Exact solutions of nonlinear diffusion equations by variational iteration method

Computers and Mathematics with Applications 54 (2007) 1112–1121 www.elsevier.com/locate/camwa

Exact solutions of nonlinear diffusion equations by variational iteration method A. Sadighi, D.D. Ganji ∗ Mazandaran University, Department of Mechanical Engineering, P.O. Box 484, Babol, Iran Received 28 September 2006; accepted 20 December 2006

Abstract In this paper, He’s variational iteration method is applied to obtain exact solutions of some nonlinear diffusion equations. The variational iteration method is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The solutions obtained are compared with those obtained by the Adomian decomposition method and homotopy perturbation method, showing excellent agreement, but the variational iteration method is more effective. He’s variational iteration method can be introduced to overcome the difficulties arising in calculating Adomian polynomials. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear diffusion equations; Variational iteration method; Lagrange multipliers; Exact solution; Adomian decomposition method

1. Introduction Most problems and scientific phenomena, such as heat transfer, fluid mechanics, plasma physics, plasma waves, thermo-elasticity and chemical physics, occur nonlinearly. Except for a limited number of these problems, we encounter difficulties in finding their exact analytical solutions. Very recently, some promising approximate analytical solutions are proposed, such as the Exp-function method [1,2], homotopy-perturbation method [3–11], and variational iteration method (VIM) [12–16]. Other methods are reviewed in Refs. [17,18]. The variational iteration method is the most effective and convenient one for both weakly and strongly nonlinear equations. This method has been shown to effectively, easily, and accurately solve a large class of nonlinear problems with components converging rapidly to accurate solutions. VIM was first proposed by He [12] and was successfully applied to various engineering problems [19–21]. In this paper, we apply VIM to obtain exact solutions of nonlinear diffusion equations in the form u t = (D(u)u x )x

(1)

subject to the initial condition: u(x, 0) = f (x) ∗ Corresponding author. Tel.: +98 111 3234205; fax: +98 111 3234205.

E-mail addresses: am [email protected] (A. Sadighi), ddg [email protected] (D.D. Ganji). c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2006.12.077

(2)

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where D(u) is the diffusion term, which plays an important role in wide range of applications in the diffusion processes. D(u) appears in several functional forms, such as power law and exponential forms [22]. In this work, we focus on some fast and slow diffusion processes which are characterized by different forms of the diffusion term. 1.1. Fast diffusion processes This process is described by a diffusion term of the form D(u) = u n ,

n < 0.

(3)

For n = −0.5, Eq. (1) models the plasma diffusion and thermal expulsion of liquid helium [23–25]. For n = −1, Eq. (1) appears in the thermal limit approximation of Carleman’s model of the Boltzman equation and the expansion into a vacuum of a thermalized electron cloud described by the isothermal Maxwellian distribution [22,26]. For n = −2, Eq. (1) is considered as a model of diffusion in high-polymeric systems [22,25]. 1.2. Slow diffusion processes This process is described by a diffusion term of the form D(u) = u n ,

n > 0.

(4)

For n = 1, Eq. (1) arises in isothermal percolation of a perfect gas through a micro-porous medium [23–25]. For n = 2, Eq. (1) is used to model a process of melting and evaporation of metals [22–24]. 1.3. Other diffusion processes Other cases of diffusion processes can be given as follows [22]: D(u) =

1 , 1 + u2

D(u) =

1 , 1 − u2

D(u) =

u2

1 . −1

(5)

2. Analysis of the variational iteration method To clarify the basic ideas of VIM, we consider the following differential equation: (6)

Lu + N u = g(t) 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 Z t u n+1 (t) = u n (t) + λ(Lu n (ξ ) + N u˜ n (ξ ) − g(ξ ))dξ

(7)

0

where λ is a general Lagrange multiplier [12,17] which can be identified optimally via the variational theory [17]. The subscript n indicates the nth approximation, and u˜ n is considered as a restricted variation [17], i.e. δ u˜ n = 0. In order to solve Eq. (1) by VIM, we construct a correction functional, as follows: Z t ˜ n )u nx )x }dτ u n+1 = u n + λ{u nτ − ( D(u (8) 0

˜ n ) is considered as a restricted variation. Its stationary conditions can be obtained as follows: where δ D(u λ0 (τ ) = 0 1 + λ(τ )|τ =t = 0.

