Numerical simulations of systems of PDEs by variational iteration method

Physics Letters A 372 (2008) 822–829 www.elsevier.com/locate/pla

Numerical simulations of systems of PDEs by variational iteration method B. Batiha a , M.S.M. Noorani a , I. Hashim a,∗ , K. Batiha b a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia b Business Networking and System Management, Philadelphia University, Jordan

Received 17 May 2007; received in revised form 6 August 2007; accepted 9 August 2007 Available online 29 August 2007 Communicated by A.R. Bishop

Abstract In this Letter, a general framework of the variational iteration method (VIM) is presented for solving systems of linear and nonlinear partial differential equations (PDEs). In VIM, a correction functional is constructed by a general Lagrange’s multiplier which can be identified via a variational theory. VIM yields an approximate solution in the form of a rapid convergent series. Comparison with the exact solutions shows that VIM is a powerful method for the solution of linear and nonlinear systems of PDEs. © 2007 Elsevier B.V. All rights reserved. Keywords: Systems of PDEs; Variational iteration method; Lagrange multiplier

1. Introduction It is well known that many phenomena in scientific fields such as solid state physics, plasma physics, fluid dynamics, mathematical biology and chemical kinetics, can be modelled by systems of linear or nonlinear PDEs. The nonlinear models of real-life problems are still difficult to solve either numerically or theoretically. A broad class of analytical solutions methods and numerical solutions methods were used to handle these problems [1–3]. The general ideas and the essential features of these systems are of wide applicability. The existing techniques encountered difficulties in terms of the size of computational work needed, especially when the system involves several PDEs. Authors in [4] and [5] used the characteristics method and the Riemann invariants method to handle systems of PDEs. Vandewalle and Piessens [6] implemented a method based on a combination of the waveform relaxation method and multigrid to solve nonlinear systems. Wazwaz [7] used the Adomian decomposition method (ADM) to handle some systems of PDEs and reaction– diffusion Brusselator model. Gu and Li [8] introduced the modified Adomian method to solve some systems of nonlinear differential equations (see also [9,10]). In recent years, a great deal of attention has been devoted to study VIM, which was first envisioned by Professor Ji-Huan He for solving a wide range of problems whose mathematical models yield differential equation or system of differential equations [11–16]. VIM has successfully been applied to many situations. For example, Soliman and Abdou [17] used VIM to solve three kinds of nonlinear PDEs, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d– Sokolov–Wilson equations. Sweilam et al. [18] applied VIM to solve the nonlinear system of Cauchy problem arising in one dimensional nonlinear thermoelasticity. Ganji et al. [19] used VIM to solve nonlinear Jaulent–Miodek, coupled KdV and coupled MKdV equations. Sweilam [20] applied VIM to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. Bildik et al. [21] used VIM to solve different types of nonlinear PDEs. Wazwaz [22] used VIM to determine rational solutions for the KdV, K(2, 2), Burgers’ and cubic Boussinesq equations. Abdou and Soliman [23] used VIM to obtain the solution * Corresponding author.

E-mail address: [email protected] (I. Hashim). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.08.032

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of Burger’s equation and coupled Burger’s equations. Batiha et al. [24] applied VIM to solve the generalized Huxley equation and in [25] they employed VIM to the generalized Burgers–Huxley equation. Tari et al. [26] applied VIM to solve nonlinear equations arising in heat transfer, in this Letter the authors developed the correction functional using the extra parameters such as cn . Tatari et al. [27] used VIM for finding the solution of a semi-linear inverse parabolic equation. Yusufoglu [28] employed VIM for construction of some compact and noncompact structures of Klein–Gordon equations. Very recently, Wazwaz [29] applied VIM to some linear and nonlinear systems of PDEs. The purpose of this Letter is to present a general framework of VIM for linear and nonlinear systems of PDEs. Several examples are given to illustrate the effectiveness of the present method. It is shown in particular that exact solutions to nonlinear systems of PDEs are possible and in so doing we extend the analysis of Wazwaz [29]. 2. VIM for systems of PDEs VIM is based on the general Lagrange’s multiplier method [30]. The main feature of the method is that the solution of a mathematical problem with linearization assumption is used as initial approximation or trial function. Then a more highly precise approximation at some special point can be obtained. This approximation converges rapidly to an accurate solution [16]. Consider the general system of PDEs, L1 (u1 , u2 , . . . , um ) + N1 (u1 , u2 , . . . , um ) = f1 ,

(1)

L2 (u1 , u2 , . . . , um ) + N2 (u1 , u2 , . . . , um ) = f2 , .. .

