Solution by Homotopy Perturbation Method of Linear and Nonlinear Diffusion Equation

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 4, April 2013)

169

Solution by Homotopy Perturbation Method of Linear and

Nonlinear Diffusion Equation

Khyati R. Desai

1

, V. H. Pradhan

2

Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat-395007, India

Abstract--In this paper exact solution of linear and nonlinear diffusion equations are obtained by Homotopy Perturbation Method. He’s Homotopy Perturbation Method is powerful and capable method to solve linear and nonlinear equation directly.

Keywords-- Homotopy Perturbation Method, Diffusion Equation.

I. INTRODUCTION

Most of the scientific problems in engineering are nonlinear. Except in a limited number of these problems, finding the exact analytical solutions of such problems are quite difficult. Therefore, there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Homotopy perturbation method is such method which is straightforward and convenient for both linear and non linear equations. It is also applicable to both partial differential equation and ordinary differential equation. The homotopy perturbation method is proposed by He in 1998 and was developed and improved by him. Homotopy perturbation method is the combination of traditional perturbation method and homotopy method so it takes full advantages of both methods.

The diffusion equation is a partial differential equation which describes density fluctuations in a material undergoing diffusion. The diffusion equation of the form

 





0

t x _x

u



D u u

 

x

L

is studied. The term

 

D u

is the diffusion term that plays a crucial role in a wide range of applications in diffusion process[1,9]. The diffusion term

D u

 

appears in several forms such as power law and exponential forms. Some of the well known diffusion processes are the fast and slow diffusion processes, where the diffusion term is of the form

 

n

D u



u

where

n



0

and

n



0

, respectively. II. HOMOTOPY PERTURBATION METHOD

To explain this method let us consider the following function:

 

0 A u



f r



r



(1) With boundary conditions of

,

u

0 B u

t





_{ }



_







(2)

Where A, B, f (r) and



are a general differential

operator, a boundary operator, a known analytical function and the boundary of the domain



respectively. Generally speaking, the operator A can be divided into a linear part L(u) and non linear part N(u). So equation (1) may written as

( )

0 L u



N u



f r



(3) By Homotopy technique [3,4], we construct a Homotopy

( , ) :

[0,1]

v r p





R

which satisfies: 0

( , )

(1

)[ ( )

( )]

[ ( )

( )]

0 H v p

p L v

L u

p A v

f r

 









(4)

[0,1],

p



r



Or

0 0

( , )

( )

[ ( )

( )]

0 H v p

L v

L u

pL u

p N v

f r











(5)

Where

p



[0,1]

is an embedding parameter, while

u

₀

is initial approximation of equation (1) which satisfies the boundary conditions from equations (4) and (5) we will have,

0

( ,0)

( )

0 H v



L v



L u



(6)

( ,1)

( )

0 H v



A v



f r



(7)

The changing process of p from zero to unity is just that

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According to Homotopy perturbation method, we can first use the embedding parameter p as a “small parameter” and assume that the solution of equation (4) and (5) can be written as a power series in p:

2

0 1 2

...

v

 

v

pv



p v



(8)

Setting p=1 yields in the approximate solution of equation (8) to:

0 1 2 1

lim

...

p

u

v





   

(9) Equation (9) is the solution of equation (1) obtained by

Homotopy perturbation method.

1 Linear Homogeneous Diffusion Equation

Consider Linear Homogeneous Diffusion Equation

0

t xx

u



u



u

 

x



t



(1.1) Boundary conditions are given by

(0, )

0 ( , )

0 u

t



u



t



(1.2) Initial Condition is given by

( , 0)

u x



Sin x

(1.3)

This is a Homogeneous Equation represent the Heat equation which is solved by Homotopy Perturbation Method as below.

