Asymptotic solution for heat convection-radiation equationFazle Mabood, Waqar A. Khan, and Ahmad Izani Md Ismail

Citation: AIP Conference Proceedings 1605, 529 (2014); doi: 10.1063/1.4887644 View online: http://dx.doi.org/10.1063/1.4887644 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 Asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in the 2D convexpolygonal domain AIP Conf. Proc. 1558, 383 (2013); 10.1063/1.4825504 Convection in rotating annulus: An asymptotic theory and numerical solutions Phys. Fluids 10, 2396 (1998); 10.1063/1.869756 Asymptotic solution of the Enskog equation Phys. Fluids A 1, 746 (1989); 10.1063/1.857369 Asymptotic solution of the diatomic Boltzmann equation Phys. Fluids 26, 50 (1983); 10.1063/1.863983 Asymptotic Expansions of Solutions of the Heat Conduction Equation in Internally Bounded Cylindrical Geometry J. Appl. Phys. 27, 1453 (1956); 10.1063/1.1722288

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Asymptotic Solution for Heat Convection-Radiation Equation

Fazle Mabood a, Waqar A. Khan b and Ahmad Izani Md Ismail a

a School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia b Department of Engineering Sciences, National University of Sciences and Technology, PN Engineering

College, Karachi, 75350 Pakistan

Abstract. In this paper, we employ a new approximate analytical method called the optimal homotopy asymptotic method (OHAM) to solve steady state heat transfer problem in slabs. The heat transfer problem is modeled using nonlinear two-point boundary value problem. Using OHAM, we obtained the approximate analytical solution for dimensionless temperature with different values of a parameter . Further, the OHAM results for dimensionless temperature have been presented graphically and in tabular form. Comparison has been provided with existing results from the use of homotopy perturbation method, perturbation method and numerical method. For numerical results, we used Runge-Kutta Fehlberg fourth-fifth order method. It was found that OHAM produces better approximate analytical solutions than those which are obtained by homotopy perturbation and perturbation methods, in the sense of closer agreement with results obtained from the use of Runge-Kutta Fehlberg fourth-fifth order method.

Keywords: OHAM; heat transfer; nonlinear ODE PACS: 40.05.+e, 40.40.+c

INTRODUCTION

Nonlinear physical phenomena are important since most physical systems are inherently nonlinear in nature. Nonlinear differential equations are generally difficult to solve and nonlinear phenomena often lead to interesting scientific phenomena such as chaos. Hence, the analysis of the approximate solutions of nonlinear differential equations is important in the study of nonlinear physical phenomena.

There are many methods for seeking approximate solutions of nonlinear equations. Examples of such methods are established methods such as homotopy perturbation method (HPM) [1-5], perturbation method (PM) [6-8] and homotopy analysis method (HAM) [9]. In this paper, we use a new approximate analytical method to investigate steady state heat transfer in slabs.

The mathematical model for the temperature distribution in a lumped system of combined convection–radiation in a slab composed of materials with variable thermal conductivity is given by the following nonlinear boundary value problem [10]:

4'' , 0, 1y y , (1) subject to boundary conditions:

(0) 0, (1) 1y y , (2) where

y = a

i a

T TT T

= Dimensionless temperature, = ( )

a

t hAV c

= Dimensionless time, = ( )aT T ,

V = Volume of the system, A = Surface area, = Density, h = Heat transfer coefficient,

iT = Initial temperature, aT = Convection environment temperature, ac = Specific heat temperature at aT , = Volumetric thermal expansion coefficient (constant).

The mathematical model for convection-radiation in slabs have been studied using HPM and PM by Khan [10] who concluded that the solution generated by HPM and PM diverges for large value of . Studies of nonlinear heat radiation equation using HAM, HPM and PM have also been conducted by Abbasbandy [11], Domairry and Nadim [12] as well as by Ganji and Rajabi [13].