(9) (10)

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The Lagrange multiplier can therefore be simply identified as λ = −1, and the following iteration formula can be obtained: Z t ˜ n )u nx )x }dτ. {u nτ − ( D(u (11) u n+1 = u n − 0

Beginning with an initial approximation, u 0 (x, t) = u(x, 0), components of the iteration formula can be easily found. 3. Numerical applications In order to assess the advantages and accuracy of VIM for solving nonlinear problems, we consider the following examples. 3.1. Example 1 First we consider a fast diffusion process, as follows [22]: u t = (u −1 u x )x

(12)

subject to the initial condition of 2c (a + x)2

u(x, 0) =

where a and c 6= 0 are arbitrary contacts. In order to solve Eq. (12), we construct the following iteration formula: Z t u n+1 = u n − {u nτ − (u˜ −1 n u nx )x }dτ.

(13)

(14)

0

Then, we start with an initial approximation u 0 (x, t) = u(x, 0) given by Eq. (13). Using the above variational formula (14), we can obtain the following result: Z t {u 0τ − (u˜ −1 (15) u1 = u0 − 0 u 0x )x }dτ. 0

Substituting u 0 (x, t) into Eq. (15) and after some simplifications, we have: u 1 (x, t) =

2c 2t + . 2 (a + x) (a + x)2

(16)

In the same way, we obtain u 2 (x, t) as follows: u 2 (x, t) =

2c 2t + . (a + x)2 (a + x)2

(17)

Continuing in this manner, we can obtain u n+1 (x, t) = u n (x, t) for n ≥ 1, which means that the exact solution of Eq. (12) is easily obtained in the form u(x, t) =

2(c + t) (a + x)2

(18)

which is exactly the same as that obtained by ADM [22]. The behavior of u(x, t) obtained by VIM at c = 1 and a = 1 is shown in Fig. 1. In order to compare the result obtained by VIM with that of HPM, we construct the following homotopy for Eq. (12) : (1 − p)u t + p(u t + u −2 u 2x − u −1 u x x ) = 0.

(19)

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Fig. 1. The behavior of u(x, t) obtained by VIM at c = 1 and a = 1, for Example 1.

Substituting u from u = u 0 + pu 1 + p 2 u 2 + · · · into Eq. (19) and rearranging based on powers of p-terms, we obtain:   ∂u 0 p 0 : u 20 =0 ∂t    2      ∂u 0 ∂ u0 ∂u 0 2 2 ∂u 1 p 1 : 2u 0 u 1 − u0 + + u =0 0 ∂t ∂x ∂t ∂x2       2      ∂u 1 ∂u 2 ∂ u1 ∂u 0 ∂u 1 2 ∂u 0 p 2 : 2u 0 u 1 + u 20 − u0 + u + 2 1 ∂t ∂t ∂t ∂x ∂x ∂x2   2   ∂u 0 ∂ u0 = 0. + 2u 0 u 2 − u1 ∂t ∂x2

(20)

(21) (22)

(23)

Solving Eqs. (21)–(23), we obtain: 2c (a + x)2 2t u1 = (a + x)2 u 2 = 0. u0 =

(24) (25) (26)

The solution of the Eq. (12), when p → 1, is as follows: u = u0 + u1 + u2 + · · ·

(27)

which is same as that obtained by VIM. 3.2. Example 2 We next consider a slow diffusion process [22] u t = (uu x )x

(28)

subject to the initial condition 1 2 x , x >0 c where c > 0 is an arbitrary constant. u(x, 0) =

(29)

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First, we construct the following iteration formula: Z t {u nτ − (u˜ n u nx )x }dτ. u n+1 = u n −

(30)

0

We start with an initial approximation, u 0 (x, t) = u(x, 0), given by Eq. (29). Using the above variational formula (30), we can obtain the following result: Z t {u 0τ − (u˜ 0 u 0x )x }dτ. (31) u1 = u0 − 0

Substituting u 0 (x, t) into Eq. (31) and after some simplifications, we have:   1 6t u 1 (x, t) = x 2 + 2 . c c Continuing in this manner, we can obtain the other components as follows:   1 6t 36t 2 72t 3 + 2+ 3 + 4 u 2 (x, t) = x 2 c c c c   2 1 6t 36t 216t 3 864t 4 2592t 5 + 2+ 3 + + + + · · · u 3 (x, t) = x 2 c c c c4 c5 c6   6t 36t 2 216t 3 1296t 4 33696t 5 31104t 6 2 1 u 4 (x, t) = x + 2+ 3 + + + + + ··· c c7 c c c4 c5 c6