Homotopy

H v p

( , ) :



[0,1]



R

for equation (1.1) is defined as [3]

0

2 2

( , )

(1

)

0 v

v

H v p

p

t

v

p

v

t

x







 

_



_















_



_









(1.4)

Suppose the solution of above homotopy is power series in

p



[0,1]

therefore equation (1.4) can be written as,

2 3

0 1 2 3

4 4 0

2 3

0 1 2 3

2 2 2 2

2 3

0 1 2 3

2 2 2 2

2 3

0 1 2 3

(1 )

...

... 0 ...

v v v v

p p p

t t t t

p

v v

p

t t

v v v v

p p p

t t t t

v v v v

p p p p

x x x x

v pv p v p v

   

 _ _ _ 

 _ _ _ _ 

  

 

_ _{ } 

 _ _ 

 

   

 _ _ _ _ 

 _ _ _ _ 

 

   

 

 _     _

   

 

    

 

 

(1.5)

Comparing powers of from both sides, we get,

0 0 0

:

v

0 p

t









(1.6)

1 1 0 0 0

2 0

0 2

:

0 v

v

p

t

v

x



_

_





 



(1.7)

2

2 2 1 1 1

1 2

:

v

0 p

v

t

x









 



(1.8)

2

3 3 2 2 2

2 2

:

v

0 p

v

t

x



_



_



_



_{ }



(1.9)

. . .

Solving all above partial differential equations we get, 0

v



Sin x

(1.10) 1

2 v

 

t Sin x

(1.11)

 

2 2

2

2 t

(3)

International Journal of Emerging Technology and Advanced Engineering

171  

3

2

6 t

v

 

Sin x

(1.13) .

. .

Solution of equation (1.1) can be written as, 0 1 2

...

u

   

v

2 3

2 (2 )

(2 )

...

2

6 u

Sin x

t Sin x

t

Sin x

 









2 3

(2 )

1 2

...

2!

3!

t

u

Sin x



t



 

_

 





_





2t

u

e



Sin x

 

(1.14)

Equation (1.14) is the exact solution of the model given by (1.1) [1].

2 Inhomogeneous Diffusion equation

Consider inhomogeneous diffusion equation

t xx

u



u



cos x

(2.1) With boundary conditions

 

(0, ) 1

t

,

1

t

u

t

 

e and u





t

  

e

 (2.2) And initial condition

u x

( , 0)



0

Homotopy

H v p

( , ) :



[0,1]



R

for inhomogeneous equation (2.1) is defined as [4]

0

2 2

( , )

(1

)

cos

0 v

v

H v p

p

t

v

p

x

t

x







 

_



_















_



_









(2.3)

p



[0,1]

2 3

0 1 2 3

4 4 0

2 3

0 1 2 3

2 2 2 2

2 3

0 1 2 3

2 2 2 2

(1 )

...

... 0 cos

v v v v

p p p

t t t t

p

v v

p

t t

v v v v

p p p

t t t t

v v v v

p p p p

x x x x

x

   

 _ _ _ 

 _ _ _ _ 

  

 

_ _{ } 

 _ _ 

 

   

 _ _ _ _ 

 _ _ _ _ 

 

   

 

 _     _

   

 



 

 

(2.4)

Comparing powers of from both sides, we get,

0 0 0

:

v

0 p

t









(2.5)

1 1 0 0

2

0 0

2

:

cos

0 v

v

p

t

v

x

t

x



_

_











(2.6)

2

2 2 1 1 1

2

:

v

0 p

t

x













(2.7)

2

3 3 2 2 2

2

:

v

0 p

t

x



_



_



_



_



(2.8)

. . .

0 v



(2.9) 1

cos

v



t

x

(2.10) 2

2

cos

2 t

v

 

x

(2.11) 3

3

cos

3 t

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International Journal of Emerging Technology and Advanced Engineering

172

. . .

Solution of equation (2.1) can be written as, 0 1 2

...

u

   

v

2 3

0 cos

cos

....

2 3!

t

u

 

t

x



x



x



2 3

cos (

...)

2!

