Semi analytical solution of second order fuzzy Riccati equation by homotopyperturbation methodA. F. Jameel and Ahmad Izani Md Ismail Citation: AIP Conference Proceedings 1605, 185 (2014); doi: 10.1063/1.4887586 View online: http://dx.doi.org/10.1063/1.4887586 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1605?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The solution of high dimensional Fredholm integral equations using homotopy perturbation method AIP Conf. Proc. 1635, 52 (2014); 10.1063/1.4903562 Numerical solution of Fredholm-Hammerstein integral equations by using optimal homotopy asymptotic methodand homotopy perturbation method AIP Conf. Proc. 1605, 90 (2014); 10.1063/1.4887570 Approximate solutions to second order parabolic equations. I: Analytic estimates J. Math. Phys. 51, 103502 (2010); 10.1063/1.3486357 Variational Iteration Method and Homotopy‐Perturbation Method for Solving Second‐Order Non‐Linear WaveEquation AIP Conf. Proc. 936, 81 (2007); 10.1063/1.2790272 The Riccati method for the Helmholtz equation J. Acoust. Soc. Am. 100, 1432 (1996); 10.1121/1.415990

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Semi Analytical Solution of Second Order Fuzzy Riccati Equation by Homotopy Perturbation Method

A.F. Jameel and Ahmad Izani Md Ismail

School of Mathematical Sciences, University Science Malaysia, 11800 USM Penang, Malaysia

Abstract. In this work, the Homotopy Perturbation Method (HPM) is formulated to find a semi-analytical solution of the Fuzzy Initial Value Problem (FIVP) involving nonlinear second order Riccati equation. This method is based upon homotopy perturbation theory. This method allows for the solution of the differential equation to be calculated in the form of an infinite series in which the components can be easily calculated. The effectiveness of the algorithm is demonstrated by solving nonlinear second order fuzzy Riccati equation .The results indicate that the method is very effective and simple to apply. Keyword: Fuzzy numbers, Fuzzy differential Equations, Homotopy Perturbation Method.

PACS: 02.30.Hq, 02.60.Lj

INTRODUCTION

The concept of the fuzzy derivative was introduced by Chang and Zadeh [1]. Later, Dubois and Prade [2] presented a concept of the fuzzy derivative based on the extension principle. Other concepts of fuzzy derivative have been discussed by Puri and Ralescu [3], Goetschel and Voxman [4], Seikkala [5] and Friedman et al. [6, 7]. Fuzzy Differential Equation (FDE) involving FIVP was rigorously treated by Kaleva [8, 9]. Fuzzy Initial value problem (FIVP) appears when the modeling of these problems was imperfect and its nature is under uncertainty. Fuzzy ordinary differential equations are suitable mathematical models to model dynamical systems in which there exist uncertainties or vagueness. These models are used in various applications including population models [10], quantum optics gravity [11], and medicine [12, 13]. HPM introduced by Him [14-17] has been used by many mathematicians and engineers to solve various functional equations. In this method the solution is considered as the sum of an infinite series which converges rapidly to the accurate solutions. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter which is considered as a small parameter. In recent year's semi analytical methods such as Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) have been used to solve fuzzy problems involving ordinary differential equations. In [18], the HPM was used to solve first order linear fuzzy initial value problems involving ordinary differential equations. The ADM was employed in [19, 20] to solve first order linear and nonlinear fuzzy initial value problems. VIM in [21] have been employed and compared with ADM to obtain the approximate solution for first order linear FIVP. In [22] VIM also was used to find semi-analytical solution for fuzzy differential equations including nonlinear first order FIVP. Abbasbandy et al. [23] used VIM to find the approximate solution for high order linear FIVP by converted it into a first order system of fuzzy differential equations. In this paper, we use HPM to obtain a semi-analytical solution of second order nonlinear FIVP involving a fuzzy Riccati equation directly. To the best of our knowledge, this is the first attempt at solving the second order FIVP directly without reducing to a first order system using the HPM. The structure of this paper is organized as follows. In section 2, some basic definitions and notations are given which will be used in other sections. In section 3, the structure of HPM is formulated for solving second FIVP. In section 4, we employ HPM on test example and finally, in section 5, we give the conclusion of this study.

Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21)AIP Conf. Proc. 1605, 185-190 (2014); doi: 10.1063/1.4887586

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PRELIMINARIES Definition 1 [2, 23]: An arbitrary fuzzy number is represented by an ordered pair of functions

( ) ( ( ), ( ))t t t for all [0,1]r which is satisfying:

(I) ( )t is normal, i.e 0t with ( ) 10t . (II) ( )t is convex fuzzy set , i.e. ( (1 ) ) min{ ( ), ( )}t s t s , , , [0,1]t s , [0,1, . (III) ( )t is a bounded left continuous non-decreasing function over [0,1] .