(2)

Lm (u1 , u2 , . . . , um ) + Nm (u1 , u2 , . . . , um ) = fm ,

(3)

where L1 , L2 , . . . , Lm are linear operators, N1 , N2 , . . . , Nm are nonlinear operators, um = um (x, y, t) and fm = fm (x, y, t). According to VIM [13–16], we can construct a correction functional as follows: t u1,n+1 = u1,n +

  λ1 L1 (u1,n , u2,n , . . . , um,n ) + N1 (u˜ 1,n , u˜ 2,n , . . . , u˜ m,n ) − f1 dτ,

(4)

  λ2 L2 (u1,n , u2,n , . . . , um,n ) + N2 (u˜ 1,n , u˜ 2,n , . . . , u˜ m,n ) − f2 dτ,

(5)

0

t u2,n+1 = u2,n + 0

.. . t um,n+1 = um,n +

  λm Lm (u1,n , u2,n , . . . , um,n ) + Nm (u˜ 1,n , u˜ 2,n , . . . , u˜ m,n ) − fm dτ,

(6)

0

where λ1 , λ2 , . . . , λm are a general Lagrangian multipliers [30] which can be identified optimally via the variational theory by using the stationary conditions, the subscript n denotes the nth-order approximation, u˜ n is considered as a restricted variation [13,14], i.e. δ u˜ n = 0. 3. Applications To illustrate the effectiveness of the present method, several test examples are considered in this section. The accuracy of the method is assessed by comparison with the exact solutions. 3.1. Linear systems 3.1.1. Example 1 Consider the 2 × 2 linear system of (1)–(3), letting u1 = u and u2 = v, ut + vx = 0,

(7)

vt + ux = 0,

(8)

with the following initial conditions: u(x, 0) = ex ,

v(x, 0) = e−x .

(9)

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In operator form (7)–(8) can be written as: L 1 = ut + v x ,

N1 = 0,

f1 = 0,

(10)

L2 = vt + ux ,

N2 = 0,

f2 = 0.

(11)

The exact solution is [7]: u(x, t) = ex cosh t + e−x sinh t,

v(x, t) = e−x cosh t − ex sinh t.

(12)

To solve Eqs. (7)–(8) by means of VIM, we construct a correction functional, t un+1 (x, t) = un (x, t) +

  λ1 (un )s + (v˜n )x ds,

(13)

  λ2 (vn )s + (u˜ n )x ds,

(14)

0

t vn+1 (x, t) = vn (x, t) + 0

where u˜ n , v˜n are considered as restricted variations, which means u˜ n = 0 and v˜n = 0. To find the optimal λ1 and λ2 , we proceed as follows: t δun+1 (x, t) = δun (x, t) + δ

  λ1 (un )s + (v˜n )x ds,

(15)

λ1 (un )s ds,

(16)

0

and consequently t δun+1 (x, t) = δun (x, t) + δ 0

which results t δun+1 (x, t) = δun (x, t) + λ1 δun (x, s) −

δun (x, s)λ1 ds.

(17)

0

Similarly t δvn+1 (x, t) = δvn (x, t) + δ

  λ2 (vn )s + (u˜ n )x ds,

(18)

λ2 (vn )s ds,

(19)

0

and consequently t δvn+1 (x, t) = δvn (x, t) + δ 0

which results t δvn+1 (x, t) = δvn (x, t) + λ2 δvn (x, s) −

δvn (x, s)λ2 ds.

(20)

0

The stationary conditions can be obtained as follows: λ1 (s) = 0,

1 + λ1 (s)|s=t = 0,

(21)

λ2 (s) = 0,

1 + λ2 (s)|s=t = 0.

(22)

The Lagrange multipliers, therefore, can be identified as λ1 (s) = λ2 (s) = −1,

(23)

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Fig. 1. (a) The exact solution for u(x, t), (b) 2-iterate VIM for u(x, t), (c) the exact solution for v(x, t), (d) 2-iterate VIM for v(x, t).

and the iteration formulas are given as t un+1 (x, t) = un (x, t) − 0 t



 (un )s + (vn )x ds,

(24)

 (vn )s + (un )x ds.