3!

t

u



x t



 

cos

(1

t

)

u



x



e

 (2.13)

Equation (2.13) is the exact solution of (2.1) [1]. 3 Fast and slow diffusion processes

3.1 Fast Diffusion Process

Consider nonlinear diffusion equation



2



t x _x

u



u u

 (3.1) With boundary conditions

2

1

1 (0, )

(1, )

1

t _t

u

t

and u

t

e

_e





(3.2)

And initial condition is





1 2 2

( , 0) 1

u x  x  (3.3)

Homotopy

H v p

( , ) :



[0,1]



R

for equation (3.1) is defined as

0

2

2 1 2

2

( , ) (1 )

2 ( )_x 0

v v H v p p

t t

v v

p v v v

t x         _  _         _  _     (3.4)

p



[0,1]

2

0 1 2

3 3 4 4 0

2 3

0 1 2 3

2 2

0 1 2

2 2 2 2

2 3

0 1 2 3

2 2 2 2

2 3

0 1 2

2 0 1 (1 ) ... ... ( ...) ... 2( ...)

v v v

p p

t t t

p

v v v

p p

t t t

v v v v

p p p

t t t t

v pv p v

v v v v

p p p p

x x x x

v pv p v

v v p p x x       _ _   _ _ _        _ _ _{ }   _ _ _     _  _  _  _               _     _            _  _    2 3 3 2 0 ... v v p x x                      _ _ _  _   _  _ _ _ _    (3.5)

Comparing powers of „p‟ both sides we get,

0 0 0

:

v

0 p

t



_



_



(3.6)

1 1 0 0 0

2 2

2 0 3 0

0 2 0

:

2

0 v

v

p

t

v

x

 



















_

_





_



_

(3.7)

2 2 1 1

2 2

3 0 2 1

1 0 2 0 2 2

4 0 3 0 1

0 1 0

:

2

6

4

0 v

v

p

t

v

v v

v

x

v

v v

v

x

   



_



_









 







_

_







 





(3.8) 2

3 3 2 2 3 0

0 2 2

2 2

4 2 0 3 1

0 1 2 1 0 2 2 2

2 2 4 0

0 2 0 2

2

2 5 0 4 0 1

1 0 1 0

2

3 1 3 0 2

0 0

:

2

3

2

6

12

2

4

0 v

v

p

v v

t

x

v

v v

x

v

v v

x

v

v v

x

v

x

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. . .

2 ₂ 0

(1

)

v

 

x

 (3.10)

2 3 2 ₂

1

2 2

5 3

2 2 2 2

2 (1

)

3

1 (1

)

(1

)

(1

)

x

v

t

x



_









_









_

_

_

_







_{ }

_









_

_





_

_

_







(3.11)

2 2 2

2 5

2 2

2 (1

)

t

t x

v

x







(3.12)

3 2 4

3 7

2 2

(1 10

4 )

6(1

)

t

x

v

x





 



(3.13)

. . .

Therefore the solution of the fast diffusion equation (3.1) is given by,

0 1 2 3

...

u

    

v

(3.14)





1 2 ₂

3 1

2 2 ₂ 2 ₂

5 2

2 2 ₂

7 3

2 2 2 4

(1

)

(1

)

(1

)

1 2

(1

)

2!

(1

) (1 10

4 ) ....

3!

v

x

t x

x

t

x

t

x



 



 







_



 

_



















(3.15)

Equation (3.15) is the Homotopy solution of equation (3.1), the infinite series converges to exact solution of the fast diffusion equation given by (3.1) [1].