(IV) ( )t is a bounded left continuous non-increasing function over [0,1] , ( ) ( ),0 1t t r . Let E be the set of all upper semi-continuous normal convex fuzzy numbers with r-level bounded intervals then we have the following definitions

Definition 2 [21]: A mapping :f J E for some interval J E is called a fuzzy process, and we denoted r-level by:

[ ( )] [ ( ; ), ( ; )], , [0,1]f t f t r f t r t T rr The r-level sets of a fuzzy number are much more effective as representation forms of fuzzy set than the above. Fuzzy sets can be defined by the families of their r-level sets based on the resolution identity theorem [11]. Definition 4 [24]: Each function :f X Y induces another function : ( ) ( )f F X F Y defined for each fuzzy interval U in X by:

1 ( ), if range ( )( )( )( )0 , if range ( )

Sup U x y fx f yf U yy f

In this case, the function f is said to be obtained by idea of Zadeh extension principle.

FUZZIFICATION AND DEFUZZIFICATION OF HPM OF 2nd ORDER FIVP

The general structures of HPM for solving crisp nonlinear problems have been presented in [14-17]. In this section we present in details the structure of HPM for the approximate solution of FIVP. The HPM is applied to solve approximately second order fuzzy differential equation. Toward this end, consider the following 2nd order FIVP:

/ / /

/ /

0 00 0

( ) ( , ( ), ( )) ( ), [ , ] 0

( ) , ( )

y t f t y t y t w t t t T

y t y y t y (1)

Here ( )y t is the fuzzy function of the crisp variable t,

/( , ( ), ( ))f t y t y t is a fuzzy function of the crisp

variable t and the fuzzy variable and / /

( )y t is the 2nd order fuzzy derivative [21, 23] of /

( ), ( )y t y t , with /

0 0,y y and ( )w t are the fuzzy numbers as mentioned in section 2. We denote the fuzzy function ( )y t by

[ , ]y y y , for all t [ 0t , T] and r [0, 1], means that the r-level sets of ( )y t can be defined as follows

[ ( )] [ ; , ; ], [ ( )] [ ( ; ), y ( ; )]r ry t y t r y t r y t y t r t r (2)

0 0 0 0 0 0[ ( )] [ ( ; ), ( ; )], [ ( ; )] [ ( ; ), ( ; )]r ry t y t r y t r y t r y t r y t r (3)

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Now for defuzzification of Eq. (1), we let /

( ) ( ), ( )t y t y t as a fuzzy function of /

( ), ( )y t y t , such that

[ ( )] [ ( ; ), ( ; )] [ ; , ; , ( ; ), ( ; )]rt t r t r y t r y t r y t r y t r (4)

Also we can write

( , ( )) ( , ( ); ), ( , ( ); )r

f t t f t t r f t t r , ( , ; ) , , ( , ; ) ,r r

f t r F t f t r G t (5)

Since / /

( ) ( , ( ))y t f t t and by using the fuzzy extension principle as in section 2, we can define the following membership function

, ( ; ); , ( ) | ( ) ( ; )

, ( ; ); , ( ) | ( ) ( ; )

f t t r r min f t r r t r

f t t r r max f t r r t r

f ,, | ( ) ( ;|| ( )| (( ) |( ) |( ) |( ) |

f , | ( ) ( ;) (| (( ) |( ) | (6)

where

( , ; ; ) , ; , ( , ; ; ) , ;f t t r r F t t r f t t r r G t t r (7)

Now we can rewrite Eq. (1) as follows

/ // / ( ; ) ( , ( )) ; , ( ; ) ( , ( )) ;y t r F t t w t r y t r G t t w t r (8)

we rewrite Eq. (8) for all t [ 0t , T] and r [0,1], in the following forms

/2 0 00 0

/2 0 00 0

( ; ) ( , ( )) ; , ( ; ) , ( ; )

( ; ) ( , ( )) ; , ( ; ) , ( ; )

L y t r F t t w t r y t r y y t r y

L y t r G t t w t r y t r y y t r y (9)

where 2 22 ,L L L are the linear operators with 2

2 2

dLdt

and F,G are nonlinear operators. According to

the HPM, we construct a homotopy form from Eq. (9) which satisfies the following relation

( ; ; ) (1 ) ; ; ; ; ; ; 02 20 0

( ; ; ) (1 ) ; ; ; ; ; ; 02 20 0

H t p r p L y t r y t r p L y t r F t t r w t r

H t p r p L y t r y t r p L y t r G t t r w t r (10)

where ;0y t r and ;0y t r are initials guessing that satisfied the initial conditions in Eq.(1) and can be

defining as follows:

; , ; ( ) ( )0 1 2 0 1 2y t r C r C r t y t r C r C r t (11)

for all r [0,1], ,1 2C r C r are the constants can be determined easily from the initial conditions (1).