(32)

(33) (34) (35)

and so on. In the same manner, the rest of components of the iteration formula (30) can be obtained. Therefore, the solution of u(x, t) in closed form is u(x, t) =

x2 c − 6t

(36)

which is exactly the same as that obtained by ADM [22]. The behavior of u(x, t) obtained by VIM at c = 1 is shown in Fig. 2. Let construct the following homotopy for Eq. (28): (1 − p)u t + p(u t − u 2x − uu x x ) = 0.

(37)

Substituting u from Eq. (20) into Eq. (37) and rearranging based on powers of p-terms, we obtain: p0 :

∂u 0 =0 ∂t

(38) 2

∂u 1 ∂ 2u0 ∂u 0 − u0 =0 − 2 ∂t ∂x ∂x     2   2  ∂u 0 ∂u 1 ∂ u1 ∂ u0 2 ∂u 2 p : −2 − u0 − u1 = 0. 2 ∂t ∂x ∂x ∂x ∂x2 p1 :







(39) (40)

Solving Eqs. (38)–(40), we obtain: x2 c 6x 2 t u1 = 2 c 36x 2 t 2 . u2 = c3 u0 =

(41) (42) (43)

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Fig. 2. The behavior of u(x, t) obtained by VIM at c = 1 for Example 2.

The solution of the Eq. (28) when p → 1, will be as follows: u = u0 + u1 + u2 + · · ·

(44)

which admits the solution obtained by VIM. 3.3. Example 3 Let’s consider another slow diffusion processes in the form of u t = (u 2 u x )x

(45)

subject to the initial condition u(x, 0) =

x +h √ 2 c

where h and c > 0 are arbitrary contacts. In order to solve Eq. (45), we construct the following iteration formula: Z t u n+1 = u n − {u nτ − (u 2n u nx )x }dτ.

(46)

(47)

0

We start with an initial approximation u 0 (x, t) = u(x, 0) given by Eq. (46). Using the above variational formula (47), we can obtain the following result: Z t u1 = u0 − {u 0τ − (u 20 u 0x )x }dτ. (48) 0

Substituting u 0 (x, t) into Eq. (48) and after some simplifications, we have:   x +h t u 1 (x, t) = √ 1+ . 2c 2 c

(49)

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Fig. 3. The behavior of u(x, t) obtained by VIM at c = 1 and h = 1.

Continuing in this manner, we can obtain the other components as follows:   t 3t 2 t3 t4 x +h 1+ + + 3+ u 2 (x, t) = √ 2c 8c2 8c 64c4 2 c   t x +h 3t 2 5t 3 13t 4 9t 5 1+ u 3 (x, t) = √ + 2+ + + + · · · 2c 8c 16c3 64c4 80c5 2 c

(50) (51)

and so on. In the same manner, the rest of components of the iteration formula (47) can be obtained. Therefore, the solution of u(x, t) in closed form is u(x, t) =

1 x +h √ 2 c−t

(52)

which is exactly the same as that obtained by ADM [22]. The behavior of u(x, t) obtained by VIM at c = 1 and h = 1 is shown in Fig. 3. Proceeding as before, we construct the following homotopy for Eq. (45): (1 − p)u t + p(u t − 2uu 2x − u 2 u x x ) = 0.

(53)

Substituting u from Eq. (20) into Eq. (53) and rearranging based on powers of p-terms, we obtain: p0 :

∂u 0 =0 ∂t

(54)

 ∂u 0 2 − 2u 0 =0 ∂x   2   2      ∂ u0 ∂u 0 ∂u 1 ∂u 0 2 2 ∂u 2 2 ∂ u1 p : − u0 − 2u 0 u 1 − 4u 0 − 2u 1 = 0. ∂t ∂x ∂x ∂x ∂x2 ∂x2 ∂u 1 p : − u 20 ∂t 1



∂ 2u0 ∂x2





(55) (56)