(25)



vn+1 (x, t) = vn (x, t) − 0

For simplicity, we take the initial approximations u0 = u(x, 0) and v0 = v(x, 0) as given by (9). The next two iterates are easily obtained from (24)–(25) and are given by u1 (x, t) = ex + te−x ,

v1 (x, t) = e−x − tex ,

1 1 u2 (x, t) = ex + te−x + ex t 2 , v2 (x, t) = e−x − tex + e−x t 2 . 2 2 Fig. 1 shows the comparisons between the 2-iterate of VIM and the exact solutions.

(26) (27)

3.1.2. Example 2 Consider the following 2 × 2 linear system of PDEs of (1)–(3), letting u1 = u and u2 = v, ut + ux − 2v = 0,

(28)

vt + vx + 2u = 0,

(29)

with the following initial conditions: u(x, 0) = sin x,

v(x, 0) = cos x.

(30)

In operator form, (28)–(29) can be written as: L1 = ut + ux − 2v,

N1 = 0,

f1 = 0,

(31)

L2 = vt + vx + 2u,

N2 = 0,

f2 = 0.

(32)

The exact solution is [7]: u(x, t) = sin(x + t),

v(x, t) = cos(x + t).

(33)

To solve Eqs. (28)–(29) by means of VIM, we construct a correction functional, t un+1 (x, t) = un (x, t) + 0

  λ1 (un )s + (u˜ n )x − 2vn ds,

(34)

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Fig. 2. (a) The exact solution for u(x, t), (b) 3-iterate VIM for u(x, t), (c) the exact solution for v(x, t), (d) 3-iterate VIM for v(x, t).

t vn+1 (x, t) = vn (x, t) +

  λ2 (vn )s + (v˜n )x + 2un ds,

(35)

0

where u˜ n , v˜n are considered as restricted variations, which means u˜ n = 0, v˜n = 0. As before, the Lagrange multipliers can be identified as λ1 (s) = λ2 (s) = −1,

(36)

and the iteration formulas are given as t un+1 (x, t) = un (x, t) −



 (un )s + (un )x − 2vn ds,

(37)

 (vn )s + (vn )x + 2un ds.

(38)

0

t vn+1 (x, t) = vn (x, t) −



0

For simplicity, we take the initial approximations u0 = u(x, 0) and v0 = v(x, 0) as given by (30). The next two iterates are easily obtained from (37)–(38) and are given by u1 (x, t) = sin x + t cos x,

v1 (x, t) = cos x − t sin x,

1 1 u2 (x, t) = sin x + t cos x + t 2 sin x, v2 (x, t) = cos x − t sin x + t 2 sin x. 2 2 Fig. 2 shows the comparisons between the 2-iterate of VIM and the exact solution (33).

(39) (40)

3.2. Nonlinear systems 3.2.1. Example 3 Consider the 2 × 2 nonlinear system of (1)–(3), letting u1 = u and u2 = v, ut + vux + u = 1,

(41)

vt − uvx − v = 1,

(42)

with the following initial conditions: u(x, 0) = ex ,

v(x, 0) = e−x .

(43)

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In operator form, (41)–(42) can be written as: L1 = ut + u,

N1 = vux ,

f1 = 1,

(44)

L2 = vt − v,

N2 = −uvx ,

f2 = 1.

(45)

The exact solution is [7]: v(x, t) = e−x+t .

u(x, t) = ex−t ,

(46)

To solve Eqs. (41)–(42) by means of VIM, we construct correction functionals, t un+1 (x, t) = un (x, t) +

  λ1 (un )s + v˜n (u˜ n )x + un − 1 ds,

(47)

  λ2 (vn )s − u˜ n (v˜n )x − vn − 1 ds,

(48)

0

t vn+1 (x, t) = vn (x, t) + 0

where u˜ n , v˜n are considered as restricted variations, which means u˜ n = 0, v˜n = 0. We note that these correction functionals are different from the ones given in [29]. Its stationary conditions can be obtained as follows: 1 + λ1 (t) = 0,

λ1 (s) − λ1 (s)|s=t = 0,

(49)

1 + λ2 (t) = 0,

λ2 (s) + λ2 (s)|s=t

(50)

= 0.

The Lagrange multipliers, therefore, can be identified as λ1 (s) = −es−t ,

λ2 (s) = −e−s+t ,

(51)

and the iteration formulas are given as t un+1 (x, t) = un (x, t) −

  es−t (un )s + vn (un )x + un − 1 ds,

(52)

  e−s+t (vn )s − un (vn )x − vn − 1 ds.

