1) limit

1198701

larr ℎ119865(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198702

larr ℎ sdot 119865(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198701

1199102

(119905119894

119903) +

(14)1198701

)

1198703

larr ℎsdot119865(119905119894

+(38)ℎ 1199101

(119905119894

119903)+(332)1198701

+(932)1198702

1199102

(119905119894

119903) + (332)1198701

+ (932)1198702

)

1198704

larr ℎ sdot 119865(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198701

minus

(72002197)1198702

+ (72962197)1198703

1199102

(119905119894

119903) + (1932

2197)1198701

minus (72002197)1198702

+ (72962197)1198703

)

1198705

larr ℎ119865(119905119894

+ℎ 1199101

(119905119894

119903)+(439216)1198701

minus81198702

+(3680

513)1198703

minus (8454104)1198704

1199102

(119905119894

119903) + (439216)1198701

minus

81198702

+ (3680513)1198703

minus (8454104)1198704

)

1198706

larr ℎ119865(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198701

+ 21198702

minus (3544

2565)1198703

+ (18594104)1198704

minus (1140)1198705

1199102

(119905119894

119903) minus

(827)1198701

+ 21198702

minus (35442565)1198703

+ (18594104)1198704

minus

(1140)1198705

)

1198711

larr ℎ sdot 119866(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198712

larr ℎ119866(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198711

1199102

(119905119894

119903) +

(14)1198711

)

1198713

larr ℎ119866(119905119894

+ (38)ℎ 1199101

(119905119894

119903) + (332)1198711

+ (932)1198712

1199102

(119905119894

119903) + (332)1198711

+ (932)1198712

)

1198714

larr ℎ119866(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198711

minus

(72002197)1198712

+ (72962197)1198713

1199102

(119905119894

119903) + (1932

2197)1198711

minus (72002197)1198712

+ (72962197)1198713

)

1198715

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) + (439216)1198711

minus 81198712

+

(3680513)1198713

minus (8454104)1198714

1199102

(119905119894

119903) + (439

216)1198711

minus 81198712

+ (3680513)1198713

minus (8454104)1198714

)

1198716

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198711

+ 21198712

minus

(35442565)1198713

+ (18594104)1198714

minus (1140)1198715

1199102

(119905119894

119903) minus (827)1198711

+ 21198712

minus (35442565)1198713

+ (1859

4104)1198714

minus (1140)1198715

)

Step 4

1199101

(119905119894+1

119903) = 1199101

(119905119894

119903)+(16135)1198701

+(665612825)1198703

+

(2856156430)1198704

minus (950)1198705

+ (255)1198706

Step 5

1199102

(119905119894+1

119903) = 1199102

(119905119894

119903)+(16135)1198711

+(665612825)1198713

+

(2856156430)1198714

minus (950)1198715

+ (255)1198716

Step 6 119905119894+1

= 119905119894

+ ℎ Write 1199101

(119905119894+1

119903) 1199102

(119905119894+1

119903) 119905119894+1

Step 7 Next 119894

Step 8 End

5 Numerical Example

Example Solve 1199101015840

= minus119910 + 119905 + 1 with initial condition 119910(0) =

(096 1 101) Then find the solution at 119905 = 01

Solution For (i)-gH differentiable case the exact solution is

1199101

(119903 119905) = 119905 + (096 + 004119903) 119890minus119905

1199102

(119903 119905) = 119905 + (101 minus 001119903) 119890minus119905

(41)

and for (ii)-gH differentiable case the exact solution is

1199101

(119903 119905) = 1 + 119905 + (minus004 + 004119903) 119890119905

1199102

(119903 119905) = 1 + 119905 + (001 minus 001119903) 119890119905

(42)

Remark 14 From Figure 1 and Table 1 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

001 (for ℎ = 0001 is nearly equal) whereas the lowerexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

Remark 15 From Figure 2 and Table 1 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the upperexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

6 Application

Problem A tank initially contains 300 gals of brine which hasdissolved in 119888 lbs of salt Coming into the tank at 3 galsmin

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

6 International Journal of Differential Equations

Clearly 1199101

(119905 120572) and 1199102

(119905 120572) converge to 1198841

(119905 120572) and 1198842

(119905 120572)respectively whenever ℎ rarr 0 that is

limℎrarr0

1199101

(119905 120572) = 1198841

(119905 120572)

limℎrarr0

1199102

(119905 120572) = 1198842

(119905 120572)

(26)

Proof Before we go to the main proof we need to know someresults

Lemma 11 Let the sequence of numbers 119882119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le 1198601003816100381610038161003816119882119899

1003816100381610038161003816 + 119861 0 le 119899 le 119873 minus 1 (27)

for some given positive constants 119860 and 119861 Then

10038161003816100381610038161198821198991003816100381610038161003816 le 119860119899 10038161003816100381610038161198820

1003816100381610038161003816 + 119861119860119899

minus 1

119860 minus 1 0 le 119899 le 119873 (28)