A new approximate analytical method named optimal homotopy asymptotic method (OHAM) has been proposed by Marinca and Nicolae [14]. OHAM has been used for solving nonlinear equations arising in heat transfer. Marinca et al. [15] studied the steady flow of fourth-grade fluid past a porous plate using OHAM.

Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21)AIP Conf. Proc. 1605, 529-534 (2014); doi: 10.1063/1.4887644

© 2014 AIP Publishing LLC 978-0-7354-1241-5/$30.00

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Islam et al. [16] used OHAM for the asymptotic solutions of Couette and Poiseuille flows of a fluid whilst Mabood et al. [17-19] examined the Riccati equation and heat transfer phenomena with the help of OHAM.

According to Herisanu and Marinca [20], the idea of OHAM is based on coupling the homotopy perturbation method and a computational algorithm intended to optimally control the convergence of the solution through an auxiliary function ( )H q which depends on a number of initially unknown parameters. The algorithm used to identify these unknown parameters can be based on the least squares method, collocation method or Galerkin method. In this study, we have used least squares method.

Basic Principles of OHAM

We describe OHAM following Marinca et al. [15], Islam et al. [16], and Mabood et al. [17-19]. Consider the following differential equation and boundary condition:

( ( )) ( ) ( ( )) 0, , 0,dvL v x g x N v x B vdx

(3)

where ,L N are linear and nonlinear operators, x denotes independent variable, ( )v x is an unknown function, ( )g x is a known function and B is a boundary operator. A deformation equation is constructed as follows:

(1 )[ ( ( , )) ( )] ( )[ ( ( , )) ( ) ( ( , ))],

,, , 0,

q L x q g x H q L x q g x N x q

x qB x q

x (4)

where 0 1q is an embedding parameter, ( )H q is an auxiliary function such that ( ) 0,H q for 0q and (0) 0H , and ,x q is an unknown function.

For 0q and 1q , we have, 0, 0x v x and , 1x v x , respectively.

Hence, as q varies from 0 to 1, the solution ( , )x q varies from 0 ( )v x to the solution ( )v x , where 0 ( )v x is obtained from Eq. (4) for 0q . The zeroth order problem is

00 0( ) 0, , 0

dvL v x g x B v

dx. (5)

The auxiliary function ( )H q is chosen in the form:

1

nk

kk

H q q C , (6)

where ,iC i are constants which are to be determined [20] .

For the solution, , , ix q C is expanded in Taylor’s series about q and is given by

01

, , , , 1,2,3,...ki k i

kx q C v x v x C q i (7)

Substituting Eq. (7) into Eq. (4), and equating the coefficients of the like powers of q equal to zero, gives the linear equations. The zeroth order problem is given by Eq. (5), and the first and second order problems are given by the Eqs. (8) and (9), respectively:

11 1 0 0 1, , 0

dvL v x C N v x B v

dx, (8)

2 1 2 0 0 1 1 1 0 1

22

, ,

, 0.

L v x L v x C N v x C L v x N v x v x

dvB v

dx

(9)

The equations for kv x are given by

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1 0 0

1

0 1 11

, ,..., ,

, 0, 2,3,...

k k k

k

i k i k i ki

kk

L v x L v x C N v x

C L v x N v x v x v x

dvB v k

dx

(10)

where 0 1, ,...,m mN v x v x v x is the coefficient of mq in the expansion of , , iN x q C about the

embedding parameter .q

0 0 0 11

, , , ,..., mi m m

mN x q C N v x N v v v q . (11)

The convergence of the series (7) is dependent upon the auxiliary constants 1 2, ,...C C . If it is convergent at 1q , one has

1 2 0 1 21

, , ,..., , , ,...,m

m i mi

v x C C C v x v x C C C1 2, 1 2v x C C1 2, 1 2 . (12)

Substituting Eq. (12) into Eq. (3), which is the general problem, results in the following residual:

1 2 1 2 1 2, , ,..., ( ( , , ,..., )) ( ) ( ( , , ,..., ))m m mR x C C C L v x C C C g x N v x C C C( , , ,..., )) ( ) ( ( , , ,..., ))m m1 2 1 2( , .. )) ( ) ( ( ,...,.. )) ( ) ( ( ..1 2 1 211 m1 2 1 2 ,...,1 2 1 21 2 11 2 1( )) ( )( )) (( )1 21 21 2 . (13)

If 0R , then vv will be the exact solution. For nonlinear problems, generally this will not be the case. To find iC ( 1,2,...,i m ), we use the method of least squares

21 2 1 2, ,..., , , ,...,

b

m ma

J C C C R x C C C dx , (14)

where ( )R L v g x N v( )v g x N v( )v g xg x( ) is the residual and

1 2

... 0m

J J JC C C

. (15)

With these constants, one can get the approximate solution of order m.

OHAM Solution

Applying the proposed method (OHAM) for Eqs. (1) and (2), the zeroth, first, second, third and fourth order problems can be described as:

0 0 0( ) 0, (0) 0, (1) 1y y y (16) 4 (2)

1 1 1 0 1 0 1 1( , ) ( ) (1 ) ( ), (0) 0, (1) 0y C C y C y y y (17) 3 (2)

2 1 1 0 1 1 1 1 1 2 2( , ) 4 ( ) ( , ) (1 ) ( , ), (0) 0, (1) 0y C C y y C C y C y y (18) 2 2 3 (2)

3 1 1 0 1 1 1 0 2 1 1 2 1 3 3( , ) 6 ( ) ( , ) 4 ( ) ( , ) (1 ) ( , ), (0) 0, (1) 0y C C y y C C y y C C y C y y (19) 3 2

1 1 1 0 1 0 1 1 2 14 1 3 (2)

1 0 3 1 1 3 1

4 ( , ) ( ) 12 ( ) ( , ) ( , )( , )

4 ( ) ( , ) (1 ) ( , )

C y C y C y y C y Cy C

C y y C C y C , 4 4(0) 0, (1) 0y y (20)

Solving Eqs. (16) - (20) in succession, we obtain 0 ( ) 1y (21)

1121( , ) (1 )

2Cy

C (22)

2 2 2 2 2 2 2 2 2 4 2 21 1 1 1 1 12 1 1

1 (3 3 3 3 5 )6

( , ) 6C C C C Cy C C C (23)

2 2 2 2 2 2 2 2 2 4 2 2 31 1 1 1 1 1 1 1 1

2 3 2 3 2 2 3 4 2 3 3 3 2 3 3 4 3 3 6 3 31 1 1 1 1 1 1 1

31 (90 90 180 180 300 360 60 90

18090 300 360 60 3

( ,

43 435 105 13

)

)

C C C C C C C C C

C C C C

y

C C C C (24)

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2 2 2 2 2 2 2 2 2 4 2 21 1 1 1 1 1 1 1

3 2 3 2 3 2 2 3 4 2 3 3 3 2 3 31 1 1 1 1 1 1

4 3 3 6 3 3 4 2 41 1 1 1

41 (360 360 1080 1080 1800 2160 360

7201080 1080 3600 4320 720 4116 5220

1260 156 360 36

,

0

( )

1

C C C C C C C C

C C C C C C C

C C

y

C C 2 4 2 2 4 4 2 41 1 1

3 4 2 3 4 4 3 4 6 3 4 4 4 2 4 41 1 1 1 1 1

4 4 4 6 4 4 8 4 41 1 1

800 2160 360

4116 5220 1260 156 3631 4724

1330 260 23 )

C C C

C C C C C C

C C C

(25)

The fourth order approximate solution by OHAM for 1q is

1 0 1 1 2 1 3 1 4 1( , ) ( ) ( , ) ( , ) ( , ) ( , )y C y y C y C y C y C1( )y( )1( , )1 . (26) We use the method of least squares to obtain the unknown convergent constant 1C in Eq. (26). For 2 , we obtain 1 0.33901C . Thus, the series solution via OHAM is

2 2 4

2 4 6

2 4 6 8

1 11 ( 0.67802 0.67803 ) (0.95409 1.41382 0.45972 )2 61 ( 42.41410 52.85860 14.49660 4.05213 )

1801 (210.40590 286.25200 93.79330 22.80815 4.86090 ).