Since is an embedding parameter, ;0y t r ;0y t r are the initial guesses approximation

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of Eq. (1), which satisfies the initial condition. For all t [ 0t , T] and r [0, 1] and from Eq. (10) we have

( ;0; ) (1 ) ; ; 0, ( ;0; ) ; ; ; ; 02 20 0H t r p L y t r y t r H t r L y t r F t t r w t r (12)

Also for the upper bound we have

( ;0; ) (1 ) ; ; 0, ( ;1; ) ; ; ; ; 02 20 0H t r p L y t r y t r H t r L y t r G t t r w t r (13)

We use the embedding parameter p as a small parameter to construct HPM series such that from Eq. (12) and (13) we can represent each ;y t r in these equations as follows:

' ' ( ) ( ); ; , ; ; ,..., ; ;0 0 0

k k n k ny t r p y t r y t r p y t r y t r p y t rk k kk k k

' ' ( ) ( )k ' ' ( ) (' ( ) ( t r) ;( )(((y t r; kp yk ' ' ( )' (' ( )p y ;; ( )( )( )( )k ( )(( )( ); ( )k kky p y yy p yp y y; , ; ; ,..., ;; , ; ; ,..., ;; , ; ; ,..., ;( )( )( )( ); ; ; ;; ; ; ;; ; ; ;( )( )( )( ) (14)

Now substituting the above equations into Eq. (12) and (13) and collecting terms of the same powers of p, we have:

0 : ; ( ) ( )0 1 2

; ; , ; ; 0 01 021 :'; 0, ; 0 1 0 1 0

; , ; 0 1222 :'; 0,2 0 2

p y t r C r C r t

y t r y t r F t t r w t rp

y t r y t r

y t r F t t rp

y t r y

L

t

L

; 0 0 .........

; , ; 0 121 :'; 0, ; 0 1 0 1 0

r

y t r F t t rkkkpy t r y t rk k

L

(15)

Similarly for the upper bound

0 : ; ( ) ( )0 1 2

; ; , ; ; 0 1 01 :'; 0, ; 0

2 0

2

1 0 1 0

; , ; 0 2 12 :'; 0,2 0

p y t r C r C r t

y t r y t r G t t r w t rp

y t r y t r

y t r G t t rp

t y

L

y

L

r ; 0 2 0 ........

; , ; 0 11 :'; 0, ; 0 1 0 1

2

0

t r

y t r G t t rkkpy t r y t r

k

k

L

k

(16)

for 0,1,...k n and r [0,1]. The approximate solution is given by setting p=1 as follows:

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1

; ; ;0

my t r t r y t rm k

ky t r;y t r t r; ;;m ky;;;; (17)

Now the exact solutions of Eq. (1) can be obtained by setting p = 1as follows:

; lim ; lim ; ;1 1 0 0

kY t r y t r p y t r y t rk kp p k klY t r; ll li t r; lim ;; lim kykk

1y ; lim;; lim; kp y ;;;;;;k ;;kp yk (18)

NUMERICAL EXAMPLE

Consider the fuzzy version of the second nonlinear order Riccati equation [25]

2'' 6 , 0

'0 0.1 0.9,1.1 0.1 , 0 0.1 0.1,1.1 0.1 , [0,1]

y t y t t t

y r r y r r r

2'' 6 2 , y t 6 26 , 6 yy

'0 0 1 0 9 1 1 0 1 0 0 1 0 1 1 1y r r y r0 0.1 0.9,1.1 0.1 , 0 0.1 0.1,1.10 1 0 9 1 1 0 1 0 0 10.1 0.9,1.1 0.1 , 0 0.10.9,1.1 0.1 0 0.10.1 0.9,1.1 0.1 0 0.1 (19)

Since Eq. (19) without analytical exact solution to show the accuracy of HPM in section (3) with five terms series solution of Eq. (19), by taking the residual error [26] such that

2''; , ; 6 , ;5 5E t r t r t r t2

tr;;r;r;r;r;;E t; ; 6; 66; 6; 6; 6; 6 (20)

By using to the HPM prouder in section (3), we can obtain the following results

TABLE 1. Accuracy of the approximation results by 5- terms of HPM of Eq. (19) at for all at

FIGURE 1. Fuzzy approximation solution of 5- terms HPM of Eq. (19) at for all at

r E(0.1;r)HPM E(0.1;r)HPM 0 1.079408243342960× 4.583994109297329×

0.2 1.265096330471671× 4.017310253900952× 0.4 1.477548697281783× 3.511815674372975× 0.6 1.719910117103928× 3.061897083100895× 0.8 1.995615175887088× 2.662359879779252× 1 2.308407723305983× 2.308407723305983×

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CONCLUSIONS In this paper, we applied Homotopy Perturbation Method has been successfully introduced and applied to solve fuzzy nonlinear second order Riccati equation to obtain an approximate solution. The problem was solved directly without it first being reduced to a first order system. The obtained results in this paper show that the HPM is a capable and accurate method for solving second order nonlinear fuzzy initial value problems even without exact analytical solution.

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