Lemma 12 Let the sequence of numbers 119882119873

119899=0

and 119881119873

119899=0

satisfy1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 119860max 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 + 119861

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 119860max

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 + 119861

(29)

for some given positive constants 119860 and 119861 and denote

119880119899

=1003816100381610038161003816119882119899

1003816100381610038161003816 +1003816100381610038161003816119881119899

1003816100381610038161003816 0 le 119899 le 119873 (30)

Then

119880119899

le 119860119899

1198800

+ 119861119860119899

minus 1

119860 minus 1

0 le 119899 le 119873 (31)

where 119860 = 1 + 2119860 and 119861 = 2119861

Let 119865(119905 119906 V) and 119866(119905 119906 V) be obtained by substituting[1199101

(119905 120572) 1199102

(119905 120572)] = [119906 V] in (21) and (23) that is

119865 (119905 119906 V) =16

1351198701

(119905 119906 V) +6656

128251198703

(119905 119906 V)

+2856156430

1198704

(119905 119906 V) minus9

501198705

(119905 119906 V)

+2

551198706

(119905 119906 V)

119866 (119905 119906 V) =16

1351198711

(119905 119906 V) +6656

128251198713

(119905 119906 V)

+2856156430

1198714

(119905 119906 V) minus9

501198715

(119905 119906 V)

+2

551198716

(119905 119906 V)

(32)

The domain where 119865 and 119866 are defined is as

119867 = (119905 119906 V) | 0 le 119905 le 119879 minusinfin lt V lt infin minusinfin lt 119906

le V (33)

Theorem 13 Let 119865(119905 119906 V) and 119866(119905 119906 V) belong to 119862119901minus1

(119870)

and let the partial derivative of 119865 and 119866 be bounded over 119870Then for arbitrary fixed 0 le 120572 le 1 the numerical solu-tion of (9) [119910

1

(119905 120572) 1199102

(119905 120572)] converges to the exact solution[1198841

(119905 120572) 1198842

(119905 120572)]

Proof (see [46]) By using Taylorrsquos theorem we get

1198841

(119905119899+1

120572) = 1198841

(119905119899

120572)

+ ℎ119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

1

(1205851198991

)

1198842

(119905119899+1

120572) = 1198842

(119905119899

120572)

+ ℎ119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

+ℎ119901+1

(119901 + 1)119884(119901+1)

2

(1205851198992

)

(34)

where 1205851198991

1205851198992

isin (119905119899

119905119899+1

)Now if we denote

119882119899

= 1198841

(119905119899

120572) minus 1199101

(119905119899

120572)

119881119899

= 1198842

(119905119899

120572) minus 1199102

(119905119899

120572)

(35)

then the above two expressions converted to

119882119899+1

= 119882119899

+ ℎ 119865 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119865 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

1

(1205851198991

)

119881119899+1

= 119881119899

+ ℎ 119866 (119905119899

1198841

(119905119899

120572) 1198842

(119905119899

120572))

minus 119866 (119905119899

1199101

(119905119899

120572) 1199102

(119905119899

120572)) +ℎ119901+1

(119901 + 1)

sdot 119884(119901+1)

2

(1205851198992

)

(36)

Hence we can write1003816100381610038161003816119882119899+1

1003816100381610038161003816 le1003816100381610038161003816119882119899

1003816100381610038161003816 + 2119871ℎmax 1003816100381610038161003816119882119899

1003816100381610038161003816 1003816100381610038161003816119881119899

1003816100381610038161003816 +ℎ119901+1

(119901 + 1)119872

1003816100381610038161003816119881119899+11003816100381610038161003816 le

10038161003816100381610038161198811198991003816100381610038161003816 + 2119871ℎmax

10038161003816100381610038161198821198991003816100381610038161003816

10038161003816100381610038161198811198991003816100381610038161003816 +

ℎ119901+1

(119901 + 1)119872

(37)

where 119872 = maxmax |119884(119901+1)

1

(119905 120572)|max |119884(119901+1)

2

(119905 120572)| for 119905 isin

[0 119879] and 119871 gt 0 is a bound for the partial derivative of 119865 and119866

Therefore we can write1003816100381610038161003816119882119899

1003816100381610038161003816 le (1 + 4119871ℎ)119899 10038161003816100381610038161198800

1003816100381610038161003816 +2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

10038161003816100381610038161198811198991003816100381610038161003816 le (1 + 4119871ℎ)

119899 100381610038161003816100381611988001003816100381610038161003816 +

2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119899

minus 1

4119871ℎ

(38)

where |1198800

| = |1198820

| + |1198810

|

International Journal of Differential Equations 7

In particular

10038161003816100381610038161198821198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

10038161003816100381610038161198811198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

(39)

Since1198820

= 1198810

= 0 we have

10038161003816100381610038161198821198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

10038161003816100381610038161198811198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

(40)