(53)

0

t vn+1 (x, t) = vn (x, t) − 0

We take the initial approximations u0 = u(x, 0) and v0 = v(x, 0) as given by (43). The next iterate is easily obtained from (52)–(53) and is given by u1 (x, t) = ex−t ,

v1 (x, t) = e−x+t ,

(54)

which is the exact solution. Further calculations will confirm that u1 = u2 = · · · = ex−t and v1 = v2 = · · · = e−x+t , i.e. we have found the exact solution. We remark that the VIM procedure given very recently by Wazwaz [29] who chose different Lagrange multipliers only yields approximate series solution of the problem. 3.2.2. Example 4 In this example, a system of three nonlinear PDEs in three unknown functions will be investigated. It is important to note that handling this system by traditional methods is complicated. Consider the following 3 × 3 nonlinear system of PDEs of (1)–(3), letting u1 = u, u2 = v and u3 = w: ut + vx wy − vy wx + u = 0,

(55)

vt + wx uy + wy ux − v = 0,

(56)

wt + ux vy + uy vx − w = 0,

(57)

with the following initial conditions: u(x, y, 0) = ex+y ,

v(x, y, 0) = ex−y ,

w(x, y, 0) = e−x+y .

(58)

The exact solution is [7]: u(x, y, t) = ex+y−t ,

v(x, y, t) = ex−y+t ,

w(x, y, t) = e−x+y+t .

(59)

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In operator form, (55)–(57) can be written as: L1 = ut + u,

N1 = v x wy − v y wx ,

f1 = 0,

(60)

L2 = vt − v,

N2 = wx uy + wy ux ,

f2 = 0,

(61)

L3 = wt − w,

N3 = ux vy + uy vx ,

f3 = 0.

(62)

To solve Eqs. (55)–(57) by means of VIM, we construct correction functionals, t un+1 (x, y, t) = un (x, y, t) +

  λ1 (un )s + (v˜n )x (w˜ n )y − (v˜n )y (w˜ n )x + un ds,

(63)

  λ2 (vn )s + (w˜ n )x (u˜ n )y + (w˜ n )y (u˜ n )x − vn ds,

(64)

0

t vn+1 (x, y, t) = vn (x, y, t) + 0

t wn+1 (x, y, t) = wn (x, y, t) +

  λ3 (wn )s + (u˜ n )x (v˜n )y + (u˜ n )y (v˜n )x − wn ds,

(65)

0

where u˜ n , v˜n and w˜ n are considered as restricted variations, which means u˜ n = 0, v˜n = 0 and w˜ n = 0. These correction functionals are different from the ones given in [29]. As in the previous example, the Lagrange multipliers can be identified as λ1 (s) = −es−t ,

λ2 (s) = −e−s+t ,

λ3 (s) = −e−s+t

(66)

and the iteration formulas are given as t un+1 (x, y, t) = un −

  es−t (un )s + (vn )x (wn )y − (vn )y (wn )x + un ds,

(67)

  e−s+t (vn )s + (wn )x (un )y + (wn )y (un )x − vn ds,

(68)

0

t vn+1 (x, y, t) = vn − 0

t wn+1 (x, y, t) = wn −

  e−s+t (wn )s + (un )x (vn )y + (un )y (vn )x − wn ds.

(69)

0

For simplicity, we take the initial approximations u0 = u(x, 0), v0 = v(x, 0) and v0 = v(x, 0) as given by (58). The next two iterates are easily obtained from (67)–(69) and are given by u1 (x, y, t) = ex+y−t ,

v1 (x, y, t) = ex−y+t ,

w1 (x, y, t) = e−x+y+t

(70)

which is the exact solution. Further calculations will confirm that u1 = u2 = · · · = ex+y−t , v1 = v2 = · · · = ex−y+t and w1 = w2 = · · · = e−x+y+t , i.e. we have found the exact solution. Again, the VIM procedure given very recently by Wazwaz [29] who chose different Lagrange multipliers only yields approximate series solution of the problem. 4. Concluding remarks In this Letter, the variation iteration method (VIM) has been successfully employed to obtain the approximate analytical and exact solutions of systems of linear and nonlinear PDEs. Comparisons with the exact solutions reveal that VIM is very effective and convenient. The VIM in particular avoids the need of finding the Adomian polynomials which can be difficult even for a computer algebra system [9,10]. It is shown that VIM is a promising tool for linear and nonlinear system of PDEs. Acknowledgements The financial support received from the Academy of Sciences Malaysia under the SAGA grant No. P24c (STGL-011-2006) is gratefully acknowledged.

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