Application of Homotopy-Perturbation Method and Variational Iteration

Int. Journal of Math. Analysis, Vol. 5, 2011, no. 18, 871 - 880

Application of Homotopy-Perturbation Method and Variational Iteration Method to Three Dimensional Diffusion Problem M. Akbarzade1 Sama Technical and Vocatinal Training college, Islamic Azad University Quchan Branch, Quchan, Iran J. Langari Sama Technical and Vocatinal Training college, Islamic Azad University Quchan Branch, Quchan, Iran Abstract Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functional using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. In this article, we have used (HPM) and (VIM) to solve the diffusion equation. Keywords: diffusion; Homotopy perturbation method (HPM); Variational iteration method (VIM).

1. Introduction Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly. And even if an exact solution is 1

Corresponding author: [email protected]

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M. Akbarzade and J. Langari

obtainable, the required calculations may be too complicated to be practical, or it might be difficult to interpret the outcome. Very recently, some promising approximate analytical solutions are proposed, such as Exp-function method [4], Adomian decomposition method [1], variational iteration method [5] and homotopy-perturbation method [2, 3 and 9]. Energy balance method is reviewed in Refs. [5, 10]. HPM is the most effective and convenient one for both linear and nonlinear equations. This method does not depend on a small parameter. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter p ∈ [0,1] , which is considered as a “small parameter”. HPM has been shown to effectively, easily and accurately solve a large class of linear and nonlinear problems with components converging rapidly to accurate solutions. HPM was first proposed by He and was successfully applied to various engineering problems [8]. The aim of this work is to employ HPM and VIM to obtain the exact solutions for diffusion equations.

2. Variational iteration method To clarify the basic ideas of He's VIM, we consider the following differential equation: Lu + Nu = g (t ), (1) Where L is a linear operator, N a nonlinear operator and g(t) an inhomogeneous term. According to VIM, we can write down a correction functional as follows: t

u n +1 (t ) = u n (t ) + ∫ λ ( Lu n (ξ ) + N u%n (ξ ) − g (ξ ))d ξ 0

(2)

Where λ is a general Lagrangian multiplier [6, 7] which can be identified optimally via the variational theory. The subscript n indicates the nth approximation and u%n is considered as a restricted variation, i.e. δ u% n = 0 .

3. Fundamentals of the homotopy-perturbation method To illustrate the basic ideas of this method, we consider the following equation [11]: A(u ) − f (r ) = 0 , r ∈ Ω (3) With the boundary condition of: ⎛ ∂u ⎞ B⎜ u , ⎟ = 0 , r ∈ Γ , (4) ⎝ ∂n ⎠

Application of homotopy-perturbation method

873

Where A is a general differential operator, B a boundary operator, f (r) a known analytical function and Γ is the boundary of the domain Ω. A can be divided into two parts which are L and N, where L is linear and N is nonlinear. Eq. (3) can therefore be rewritten as follows: L(u ) + N (u ) − f (r ) = 0 , r ∈ Ω , (5) Homotopy perturbation structure is shown as follows: H (ν , p ) = (1 − p )[L(ν ) − L(u 0 )] + p[A(ν ) − f (r )] = 0 (6) Where: ν (r , p ) : Ω × [0,1] → R (7) In Eq. (6), p ∈ [0,1] is an embedding parameter and u 0 is the first approximation that satisfies the boundary condition. We can assume that the solution of Eq. (6) can be written as a power series in p, as following: ν = ν 0 + pν 1 + p 2ν 2 + p 3ν 3 + ... (8)

And the best approximation for solution is: u = lim p→1 ν = ν 0 +ν 1 + ν 2 + ν 3 + ...