Thus if ℎ rarr 0 we get119882119873

rarr 0 and119881119873

rarr 0 which completesthe proof

44 Algorithm for Finding the Numerical Solution

Step 1 119865(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

119866(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

Step 2 Read 119905(0) 1199101

(0) 1199102

(0) ℎ limit

Step 3 For 119894 = 0(1) limit

1198701

larr ℎ119865(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198702

larr ℎ sdot 119865(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198701

1199102

(119905119894

119903) +

(14)1198701

)

1198703

larr ℎsdot119865(119905119894

+(38)ℎ 1199101

(119905119894

119903)+(332)1198701

+(932)1198702

1199102

(119905119894

119903) + (332)1198701

+ (932)1198702

)

1198704

larr ℎ sdot 119865(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198701

minus

(72002197)1198702

+ (72962197)1198703

1199102

(119905119894

119903) + (1932

2197)1198701

minus (72002197)1198702

+ (72962197)1198703

)

1198705

larr ℎ119865(119905119894

+ℎ 1199101

(119905119894

119903)+(439216)1198701

minus81198702

+(3680

513)1198703

minus (8454104)1198704

1199102

(119905119894

119903) + (439216)1198701

minus

81198702

+ (3680513)1198703

minus (8454104)1198704

)

1198706

larr ℎ119865(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198701

+ 21198702

minus (3544

2565)1198703

+ (18594104)1198704

minus (1140)1198705

1199102

(119905119894

119903) minus

(827)1198701

+ 21198702

minus (35442565)1198703

+ (18594104)1198704

minus

(1140)1198705

)

1198711

larr ℎ sdot 119866(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198712

larr ℎ119866(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198711

1199102

(119905119894

119903) +

(14)1198711

)

1198713

larr ℎ119866(119905119894

+ (38)ℎ 1199101

(119905119894

119903) + (332)1198711

+ (932)1198712

1199102

(119905119894

119903) + (332)1198711

+ (932)1198712

)

1198714

larr ℎ119866(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198711

minus

(72002197)1198712

+ (72962197)1198713

1199102

(119905119894

119903) + (1932

2197)1198711

minus (72002197)1198712

+ (72962197)1198713

)

1198715

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) + (439216)1198711

minus 81198712

+

(3680513)1198713

minus (8454104)1198714

1199102

(119905119894

119903) + (439

216)1198711

minus 81198712

+ (3680513)1198713

minus (8454104)1198714

)

1198716

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198711

+ 21198712

minus

(35442565)1198713

+ (18594104)1198714

minus (1140)1198715

1199102

(119905119894

119903) minus (827)1198711

+ 21198712

minus (35442565)1198713

+ (1859

4104)1198714

minus (1140)1198715

)

Step 4

1199101

(119905119894+1

119903) = 1199101

(119905119894

119903)+(16135)1198701

+(665612825)1198703

+

(2856156430)1198704

minus (950)1198705

+ (255)1198706

Step 5

1199102

(119905119894+1

119903) = 1199102

(119905119894

119903)+(16135)1198711

+(665612825)1198713

+

(2856156430)1198714

minus (950)1198715

+ (255)1198716

Step 6 119905119894+1

= 119905119894

+ ℎ Write 1199101

(119905119894+1

119903) 1199102

(119905119894+1

119903) 119905119894+1

Step 7 Next 119894

Step 8 End

5 Numerical Example

Example Solve 1199101015840

= minus119910 + 119905 + 1 with initial condition 119910(0) =

(096 1 101) Then find the solution at 119905 = 01

Solution For (i)-gH differentiable case the exact solution is

1199101

(119903 119905) = 119905 + (096 + 004119903) 119890minus119905

1199102

(119903 119905) = 119905 + (101 minus 001119903) 119890minus119905

(41)

and for (ii)-gH differentiable case the exact solution is

1199101

(119903 119905) = 1 + 119905 + (minus004 + 004119903) 119890119905

1199102

(119903 119905) = 1 + 119905 + (001 minus 001119903) 119890119905

(42)

Remark 14 From Figure 1 and Table 1 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

001 (for ℎ = 0001 is nearly equal) whereas the lowerexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

Remark 15 From Figure 2 and Table 1 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the upperexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

6 Application

Problem A tank initially contains 300 gals of brine which hasdissolved in 119888 lbs of salt Coming into the tank at 3 galsmin

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

International Journal of Differential Equations 7

In particular

10038161003816100381610038161198821198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

10038161003816100381610038161198811198731003816100381610038161003816 le (1 + 4119871ℎ)

119873 100381610038161003816100381611988001003816100381610038161003816

+2ℎ119901+1

(119901 + 1)119872

(1 + 4119871ℎ)119879ℎ

minus 1

4119871ℎ

(39)

Since1198820

= 1198810

= 0 we have

10038161003816100381610038161198821198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

10038161003816100381610038161198811198731003816100381610038161003816 le 119872

1198904119871119879

minus 1

2119871 (119901 + 1)ℎ119901

(40)