720

y 11 (12

(y ((

(27)

Results and Discussion

In Tables 1, 2, 3 and Fig. 1, we compare the solution obtained by OHAM with other approximate solutions obtained by HPM [10], PM [10] and with Runge-Kutta Fehlberg fourth-fifth (RKF-45) order numerical method for different values of parameter , further, the absolute error is also provided accordingly. We define error to be the absolute difference between results obtained and RKF-45 numerical method. As shown in Tables 1 and 2, the OHAM error is less than that of HPM and PM for 0.2, 0.6 . HPM and PM also show less accurate results than OHAM for larger values of (see Table 3). The point about inaccurate results for a larger value of was also raised by Khan [10]. Here, it is quite clear that if 0 , OHAM produces y = 1, which is the exact solution of Eq. (1) for 0 . In Table 4, we provided the values of 1C for different values of parameter . Thus OHAM provides a convenient way to control the convergence of the obtained approximation series and adjust convergence regions.

TABLE (1). Comparison of results and error of dimensionless temperature.

0.2 Ab. Error OHAM HPM [10] PM [10] Numerical OHAM HPM PM

0.0 0.92375 0.89333 0.93333 0.92325 5.0×10-5 2.9×10-2 1.0×10-1 0.1 0.92447 0.89433 0.93393 0.92397 5.0×10-5 2.9×10-2 9.9×10-3 0.2 0.92665 0.89734 0.93574 0.92616 4.9×10-4 2.8×10-2 9.5×10-3 0.3 0.93029 0.90239 0.93878 0.92982 4.7×10-4 2.7×10-2 8.9×10-3 0.4 0.93543 0.90950 0.94310 0.93497 4.6×10-4 2.5×10-2 8.1×10-3 0.5 0.94208 0.91875 0.94875 0.94165 4.3×10-4 2.9×10-2 7.0×10-3 0.6 0.95030 0.93019 0.95579 0.94991 3.9×10-4 1.9×10-2 5.8×10-3 0.7 0.96013 0.94393 0.96433 0.95980 3.3×10-4 1.5×10-2 4.5×10-3 0.8 0.97163 0.96006 0.97446 0.97138 2.5×10-4 1.1×10-1 3.0×10-3

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TABLE (2). Comparison of results and errors of dimensionless temperature. 0.6 Ab. Error OHAM HPM [10] PM [10] Numerical OHAM HPM PM

0.0 0.94160 0.64000 1.00000 0.83454 0.10706 0.19454 0.16546 0.1 0.94165 0.64300 0.99940 0.83599 0.10566 0.19299 0.16341 0.2 0.94186 0.65209 0.99769 0.84039 0.10147 0.18830 0.15730 0.3 0.94245 0.66748 0.99508 0.84777 0.09468 0.18029 0.14731 0.4 0.94376 0.68953 0.99193 0.85827 0.08549 0.16874 0.13366 0.5 0.94626 0.71875 0.98875 0.87202 0.07424 0.15327 0.11673 0.6 0.95054 0.75577 0.98617 0.88925 0.06129 0.13348 0.09692 0.7 0.95725 0.80140 0.98500 0.91023 0.02191 0.10883 0.04966 0.8 0.96716 0.85657 0.98617 0.93534 0.00209 0.07877 0.02111

TABLE (3). Comparison of results and errors of dimensionless temperature. 2 Ab. Error OHAM HPM [10] PM [10] Numerical OHAM HPM PM