(9)

4. The application of VIM and HPM in diffusion equations In order to assess the accuracy of VIM and HPM for solving diffusion equations, we will consider the four following examples. Consider the following general form of unsteady three dimension diffusion equation with constant coefficient, with the indicated boundary conditions: ∂u ∂ 2u ∂ 2u ∂ 2u = + + ∂t ∂x 2 ∂y 2 ∂z 2 And: u ( x, y, z, 0) = f ( x, y, z ), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

(9a)

(9b)

4.1. Example 1

Consider the following equation with the indicated boundary conditions: ∂u ∂ 2u ∂ 2u = + ∂t ∂x 2 ∂y 2 With the boundary condition of:

u (x , y , 0) = (1 − y )e x

(10a)

(10b)

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4.1.1. Variational iteration method

In order to solve Eq. ((10a) and (10b)) using VIM, we construct a correction functional, as follows: t

u n +1 = u n + ∫ λ ( 0

∂u n ∂ 2u n ∂ 2u n )d τ − − ∂τ ∂x 2 ∂y 2

(11)

Where u n = u n (x , y , t ). The Lagrangian multiplier can be identified as λ = -1. As a result, we obtain the following iteration formula: t

un +1 = un + ∫ −( 0

∂un ∂ 2un ∂ 2un − 2 − 2 )dτ ∂τ ∂x ∂y

(12)

Where un = un ( x, y, t ). Now we start with an arbitrary boundary approximation that satisfies the boundary condition: u0 ( x, y, t ) = (1 − y )e x , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

(13)

Using the above variational formula (12), we have ∂u0 ∂ 2u0 ∂ 2u0 − 2 − 2 )dτ u1 = u0 − ∫ ( 0 ∂τ ∂x ∂y t

(14)

Substituting Eq.(13) into Eq. (14)and after simplifications, we have u1 (x , y , t ) = (1 − y + t − ty )e x

(15)

In the same way, we obtain u 2 (x , y , t ), u 3 (x , y , t ), u 4 ( x , y , t ) as follows: 1 1 1 1 1 1 u2 = (1 − y + t − ty + t 2 − t 2 y )e x , u3 = (1 − y + t − ty + t 2 − t 2 y + t 3 − t 3 y )e x , 2 2 2 2 6 6 (16) 1 2 1 2 1 3 1 3 1 4 1 4 x u4 = (1 − y + t − ty + t − t y + t − t y + t − t y )e 2 2 6 6 24 24 And so on. In the same manner the rest of the components of the iteration formula can be obtained: 1 1 1 u n (x , y , t ) = (1 − y )(1 + t + t 2 + t 3 + t 4 + ...)e x (17) 2 6 24

We know that (1+ t + 1 t 2 + 1 t 3 + 1 t 4 + ...) is the Taylor series of e t : 2

u (x , y , t ) = (1 − y )e

6

24

x +t

4.1.2. Homotopy-perturbation method

We apply homotopy-perturbation to Eq. ((10a) and (10b)), as follows:

(18)


875

H (v , p ) = (1 − p ) [ L (v ) − L (u 0 )] + p [ A (v ) − f (r )] = 0,

(19)

Where L(v) is the linear part of the equation and L(u0) the boundary approximation. We consider ν as v = v 0 + pv 1 + pv 2 + K (20) Substituting Eq. (20) into Eq. (19) and rearranging based on powers of p-terms, we have the coefficient of p 0 : ∂u0 ( x, y , t ) = 0 ∂t With boundary condition: u 0 (x , y , 0) = (1 − y )e x

(21)

(22)

And solution for u 0 u 0 (x , y , t ) = (1 − y )e x

(23)

The coefficient of p1 : ∂ u1 ( x, y, t ) − (1 − y )e x = 0, u1 ( x, y, 0) = 0 ∂t And solution for u1 : u1 (x , y , t ) = t (1 − y )e x

(24)

(25)

The coefficient of p 2 : ∂ u2 ( x, y, t ) − t (1 − y )e x = 0, u2 ( x, y, 0) = 0 ∂t And solution for u2 : 1 u 2 (x , y , t ) = t 2 (1 − y )e x 2 Similary with the coefficient of p 3 we can obtain: 1 u 3 (x , y , t ) = t 3 (1 − y )e x 6 And with the coefficient of p 4 we can obtain: 1 4 t (1 − y )e x 24 The final solution: u 4 (x , y , t ) =