Thus if ℎ rarr 0 we get119882119873

rarr 0 and119881119873

rarr 0 which completesthe proof

44 Algorithm for Finding the Numerical Solution

Step 1 119865(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

119866(119905 1199101

1199102

) larr ldquoFunction to be suppliedrdquo

Step 2 Read 119905(0) 1199101

(0) 1199102

(0) ℎ limit

Step 3 For 119894 = 0(1) limit

1198701

larr ℎ119865(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198702

larr ℎ sdot 119865(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198701

1199102

(119905119894

119903) +

(14)1198701

)

1198703

larr ℎsdot119865(119905119894

+(38)ℎ 1199101

(119905119894

119903)+(332)1198701

+(932)1198702

1199102

(119905119894

119903) + (332)1198701

+ (932)1198702

)

1198704

larr ℎ sdot 119865(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198701

minus

(72002197)1198702

+ (72962197)1198703

1199102

(119905119894

119903) + (1932

2197)1198701

minus (72002197)1198702

+ (72962197)1198703

)

1198705

larr ℎ119865(119905119894

+ℎ 1199101

(119905119894

119903)+(439216)1198701

minus81198702

+(3680

513)1198703

minus (8454104)1198704

1199102

(119905119894

119903) + (439216)1198701

minus

81198702

+ (3680513)1198703

minus (8454104)1198704

)

1198706

larr ℎ119865(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198701

+ 21198702

minus (3544

2565)1198703

+ (18594104)1198704

minus (1140)1198705

1199102

(119905119894

119903) minus

(827)1198701

+ 21198702

minus (35442565)1198703

+ (18594104)1198704

minus

(1140)1198705

)

1198711

larr ℎ sdot 119866(119905119894

1199101

(119905119894

119903) 1199102

(119905119894

119903))

1198712

larr ℎ119866(119905119894

+ (14)ℎ 1199101

(119905119894

119903) + (14)1198711

1199102

(119905119894

119903) +

(14)1198711

)

1198713

larr ℎ119866(119905119894

+ (38)ℎ 1199101

(119905119894

119903) + (332)1198711

+ (932)1198712

1199102

(119905119894

119903) + (332)1198711

+ (932)1198712

)

1198714

larr ℎ119866(119905119894

+ (1213)ℎ 1199101

(119905119894

119903) + (19322197)1198711

minus

(72002197)1198712

+ (72962197)1198713

1199102

(119905119894

119903) + (1932

2197)1198711

minus (72002197)1198712

+ (72962197)1198713

)

1198715

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) + (439216)1198711

minus 81198712

+

(3680513)1198713

minus (8454104)1198714

1199102

(119905119894

119903) + (439

216)1198711

minus 81198712

+ (3680513)1198713

minus (8454104)1198714

)

1198716

larr ℎ119866(119905119894

+ ℎ 1199101

(119905119894

119903) minus (827)1198711

+ 21198712

minus

(35442565)1198713

+ (18594104)1198714

minus (1140)1198715

1199102

(119905119894

119903) minus (827)1198711

+ 21198712

minus (35442565)1198713

+ (1859

4104)1198714

minus (1140)1198715

)

Step 4

1199101

(119905119894+1

119903) = 1199101

(119905119894

119903)+(16135)1198701

+(665612825)1198703

+

(2856156430)1198704

minus (950)1198705

+ (255)1198706

Step 5

1199102

(119905119894+1

119903) = 1199102

(119905119894

119903)+(16135)1198711

+(665612825)1198713

+

(2856156430)1198714

minus (950)1198715

+ (255)1198716

Step 6 119905119894+1

= 119905119894

+ ℎ Write 1199101

(119905119894+1

119903) 1199102

(119905119894+1

119903) 119905119894+1

Step 7 Next 119894

Step 8 End

5 Numerical Example

Example Solve 1199101015840

= minus119910 + 119905 + 1 with initial condition 119910(0) =

(096 1 101) Then find the solution at 119905 = 01

Solution For (i)-gH differentiable case the exact solution is

1199101

(119903 119905) = 119905 + (096 + 004119903) 119890minus119905

1199102

(119903 119905) = 119905 + (101 minus 001119903) 119890minus119905

(41)

and for (ii)-gH differentiable case the exact solution is

1199101

(119903 119905) = 1 + 119905 + (minus004 + 004119903) 119890119905

1199102

(119903 119905) = 1 + 119905 + (001 minus 001119903) 119890119905

(42)

Remark 14 From Figure 1 and Table 1 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

001 (for ℎ = 0001 is nearly equal) whereas the lowerexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

Remark 15 From Figure 2 and Table 1 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the upperexact solution ((ii)-gH case) is approximately equal to thenumerical solution when we take the step length ℎ = 01

6 Application

Problem A tank initially contains 300 gals of brine which hasdissolved in 119888 lbs of salt Coming into the tank at 3 galsmin

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

8 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

085 09 095 1 105 11 115

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

Figure 1 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 01

1

09

08

07

06

05

04

03

02

01

0

094 098 102 106 11 114

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 2 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 01