3.2 Slow Diffusion Equation

The nonlinear slow diffusion equation of the form[2] 2

( ) , 0 1, 0

t x x

u  u u  x t (3.16) With boundary conditions

 

₂

 

1

₂

0,

,

1,

2

2 a

a

u

t

u

t

c

t

c

t







(3.17)

And with initial condition

 

, 0

2 x

a

u x

c





(3.18) Governing slow diffusion processes, where a and c are

arbitrary constants. This equation models a slow diffusion process such as evaporation and melting [5]. The exact solution of equation (3.16) is [1,2],

 

2

,

2 x

a

u x t

t

c

t









(3.19)

Equation (3.16) can be written as

 

2 ₂

2

t x xx

u



u u



u u

(3.20) By Homotopy technique, define a Homotopy

 

,

:

[0,1]

V r p





R

for the equation (3.20) as follows:

 

0

2 ₂

2 2

,

(1

)

2

0 u

v

H v p

p

t

v

p

v

t

x







 

_



_









_

_

_

_

_









_

_

























(3.21)

Assuming that the solution of equation (3.21) is given by,

2 3

0 1 2 3

...

(6)

International Journal of Emerging Technology and Advanced Engineering

174 



2

0 1 2

3 3 0

2 3

0 1 2 3

2 3

0 1 2 3

2

2 3

0 1 2 3

2 3 2

0 1 2 3

2 2 2

2 3

0 1 2

2 2 2

1

...

2( ...)

...

( ...)

v v v

p p

t t t

p

v v

p

t t

v v v v

p p p

t t t t

v pv p v p v

v v v v

p p p p

x x x x

v pv p v p v

v v v

p p p

x x x

    _ _  _ _ _       _ _{ }   _ _     _  _  _                  _     _             _  _  _     2 3 2 0 .. v x                            _      (3.23)

Comparing the powers of p both sides we get,

0 0 0

:

v

0 p

t



_



_



(3.24)

1 1 0 0 0

2 ₂

2

0 0

0 0 2

:

2

0 v

v

p

t

v

x



_

_









_

_











(3.25)

2 2 1 1 0 1

0

2 2 2

2

0 0 1

1 0 1 2 0 2

:

4

2

0 v

v

p

v

t

x

v

v v

v

x



_



_



_



 









_

_











(3.26) 2

3 3 2 2 0

2 2 0 1 1 0 1 2 2 2 2

0 2 2 0

0 0 2 1 2

:

2

4

0 v

v

p

v

t

x

v

x

v

x



_



_



_





































_

_



_

_

_

_





















 



(3.27) . . .

Solving all the above partial differential equations we get, 0

2 x

a

v

c





(3.28)





1

₄

3

x a t

v

c





(3.29)





2

2 5

3

16 x

a t

v

c





(3.30)





3

3 7

15

96 x

a t

v

c





(3.31) . . .

So, the solution of equation (3.16) is written as, 0 1 2 3

...

u

    

v

(3.32)













3 2 3 5 7

2

4

3

15 ...

16

96 x

a t

x

a

u

c

x

a t

x

a t

c







(3.33)





3

2 3

5 7

1

2

3

15 ..

8

32 t

x

a

_c

u

t

c



_

















_











(3.34)





2

2 x

a

u

c

t







(3.35) The equation (3.35) is the solution of equation (3.16)

obtained by homotopy perturbation method which is same as the exact solution given by equation (3.19).

III. CONCLUSION

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International Journal of Emerging Technology and Advanced Engineering

175

Homotopy Perturbation Method is successful method to solve linear-nonlinear problems and gives quickly convergent approximations that lead to exact solution. Homotopy perturbation method solved nonlinear problems directly without linearizing the problem.

REFERENCES

[1] Abdul-Majid Wazwaz, 2007 “The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations” computers and mathematics with applications 54 933-939.

[2] Gurhan Gurarslan and Murat Sari 2009 “Numerical solutions of linear and nonlinear diffusion equation by a differential quadrature method” Int. J. Nume. Meth. Biomed. Engg. Vol. 27 Pp-69-77. [3] J.H. He, 2003 “ Homotopy perturbation method: A new nonlinear

analytical technique”, Applied Mathematics and Computation, vol. 135, pp. 73-79.

[4] J.H. He, 2006 “some asymptotic methods for strongly nonlinear equations”, International journal of modern physics B 20 (10) pp 1141-1199.