(26)

(27)

(28)

(29)

1 1 1 u n (x , y , t ) = (1 − y )e x + t (1 − y )e x + t 2 (1 − y )e x + t 3 (1 − y )e x + t 4 (1 − y )e x + ... (30) 2 6 24 Or:

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1 1 1 u n (x , y , t ) = (1 − y )e x (1 + t + t 2 + t 3 + t 4 + ...) 2 6 24

(31)

We know that (1+ t + 1 t 2 + 1 t 3 + 1 t 4 + ...) is the series Taylor of e t : 2

u (x , y , t ) = (1 − y )e

6

24

( x +t )

(32)

4.2 Example 2

Let us solve the following partial differential equation: ∂u ∂ 2u ∂ 2u = + ∂t ∂x 2 ∂y 2

(33)

With the boundary condition of: u (x , y , 0) = e x + y . 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

(34)

4.2.1. Variational iteration method

First we construct a correction functional which reads: t

u n +1 = u n + ∫ λ ( 0

∂u n ∂ 2u n ∂ 2u n − − )d τ , ∂τ ∂x 2 ∂y 2

(35)

The Lagrangian multiplier can be identified as λ = -1. Now we start with an arbitrary boundary approximation as follows: u 0 (x , y , 0) = e x + y . 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

(36)

By the above variational formula (35), we can obtain the following result: ∂u 0 ∂ 2u 0 ∂ 2u 0 )d τ , − − 0 ∂τ ∂x 2 ∂y 2 Substituting Eq. (36) into Eq. (37) and after some simplifications, we have: t

u1 = u 0 − ∫ (

(37)

u1 (x , y , t ) = (1 + 2t )e x + y :

(38)

In the same way, we obtain u 2 (x , y , t ), u 3 (x , y , t ), u 4 ( x , y , t ) as 4 u2 ( x, y, t ) = (1 + 2t + 2t 2 )e x + y , u3 ( x, y, t ) = (1 + 2t + 2t 2 + t 3 )e x + y , 3 4 2 u4 ( x, y, t ) = (1 + 2t + 2t 2 + t 3 + t 4 )e x + y 3 3 And: 4 2 u n (x , y , t ) = (1 + 2t + 2t 2 + t 3 + t 4 + ...)e x + y 3 3

(39)

(40)


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We know that (1 + 2t + 2t 2 + 4 t 3 + 2 t 4 + ...) is the Taylor series of e 2t : 3

u (x , y , t ) = e

3

2t + x + y

(41)


We apply homotopy-perturbation to Eq. ((10a) and (10b)), as follows:

H (v , p ) = (1 − p ) [ L (v ) − L (u 0 )] + p [ A (v ) − f (r )] = 0,

(42)

Where L(v) is the linear part of the equation and L(u0) the boundary approximation. We consider ν as: v = v 0 + pv 1 + pv 2 + K (43) Substituting Eq. (43) into Eq. (42) and rearranging based on powers of pterms, we have the coefficient of p 0 : ∂u 0 (x , y , t ) = 0, ∂t With boundary condition:

(44)

u 0 (x , y , 0) = e x + y

(45)

And solution for u0 : u 0 (x , y , t ) = e x + y

(46)

In the same way, we obtain u1 ( x, y, t ), u2 ( x, y, t ), u3 ( x, y, t ), u4 ( x, y, t ) as: 4 2 u1 ( x, y, t ) = 2te x + y , u2 ( x, y, t ) = 2t 2 e x + y , u3 ( x, y, t ) = t 3e x + y , u4 ( x, y, t ) = t 4 e x + y (47) 3 3 The final solution: 4 2 u n (x , y , t ) = e x + y + 2te x + y + 2t 2e x + y + t 3e x + y + t 4e x + y + ... (48) 3 3 Or: 4 2 u n (x , y , t ) = e x + y (1 + 2t + 2t 2 + t 3 + t 4 + ...) (49) 3 3 We know that (1 + 2t + 2t 2 + 4 t 3 + 2 t 4 + ...) is the Taylor series of e 2t : 3

u (x , y , t ) = e

(2t + x + y )