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

International Journal of Differential Equations 9

Table1Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119905=

01

119903

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009686

10139

10558

11111

10597

11149

09696

10201

09610

10110

01

09723

10130

10602

11099

1064

211138

09737

10191

09650

10100

02

09759

10121

1064

611088

10686

11127

09777

10181

09690

10090

03

09795

10112

10691

11077

10730

11116

09818

10171

09730

10080

04

09831

10103

10735

1106

610

774

11105

09858

10161

09770

10070

05

09867

10094

10779

11055

10818

11094

09898

10151

09810

1006

006

09904

10085

10823

1104

410

862

11083

09939

10141

09850

10050

07

09940

10076

10867

11033

10906

11072

09979

10131

09890

1004

008

09976

1006

610

912

11022

10951

11061

10020

10121

09930

10030

09

10012

10057

10956

11011

10995

11050

1006

010

11009970

10020

110

048

1004

81100

01100

011039

11039

10100

10100

10010

10010

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

10 International Journal of Differential Equations

Lower exact solution for (i)-gH caseLower exact solution for (ii)-gH caseLower numerical solution for step length 01

Lower numerical solution for step length 001

Lower numerical solution for step length 0001

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3

Figure 3 Comparison of lower exact solutions and numerical solution for different step lengths at 119905 = 04

is brine with concentration 119896 lbs saltgals and the well stirredmixture leaves at the rate 3 galsmin Let119910(119909) lbs be the salt inthe tank at any time 119905 ge 0 Then 119889119910(119909)119889119909+ (1100)119910(119909) = 119896119909 isin [0 05] with 119910(0) = 119888 if the initial condition is beingmodeled as fuzzy numbers 119888 = (1 2 3) and 119896 = (1 2 4) Findsolution at 119909 = 04

Solution For (i)-gH differentiable case the exact solution is

1199101

(119909 120572) =(1 + 120572)

100(1 + 99119890

minus(1100)119909

)

1199102

(119909 120572) =(2 minus 120572)

50+

(148 minus 49120572)

50119890minus(1100)119909

(43)

For (ii)-gH differentiable case the exact solution is

1199101

(119909 120572) = (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (100 + 100120572)

1199102

(119909 120572) = minus (149 minus 149120572) 119890(1100)119909

+ (minus248 + 50120572) 119890minus(1100)119909

+ (400 minus 200120572)

(44)

Remark 16 From Figure 3 and Table 2 we conclude that thelower exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length ℎ =

0001 (for ℎ = 001 is nearly equal) whereas the lower exactsolution ((ii)-gH case) is not equal to any numerical solution

Remark 17 From Figure 4 and Table 2 we conclude that theupper exact solution ((i)-gH case) is approximately equal tothe numerical solution when we take the step length of ℎ =

001 and ℎ = 0001 For ℎ = 01 it is nearly equal whereasthe upper exact solution ((ii)-gH case) is not equal to anynumerical solution

7 Conclusion

In this paper we applied Runge-Kutta-Fehlberg method forfinding the numerical solution of first-order linear differentialequation in fuzzy environment The numerical solution iscompared with the exact solution ((i)-gH and (ii)-gH bothcases) The results presented in the contribution show thatRunge-Kutta-Fehlberg method is a powerful mathemati-cal tool for solving first-order linear differential equationin fuzzy environment The convergence of Runge-Kutta-Fehlberg method has been discussed The process methodis applied to a mechanical problem in fuzzy environmentwhich shows that it is a promising method to solve the saidtypes of differential equation In the future we can applythese methods for solving higher order linear and nonlineardifferential equation in fuzzy environment

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

International Journal of Differential Equations 11

Table2Com

paris

onof

thee

xactsolutio

nsandnu

mericalsolutio

nsford

ifferentstepleng

thsa

t119909=

04

120572

Exactsolutionfor(i)-gH

differentiablec

ase

Exactsolutionfor(ii)-gH

differentiablec

ase

Num

ericalsolutio

nforℎ

=01by

RKFmetho

dNum

ericalsolutio

nforℎ

=001

byRK

Fmetho

dNum

ericalsolutio

nforℎ

=0001

byRK

Fmetho

d1198841

1198842

1198841

1198842

1199101

1199102

1199101

1199102

1199101

1199102

009960

29882

25872

33928

11039

33117

10100

30301

10010

30030

01

10957

28886

26075

33326

12143

32013

11110

29291

11011

29029

02

11953

27890

26279

32723

13247

30909

12120

28281

12012

28028

03

12949

26894

264

8232121

14351

29805

13130

27271

13013

27027

04

13945

25897

266

8531519

15455

28701

14141

26261

14014

26026

05

14941

24901

26888

30916

16558

27597

15151

25251

15015

25025

06

15937

23905

27091

30314

17662

26493

16161

24241

16016

24024

07

16933

22909

27295

29711

18766

25390

17171

23231

17017

23023

08

17929

21913

27498

29109

19870

24286

18181

22221

18018

22022

09

18925

20917

27701

28507

20974

23182

19191

21211

19019

21021

119

921

19921

27904

27904

22078

22078

20201

20201

20020

20020

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

12 International Journal of Differential Equations

1

09

08

07

06

05

04

03

02

01

0

0 05 1 15 2 25 3 35 4

Upper numerical solution for step length 0001

Upper numerical solution for step length 001

Upper numerical solution for step length 01

Upper exact solution for (ii)-gH caseUpper exact solution for (i)-gH case

Figure 4 Comparison of upper exact solutions and numerical solution for different step lengths at 119905 = 04

