Variation of Parameters Method for Nonlinear Diffusion Equations

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International Journal of Modern Applied Physics, 2013, 3(1): 48-56 International Journal of Modern Applied Physics ISSN: 2168-1139 Journal homepage:www.ModernScientificPress.com/Journals/ijmep.aspx Florida, USA Article Variation of Parameters Method for Nonlinear Diffusion Equations

Rahmatullah and Syed Tauseef Mohyud-Din*

Department of Mathematics, HITEC University, Taxila Cantt, Pakistan

*Author to whom correspondence should be addressed; E-Mail: [email protected]

Article history: Received 30 July 2013, Received in revised form 30 August 2003, Accepted 10 December 2013, Published 27 December 2013.

Abstract: In this paper, we apply Variation of Parameters Method (VPM) to solve nonlinear diffusion equations which arise very frequently in various models related to different branches of physics. The suggested algorithm proved to be very efficient and finds the solution without any discretization, linearization, perturbation or restrictive assumptions. Numerical results reveal the complete reliability of the proposed VPM.

Keywords Variation of Parameters Method; nonlinear problems; diffusion equation; exact solution.

PACS: 02.30 Jr, 02.00.00.

1. Introduction

Diffusion [1-3, 10-15] is an important phenomenon in physical models and one of the important equations translating such models is the following nonlinear partial differential equation:

푢푡=(퐷(푢)푢푥)푥, (1) subject to the initial condition 푢(푥, 0) = 푓(푥), (2)

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern App. Physics. 2013, 3(1): 48-56 49 where D(u) denotes the diffusion term which plays a key role in many applications of diffusion processes and appears in several functional forms including power laws and exponential forms. Moreover, slow diffusion process is represented [1-3, 10-15]: 퐷(푢) = 푢푛 푛 > 0, (3) and fast diffusion processes is described as: 퐷(푢) = 푢푛, 푛 < 0. (4) Due to its physical importance, several techniques including decomposition [15, 16], Variational Iteration [11], Homotopy Perturbation [11, 12] have been applied by various researchers to find appropriate solutions. Inspired and motivated by the ongoing research in this area, we apply Variation of Parameters Method (VPM) [4-10] to find solutions of such equations. It is worth mentioning that Ma et. al. [4-10] used Variation of Parameters (VPM) for solving involved non-homogeneous partial differential equations and obtained solution formulas helpful in constructing the existing solutions coupled with a number of other new solutions including rational solutions, solitons, positions, negatons, breathers, complextions and interaction solutions of the KdV equations. It is observed that proposed scheme is highly efficient, very reliable and accurate. Moreover, suggested algorithm is also compatible with the physical nature of such problems and hence can be extended to find appropriate solutions of wide range of problems with diversified physical nature.

2. Variation of Parameters Method (VPM) [4-10]

Consider the following second-order partial differential equation

yfttxyzxxyyzz t x( y ,,, z ,,,,,,), yyyyyy (5) where t such that  t   is time, and f is linear or non-linear function of yyyyyyy,,,,,,xyzxxyyzz .

The homogeneous solution of (1) is

y t,,,, x y z  A Bt where A and B are functions of x,, y z and t . Using Variation of parameters method we have following system of equations

AB 0, tt

B  f , t

and hence

tt A x,,,,,,,,,,,, y z tDx  y zsfdsBx y z tCx y zfds    00 therefore,

t y x,, y ,,, z ty ,0,, xy ,0( ztyx , ,, y ,,,,,), ztsf s x y z yy yyyyds     txyzxxyyzz    0 which can be solved iteratively as [4-10]

t yk1 xyzt,,, yxyz ,,,0  tyxyz ,,,0  tsfsxyzy  (,,,,,,, k y k y k y k , y k , y k ), ds     t    x y z xx yy zz 0 k  0,1,2, .

3. Numerical Examples

In this section, we apply Variation of Parameters Method (VPM) to solve partial differential equations describing various types of diffusion process. Example 3. 1 Consider the fast diffusion process as follows −1 푢푡=(푢 푢푥)푥, (6) subject to the initial condition 2푐 푢(푥, 0) = , (7) (푎+푥)2 where ‘a’ and c≠0 are arbitrary constants. Applying Variation of Parameters Method (VPM), we get 푡 푢 = 퐴 (푥, 푡) + (푢 −1 푢 ) 푑푠. (8) 푛+1 1 ∫0 푛 푛푥 푥 2푐 Here 퐴 (푥, 푡) = 푢 (푥, 푡) = . 1 0 (푎+푥)2

Consequently, following approximants are obtained 2푐 2푡 푢 (푥, 푡) = + , (9) 1 (푎+푥)2 (푎+푥)2 2푐 2푡 푢 (푥, 푡) = + , 2 (푎+푥)2 (푎+푥)2 .