3

(50)

4.3 Example 3

Consider the following partial differential equation, with specified boundary conditions:

878


∂u ∂ 2u ∂ 2u ∂ 2u = + + ∂t ∂x 2 ∂y 2 ∂z 2 With the boundary condition of:

(51)

u (x , y , z , 0) = (1 − y )e x + z . 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

(52)

4.3.1 Variational iteration method

First we construct a correction functional which reads: ∂u n ∂ 2u n ∂ 2u n ∂ 2u n )d τ , u n +1 = u n + ∫ λ ( − − − 0 ∂τ ∂x 2 ∂y 2 ∂z 2 t

(53)

The Lagrangian multiplier can be identified as λ =-1. Similary, we can obtain u1 ( x, y, t ), u2 ( x, y, t ), u3 ( x, y, t ) as follows:

u1 ( x , y , z , t ) = (1 − y + 2 t − 2 ty ) e x + z , u 2 ( x , y , z , t ) = (1 − y + 2 t − 2 ty + 2 t 2 − 2 t 2 y ) e x + z u 3 ( x , y , z , t ) = (1 − y + 2 t − 2 ty + 2 t 2 − 2 t 2 y +

(54)

4 3 4 3 t − t y )e x + z 3 3

And:

u n ( x , y , z , t ) = (1 − y )(1 + 2t + 2t 2 +

4 3 t + ...)e x + z 3

(55)

4 3

We know that (1 + 2t + 2t 2 + t 3 + ...) is the Taylor series of e 2t :

u ( x , y , z , t ) = (1 − y ) e ( 2 t + x + z )

(56)


After applying HPM and rearranging based on powers of p-terms, we apply homotopy-perturbation to Eq. ((10a) and (10b)), as follows:

H (v , p ) = (1 − p ) [ L (v ) − L (u 0 )] + p [ A (v ) − f (r )] = 0,

(57)

Where L(v) is the linear part of the equation and L(u0) the boundary approximation. We consider ν as: v = v 0 + pv 1 + pv 2 + K (58) Substituting Eq. (58) into Eq. (57) and rearranging based on powers of p-terms, we have the coefficient of p 0 : ∂u 0 = 0, u 0 (x , y , z , 0) = (1 − y )e x + z (59) ∂t Similary, we can obtain u0 ( x, y, t ), u1 ( x, y, t ), u2 ( x, y, t ), u3 ( x, y, t ) as follows:


879

u0 ( x, y, z, t ) = (1 − y )e x + z , u1 ( x, y, z , t ) = 2t (1 − y )e( x + z ) , u2 ( x, y, z , t ) = 2t 2 (1 − y )e( x + z ) , (60)

2 u4 ( x, y, z , t ) = t 4 (1 − y )e( x + z ) 3 The final solution: 4 2 u n ( x , y , z , t ) = (1 − y )e x + z (1 + 2t + 2t 2 + t 3 + t 4 + ...) 3 3 4 We know that (1 + 2t + 2t 2 + t 3 + ...) is the Taylor series of e 2t : 3

u (x , y , z , t ) = (1 − y )e ( x + z + 2t )

(61)

(62)

5. Conclusion Comparing the two methods of HPM and VIM shows that they have got the same answers. The homotopy perturbation method does not need a small parameter. The VIM is to construct correction functional using general Lagrange multipliers identified optimally via the variational theory, and the boundary approximations can be freely chosen with unknown constants. An interesting point about VIM and HPM is that with the fewest number of iterations or even in some cases, once, it can converge to correct results.

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[6]

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