Competing Interests

The authors declare that there are no competing interests

References

[1] A Kandel and W J Byatt ldquoFuzzy differential equationsrdquo inProceedings of the International Conference Cybernetics andSociety pp 1213ndash1216 Tokyo Japan 1978

[2] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemsrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 5 pp 117ndash137 1997

[3] P Diamond ldquoTime-dependent differential inclusions cocycleattractors and fuzzy differential equationsrdquo IEEE Transactionson Fuzzy Systems vol 7 no 6 pp 734ndash740 1999

[4] P Diamond ldquoStability and periodicity in fuzzy differentialequationsrdquo IEEE Transactions on Fuzzy Systems vol 8 no 5pp 583ndash590 2000

[5] P Diamond and P Watson ldquoRegularity of solution sets fordifferential inclusions quasi-concave in a parameterrdquo AppliedMathematics Letters vol 13 no 1 pp 31ndash35 2000

[6] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[7] B Bede I J Rudas and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007

[8] M Chen C Wu X Xue and G Liu ldquoOn fuzzy boundary valueproblemsrdquo Information Sciences vol 178 no 7 pp 1877ndash18922008

[9] A Setia Y Liu and A S Vatsala ldquoNumerical solution of frac-tional integro-differential equations with nonlocal boundaryconditionsrdquo Journal of Fractional Calculus and Applications vol5 no 2 pp 155ndash165 2014

[10] Y Liu ldquoAn efficient algorithm for solving differential equationsto facilitate modeling and simulation of aerospace systemsrdquoSAE Technical Paper 2015-01-2402 2015

[11] A Setia Y Liu and A S Vatsala ldquoSolution of linear fractionalFredholm integro-differential equation by using second kindChebyshev waveletrdquo in Proceedings of the 11th InternationalConference on Information Technology (ITNG rsquo14) pp 465ndash469Las Vegas Nev USA April 2014

[12] M Ma M Friedman and A Kandel ldquoNumerical solutions offuzzy differential equationsrdquo Fuzzy Sets and Systems vol 105no 1 pp 133ndash138 1999

[13] T Allahviranloo N Ahmady and E Ahmady ldquoTwo stepmethod for fuzzy differential equationsrdquo International Mathe-matical Forum vol 1 no 17ndash20 pp 823ndash832 2006

[14] T Allahviranloo N Ahmady and E Ahmady ldquoNumericalsolution of fuzzy differential equations by predictor-correctormethodrdquo Information Sciences vol 177 no 7 pp 1633ndash16472007

[15] T Allahviranloo E Ahmady andN Ahmady ldquoNth-order fuzzylinear differential equationsrdquo Information Sciences vol 178 pp1309ndash1324 2008

[16] S Pederson and M Sambandham ldquoNumerical solution ofhybrid fuzzy differential equation IVPs by a characterizationtheoremrdquo Information Sciences vol 179 no 3 pp 319ndash328 2009

[17] S Effati and M Pakdaman ldquoArtificial neural network approachfor solving fuzzy differential equationsrdquo Information Sciencesvol 180 no 8 pp 1434ndash1457 2010

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

International Journal of Differential Equations 13

[18] C Duraisamy and B Usha ldquoAnother approach to solution offuzzy differential equations bymodified Eulerrsquosmethodrdquo inPro-ceedings of the International Conference on Communication andComputational Intelligence (INCOCCI rsquo10) pp 52ndash55 KonguEngineering College Erode India December 2010

[19] H Saberi Najafi F Ramezani Sasemasi S Sabouri Roudkoliand S Fazeli Nodehi ldquoComparison of two methods for solvingfuzzy differential equations based on Euler method and Zadehsextensionrdquo The Journal of Mathematics and Computer Sciencevol 2 no 2 pp 295ndash306 2011

[20] M Rostami M Kianpour and E bashardoust ldquoA numericalalgorithm for solving nonlinear fuzzy differential equationsrdquoThe Journal of Mathematics and Computer Science vol 2 no4 pp 667ndash671 2011

[21] M Sh Dahaghin andMMMoghadam ldquoAnalysis of a two-stepmethod for numerical solution of fuzzy ordinary differentialequationsrdquo Italian Journal of Pure andAppliedMathematics vol27 pp 333ndash340 2010

[22] K Batiha ldquoNumerical solutions of the multispecies predator-prey model by variational iteration methodrdquo Journal of Com-puter Science vol 3 no 7 pp 523ndash527 2007

[23] M Z Ahmad and M K Hasan ldquoA new fuzzy version of Eulerrsquosmethod for solving differential equations with fuzzy initialvaluesrdquo Sains Malaysiana vol 40 no 6 pp 651ndash657 2011