0.0 0.87659 -0.6666 3.3333 0.69431 0.18228 1.36091 2.63899 0.1 0.87661 -0.6566 3.3034 0.69664 0.17997 1.35324 2.60676 0.2 0.87677 -0.6256 3.2144 0.70369 0.17308 1.32929 2.51071 0.3 0.87756 -0.5712 3.0687 0.71566 0.16910 1.28686 2.35304 0.4 0.87971 -0.4896 2.8704 0.73289 0.14682 1.22249 2.13751 0.5 0.88424 -0.3750 2.6250 0.75591 0.12833 1.13091 1.86909 0.6 0.89246 -0.2202 2.3397 0.78548 0.10698 1.00568 1.55422 0.7 0.90598 -0.0166 2.0234 0.82271 0.08327 0.83931 1.20069 0.8 0.92673 0.2464 1.6864 0.86916 0.05757 0.62276 0.81724

TABLE (4). Values of convergent constant 1C for various values of . 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1C -0.5955 -0.9019 -0.8214 -0.6787 -0.5854 -0.5112 -0.4530 -0.4076 -0.3704 -0.3390

y

0 0.5 1

0.87

0.9

0.93

0.96

0.99 0.61.01.42.0

FIGURE 1. Results using OHAM for various values of .

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Conclusion The optimal homotopy asymptotic method has been used to obtain the solution of the heat convection-radiation in a slab. Comparison for dimensionless temperature has been made between the solutions obtained using OHAM with HPM, PM and with RKF-45 method. It is found that the OHAM solution is much better (in the sense of better agreement with the numerical method) than those obtained by HPM and PM. In particular, OHAM gives accurate values for larger values of than obtained by HPM and PM.

ACKNOWLEDGMENTS The authors acknowledge the financial support from School of Mathematical Sciences, USM, and the first author acknowledges the USM fellowship.

REFERENCES 1. J. H. He, Physics Letters A 350, 87-88 (2006). 2. K. Alireza, A. K. Golmankhaneh and D. Baleanu, Signal Processing 91(3), 446-451 (2011). 3. A. K. Golmankhaneh, T. Khatuni, N. A. Proghoveh and D. Baleanu, Central European Journal of

Physica 10(4), 966-976 (2012). 4. A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu, Romanian Reports on Physics 63(3), 609-

623 (2011). 5. D. D. Ganji, Physics Letters A. 355, 337-341 (2006). 6. A. Aziz and T. Y. Na, Perturbation Method in Heat Transfer, Washington, DC: Hemisphere Publishing

Corporation, 1984. 7. A. H. Nayfeh, Perturbation Methods, New York: Wiley, 1973. 8. M. V. Dyke, Perturbation Methods in Fluid Mechanics, Stanford CA: Parabolic Press, 1975. 9. S. J. Liao, Comm. Nonlinear Sci. Numer. Simul. 14, 983-997 (2009). 10. R. A. Khan, Applied Mathematics and Computation 212, 287-295 (2009). 11. S. Abbasbandy, Int. Commun. Heat and Mass Transfer 34(3), 380-387 (2007). 12. G. Domairry and N. Nadim, Int. Commu. Heat and Mass Transfer 35(1), 93-102 (2008). 13. D. D. Ganji and A. Rajabi, Int. Commu. Heat and Mass Transfer 33(3), 391-400 (2006). 14. V. Marinca and H. Nicolae, Int. Commun. Heat and Mass Transfer 35, 710-715 (2008). 15. V. Marinca, H. Nicolae, C. Bota and B. Marinca, Appl. Math. Lett. 22(2), 245-251 (2009). 16. S. Islam, R. A. Shah and A. Ishtiaq, Int. J. Nonlinear Sc. Num. Simul. 11(6), 389-400 (2010). 17. F. Mabood, A. I. M. Ismail and I. Hashim, Sains Malaysiana 42(6), 863-867 (2013). 18. F. Mabood, W. A. Khan and A. I. M. Ismail, Heat Transfer-Asian Research, 43(2), 124-133 (2014). 19. F. Mabood, W. A. Khan and A. I. M. Ismail, Mathematical Problems in Engineering, 7 pages (2013). 20. V. Marinca, H. Nicolae and I. Neme, Central European J. Phys. 6(3), 648-653 (2008).

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