. .. The closed form solution is given as 2(푐+푡) 푢(푥, 푡) = . (10) (푎+푥)2

Fig 1. Graphic depiction of solution of (5, 6)

Example 3.2 Consider a slow diffusion process

푢푡 = (푢푢푥)푥 , (11) subject to the initial condition

1 2 푢(푥, 0) = 푥 , 푥 > 0, (12) 푐 where 푐 > 0 is an arbitrary constant. Applying Variation of Parameters Method (VPM), we get 푡 푢 = 퐴 (푥, 푡) + (푢 푢 ) 푑푠, (13) 푛+1 1 ∫0 푛 푛푥 푥 1 2 Here 퐴 (푥, 푡) = 푢 (푥, 푡) = 푥 1 0 푐 푡 푢 (푥, 푡) = 푢 + (푢 푢 ) 푑푠 1 0 ∫0 0 0푥 푥

2 1 6푡 푢 (푥, 푡) = 푥 ( + ) , (14) 1 푐 푐2

Consequently, following approximants are obtained

2 3 2 1 6푡 36푡 72푡 푢 (푥, 푡) = 푥 ( + + + ) , (15) 2 푐 푐2 푐3 푐4 2 3 4 5 2 1 6푡 36푡 216푡 864푡 2592푡 푢 (푥, 푡) = 푥 ( + + + + + + ⋯ ), (16) 3 푐 푐2 푐3 푐4 푐5 푐6 and so on. In the same way rest of the components can be obtained. Therefore solution of equation (1) in the closed form is

푥2 푢(푥, 푡) = . (17) 푐−6푡

Fig. 2: Graphic depiction of solution of (11, 12)

Example 3.3 Consider another slow diffusion process in the form of

2 푢푡=(푢 푢푥)푥 , (18) subject the Initial condition

푥+ℎ 푢(푥, 0) = . (19) 2√푐

Applying Variation of Parameters Method (VPM), we get

푡 푢 = 퐴 (푥, 푡) + (푢 2 푢 ) 푑푠, (20) 푛+1 1 ∫0 푛 푛푥 푥

푥+ℎ Here 퐴 (푥, 푡) = 푢 (푥, 푡) = . 1 0 2√푐

Consequently, following approximants are obtained

푡 푢 (푥, 푡) = 푢 + (푢 2 푢 ) 푑푠 (21) 1 0 ∫0 0 0푥 푥

푥+ℎ 푡 푢 (푥, 푡) = (1 + ) , (22) 1 2√푐 2푐 푥+ℎ 푡 3푡2 푡3 푡4 푢 (푥, 푡) = (1 + + + + ) , (23) 2 2√푐 2푐 8푐2 8푐3 64푐4

푥+ℎ 푡 3푡2 5푡3 13푡4 9푡5 푢 (푥. 푡) = (1 + + + + + + ⋯ ), (24) 3 2√푐 2푐 8푐2 16푐3 64푐4 18푐5

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern App. Physics. 2013, 3(1): 48-56 53 and so on. In the same way rest of the components can be obtained. Therefore solution of equation (18) in the closed form is 1 푥+ℎ 푢(푥, 푡) = . (25) 2 √푐−푡

Fig. 3: Graphic depiction of solution of (18,19)

Example 3.4 Consider the next example where

1 푢 = ( 푢 ) , (26) 푡 1+푢2 푥 푥 subject to the initial condition

푢(푥, 0) = tan 푥. (27)

Applying Variation of Parameters Method (VPM), we get

푡 1 푢푛+1 = 퐴1(푥, 푡) + ∫ ( 2 푢푛푥)푥 푑푠, (28) 0 1+푢푛

Here 퐴1(푥, 푡) = 푢0(푥, 푡) = tan 푥,

푡 1 푢1(푥, 푡) = 푢0 + ∫ ( 2 푢0푥)푥 푑푠 0 1+푢0

푢1(푥, 푡) = tan 푥, (29)

In the same way, we find

푢2(푥, 푡) = tan 푥, (30) and so on. Therefore solution of equation (26) in the closed form is

푢(푥, 푡) = tan 푥. (31)

Fig 4: Graphic depiction of solution of (26, 27)

Example 3.5 Finally, we consider another problem of the following form

1 푢 = ( 푢 ) , (32) 푡 푢2−1 푥 푥 subject to the initial condition

푢(푥, 0) = − coth 푥. (33)

Proceeding as before, we get

푡 1 푢푛+1 = 퐴1(푥, 푡) + ∫ ( 2 푢푛푥)푥 푑푠, (34) 0 푢푛 −1

Here 퐴1(푥, 푡) = 푢0(푥, 푡) = − coth 푥

푡 1 푢1(푥, 푡) = 푢0 + ∫ ( 2 푢0푥)푥 푑푠, 0 푢0 −1

푢1(푥, 푡) = − coth 푥 , (35)

푢2(푥, 푡) = − coth 푥, (36) and so on. Hence the closed form solution is given as

푢(푥, 푡) = − coth 푥. (37)

Graph of above solution is shown in figure 5.

Fig 5: Graphic depiction of solution of (32, 33)

4. Conclusion

Variation of Parameters Method (VPM) is applied to solve nonlinear diffusion equations. The suggested algorithm which is applied without any discretization, linearization, perturbation or restrictive assumptions is very effective, reliable and accurate. Moreover, proposed scheme (VPM) can be implemented on other physical problems also.

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