[24] V Nirmala and S Chenthur Pandian ldquoA method to improvethe numerical solution of fuzzy initial value problemsrdquo Inter-national Journal of Mathematical Archive vol 3 no 3 pp 1231ndash1240 2012

[25] M Shafiee S Abbasbandy and T Allahviranloo ldquoPredictor-corrector method for nonlinear fuzzy volterra integral equa-tionsrdquo Australian Journal of Basic and Applied Sciences vol 5no 12 pp 2865ndash2874 2011

[26] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[27] H Jafari M Saeidy and D Baleanu ldquoThe variational iterationmethod for solving n-th order fuzzy differential equationsrdquoCentral European Journal of Physics vol 10 no 1 pp 76ndash852012

[28] S Tapaswini and S Chakraverty ldquoA new approach to fuzzyinitial value problem by improved Euler methodrdquo Fuzzy Infor-mation and Engineering vol 4 no 3 pp 293ndash312 2012

[29] M A Ahmed and F S Fadhel ldquoSolution of fuzzy initialvalue problemsusing Least square andAdomiandecompositionmethodrdquo Journal of Kufa for Mathematics and Computer vol 1no 4 pp 49ndash60 2011

[30] R K Saikia ldquoSolution of differential equation by Eulerrsquosmethodusing fuzzy conceptrdquo International Journal of Computer Tech-nology amp Applications vol 3 no 1 pp 226ndash230 2012

[31] R Ezzati F Mokhtari and M Maghasedi ldquoNumerical solutionof volterra-fredholm integral equations with the help of inverseand direct discrete fuzzy transforms and collocation techniquerdquoInternational Journal of Industrial Mathematics vol 4 no 3 pp221ndash229 2012

[32] L Wang and S Guo ldquoAdomian method for second-order fuzzydifferential equationrdquo World Academy of Science Engineeringand Technology vol 76 pp 979ndash982 2011

[33] O S Fard ldquoAn iterative scheme for the solution of generalizedsystem of linear fuzzy differential equationsrdquo World AppliedSciences Journal vol 7 no 12 pp 1597ndash1604 2009

[34] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 46 pp 2269ndash2280 2009

[35] B Asady and M Alavi ldquoNumerical method for solving nth-order linear fuzzy differential equationsrdquoAdvances in Fuzzy Setsand Systems vol 6 no 2 pp 95ndash106 2010

[36] C Duraisamy and B Usha ldquoNumerical solution of fuzzy differ-ential equations by Runge-Kutta method of order threerdquo Inter-national Journal of Advanced Scientific and Technical Researchvol 1 no 1 pp 1ndash13 2011

[37] V Nirmala and S Chenthur Pandian ldquoNumerical solution offuzzy differential equation by fourth order Runge-Kuttamethodwith higher order derivative approximationsrdquo European Journalof Scientific Research vol 62 no 2 pp 198ndash206 2011

[38] T Jayakumar DMaheskumar andK Kanagarajan ldquoNumericalsolution to fuzzy differential equations by Runge Kutta methodof order fiverdquo Applied Mathematical Sciences vol 6 no 57ndash60pp 2989ndash3002 2012

[39] B Ghazanfari and A Shakerami ldquoNumerical solutions of fuzzydifferential equations by extended Runge-Kutta-like formulaeof order 4rdquo Fuzzy Sets and Systems vol 189 pp 74ndash91 2012

[40] K Kanagarajan and M Sambath ldquoNumerical solution of fuzzydifferential equations by third order Runge-Kutta methodrdquoInternational Journal of Applied Mathematics and Computationvol 2 no 4 pp 1ndash8 2010

[41] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy differential equations by Ruge-Kutta Fehlbergmethodrdquo International Journal of Mathematical Analysis vol 6no 53-56 pp 2619ndash2632 2012

[42] Z Akbarzadeh Ghanaie and M Mohseni Moghadam ldquoSolvingfuzzy differential equations by Runge-Kutta methodrdquoThe Jour-nal of Mathematics and Computer Science vol 2 no 2 pp 208ndash221 2011

[43] MGhanbari ldquoSolution of the first order linear fuzzy differentialequations by some reliable methodsrdquo Journal of Fuzzy SetValuedAnalysis vol 2012 Article ID jfsva-00126 20 pages 2012

[44] V Nirmala and S Chenthur Pandian ldquoNew multi-step Runge-Kutta method for solving fuzzy differential equationsrdquo Mathe-matical Theory and Modeling vol 1 no 3 pp 16ndash22 2011

[45] N Saveetha and S Chenthur Pandian ldquoNumerical solution offuzzy hybrid differential equation by third order Runge KuttaNystrom methodrdquo Mathematical Theory and Modeling vol 2no 4 pp 8ndash17 2012

[46] N M Deshmukh ldquoA new approach to solve Fuzzy differentialequation by using third order Runge-Kutta methodrdquo ISSN No-2031-5063 Vol 1 Issue 3 pp 1ndash4 2011