Solving Eqs. (54)–(56), we obtain: u0 =

x +h √ 2 c

(x + h) √ t 4 c3 3(x + h) 2 u2 = √ t . 16 c5 u1 =

(57) (58) (59)

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The solution of the Eq. (45), when p → 1, reads u = u0 + u1 + u2 + · · ·

(60)

which admits the solution obtained by VIM. 3.4. Example 4 As another example, we study the diffusion process where the diffusion term is defined by D(u) =

1 . 1 + u2

Substituting Eq. (61) into Eq. (1), we obtain:   1 ux . ut = 1 + u2 x

(61)

(62)

subject to the initial condition u(x, 0) = tan x. Proceeding as before, we construct the iteration formula, as follows:    Z t 1 u n+1 = u n − u nτ − u dτ. nx 1 + u˜ 2n 0 x

(63)

(64)

We start with an initial approximation u 0 (x, t) = u(x, 0) given by Eq. (63). Using the above variational formula (64), we can obtain the following result: ! ) Z t( 1 u1 = u0 − u 0τ − u 0x dτ. (65) 1 + u˜ 20 0 x Substituting u 0 (x, t) into Eq. (65) and after some simplifications, we have: u 1 (x, t) = tan x.

(66)

In the same way, we obtain u 2 (x, t) as: u 2 (x, t) = tan x.

(67)

Continuing in this manner, we can obtain u n (x, t) = tan x for n ≥ 1, which means that the solution of Eq. (62) is easily obtained in the form u(x, t) = tan x

(68)

which is exactly the same as that obtained by ADM, and can be verified through substitution [22]. 3.5. Example 5 Finally, we examine the diffusion process where the diffusion term is defined by D(u) =

1 . u2 − 1

Substituting Eq. (69) into Eq. (1), we obtain:   1 ut = ux . u2 − 1 x

(69)

(70)

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Subject to the initial condition u(x, 0) = − coth x. Proceeding as before, we construct the iteration formula as follows:    Z t 1 u nτ − u nx dτ. u n+1 = u n − u 2n − 1 0 x

(71)

(72)

We start with an initial approximation u 0 (x, t) = u(x, 0) given by Eq. (71). Using the above variational formula (72), we can obtain the following result: ! ) Z t( 1 u1 = u0 − u 0τ − u 0x dτ. (73) u 20 − 1 0 x Substituting u 0 (x, t) into Eq. (73) and after some simplifications, we have: u 1 (x, t) = − coth x.

(74)

In the same way, we obtain u 2 (x, t) as: u 2 (x, t) = − coth x

(75)

Continuing in this manner, we can obtain that u n (x, t) = − coth x for n ≥ 1, which means that the exact solution of Eq. (70) is easily obtained in the form u(x, t) = − coth x

(76)

which is exactly the same as that obtained by ADM, and can be verified through substitution [22]. 4. Conclusion In this paper, VIM has been successfully used to obtain exact solutions of nonlinear diffusion equations. It is obvious that VIM is a very powerful and efficient technique for finding analytical solutions for wide classes of nonlinear problems. It is worth pointing out that this method presents a rapid convergence for the solutions. Compared with ADM, VIM is an effective and convenient method. In conclusion, VIM provides us with highly accurate numerical solutions for nonlinear problems. It also has many merits in comparison with other methods, such as: 1. VIM does not require small parameters, which are needed by perturbation method. 2. VIM avoids linearization and physically unrealistic assumptions. 3. VIM can overcome the difficulties arising in the calculation of Adomian polynomials. References [1] J.H. He, X.-H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (3) (2006) 700–708. [2] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons and Fractals (in press). [3] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of NonLinear Mechanics 35 (1) (2000) 37–43. [4] J.H. He, New interpretation of homotopy perturbation method, International Journal of Modern Physics B 20 (18) (2006) 2561–2568. [5] J.H. He, Homotopy perturbation method: A new nonlinear analytical technique, Applied Mathematics and Computation 135 (1) (2003) 73–79. [6] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (1) (2004) 287–292. [7] J.H. He, Periodic solutions and bifurcations of delay-differential equations, Physics Letters A 347 (4–6) (2005) 228–230. [8] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals 26 (3) (2005) 695–700. [9] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 26 (3) (2005) 827–833. [10] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Science and Numerical Simulation 6 (2) (2005) 207–208. [11] J.H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (1–2) (2006) 87–88.

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