[47] C Duraisamy and B Usha ldquoNumerical solution of fuzzydifferential equations by Runge-Kutta method of order fourrdquoEuropean Journal of Scientific Research vol 67 no 2 pp 324ndash337 2012

[48] T Jayakumar and K Kanagarajan ldquoNumerical solution forhybrid fuzzy system by Runge Kutta method of order fiverdquoApplied Mathematical Sciences vol 6 no 72 pp 3591ndash36062012

[49] J J Buckley T Feuring and Y Hayashi ldquoLinear systems of firstorder ordinary differential equations fuzzy initial conditionsrdquoSoft Computing vol 6 no 6 pp 415ndash421 2002

[50] M Oberguggenberger and S Pittschmann ldquoDifferential equa-tions with fuzzy parametersrdquo Mathematical and ComputerModelling of Dynamical Systems vol 5 no 3 pp 181ndash202 1999

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013

14 International Journal of Differential Equations

[51] J Casasnovas and F Rossell ldquoAveraging fuzzy biopolymersrdquoFuzzy Sets and Systems vol 152 no 1 pp 139ndash158 2005

[52] M S El Naschie ldquoFrom experimental quantum optics toquantum gravity via a fuzzy Kahler manifoldrdquo Chaos Solitonsamp Fractals vol 25 no 5 pp 969ndash977 2005

[53] A Bencsik B Bede J Tar and J Fodor ldquoFuzzy differentialequations in modeling hydraulic differential servo cylindersrdquoin Proceedings of the 3rd Romanian-Hungarian Joint Symposiumon Applied Computational Intelligence (SACI rsquo06) TimisoaraRomania 2006

[54] H Zarei A V Kamyad and A A Heydari ldquoFuzzy modelingand control ofHIV infectionrdquoComputational andMathematicalMethods inMedicine vol 2012 Article ID 893474 17 pages 2012

[55] G L Diniz J F R Fernandes J F C A Meyer and L CBarros ldquoA fuzzy Cauchy problem modelling the decay of thebiochemical oxygen demand in waterrdquo in Proceedings of theJoint 9th IFSA World Congress and 20th NAFIPS InternationalConference IEEE Vancouver Canada July 2001

[56] M Z Ahmad and B De Baets A Predator-Prey Model withFuzzy Initial Populations IFSA-EUSFLAT 2009

[57] L C Barros R C Bassanezi and P A Tonelli ldquoFuzzymodellingin population dynamicsrdquoEcologicalModelling vol 128 no 1 pp27ndash33 2000

[58] B Bede I J Rudas and J Fodor ldquoFrictionmodel by using fuzzydifferential equationsrdquo in Proceedings of the 12th InternationalFuzzy Systems Association World Congress (IFSA rsquo07) CancunMexico June 2007 vol 4529 of Lecture Notes in ComputerScience pp 23ndash32 Springer Berlin Germany 2007

[59] S P Mondal S Banerjee and T K Roy ldquoFirst order linearhomogeneous ordinary differential equation in fuzzy environ-mentrdquo International Journal of Pure and Applied Sciences andTechnology vol 14 no 1 pp 16ndash26 2013

[60] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environmentrdquo Mathe-matical theory and Modeling vol 3 no 1 pp 85ndash95 2013

[61] S P Mondal and T K Roy ldquoFirst order linear homogeneousordinary differential equation in fuzzy environment based onlaplace transformrdquo Journal of Fuzzy Set Valued Analysis vol2013 Article ID jfsva-00174 18 pages 2013

[62] S P Mondal and T K Roy ldquoFirst order linear homogeneousfuzzy ordinary differential equation based on lagrange multi-plier methodrdquo Journal of Soft Computing and Applications vol2013 Article ID jsca-00032 17 pages 2013

[63] S PMondal andTK Roy ldquoFirst order linear non homogeneousordinary differential equation in fuzzy environment basedon Laplace transformrdquo Journal of Mathematics and ComputerScience vol 3 no 6 pp 1533ndash1564 2013

[64] S PrasadMondal and T Kumar Roy ldquoFirst order homogeneousordinary differential equation with initial value as triangularintuitionistic fuzzy numberrdquo Journal of Uncertainty in Mathe-matics Science vol 2014 Article ID jums-00003 17 pages 2014

[65] S P Mondal and T K Roy ldquoSystem of differential equationwith initial value as triangular intuitionistic fuzzy number andits applicationrdquo International Journal of Applied and Computa-tional Mathematics vol 1 no 3 pp 449ndash474 2015

[66] S Chakraverty and S Tapaswini ldquoStanderd orthogonal poly-nomial based solution of fuzzy differential equationrdquo TheInternational Journal of Fuzzy Computation and Modelling vol1 no 1 2014

[67] S Tapaswini and S Chakraverty ldquoNumerical solution offuzzy arbitrary order predator-prey equationsrdquo Applications ampApplied Mathematics vol 8 no 2 pp 647ndash672 2013