Variational Iteration Method for Solving Some Models of Nonlinear ...

Int. J. Pure Appl. Sci. Technol., 4(1) (2011), pp. 30-40

International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in Research Paper

Variational Iteration Method for Solving Some Models of Nonlinear Partial Differential Equations A.S.J.AL-Saif1,* and T.A.K. Hattim2 1, 2

Department of Mathematics, College of Education, University of Basrah, Basrah, Iraq.

* Corresponding author, e-mail: ([email protected]) (Received: 16-12-2010; Accepted: 08-04-2011)

Abstract: In this paper, the variational iteration method is implemented for solving nonlinear initial value problems. Analytic solutions of the nonlinear partial differential equations with initial data are obtained. It has been shown that the method is quite efficient and is practically well suited for use in these problems. Several examples are given to verify the accuracy and efficiency of the proposed technique. Keywords: Variational iteration method, Nonlinear initial value problems.

1. Introduction In 1978, Inokuti et al. [14] proposed a general Lagrange multiplier method to solve non-linear problems, which was the first proposed to solve problems in quantum mechanics. In 1998, the Lagrange multiplier method is modified by He [8-12] into an iteration method that is called variational iteration method (VIM). It is used to solve effectively, easily, and accurately a large class of non-linear problems with approximations converging rapidly to accurate solutions, where the approximate solution of the VIM in the main is readily obtained upon using the obtained Lagrange multiplier and on the selective initial approximate. The variational iteration method changes the differential equation to a recurrence sequence of functions, where the limit of that sequence is considered as the solution of the partial differential equations. The main advantage of the method is that it can be applied directly to all types of nonlinear differential and integral equations, homogeneous or inhomogeneous, with constant or variable coefficients [1, 15-17]. Moreover, the proposed method is capable


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of greatly reducing the size of computational work while still maintaining high accuracy of the numerical solution. In this work, the non-linear problems have the following equations; ut − u xx + 2u 3 = 0 u t − uu x = 0

∂ m 1 m+1 u −u = 0 (u u x ) − ∂x m +1 And the non-linear problems have the following system of equations; u t − u xx − 2uu x + (uv) x = 0 ut −

vt − v xx − 2vv x + (uv) x = 0

(1) (2)

(3)

(4)

ut − uu x + vu y = 0

(5) vt − uv x + vv y = 0 will be studied by implementing VIM, where u (and v ) is the solution of equation, and m > 0. Equation (1) is one of a class of non-linear heat transfer equations, arises in several important physical, chemical reactions and mathematical biology applications [2, 8, 9, 22]. This class of equation is solved by using Lie group method of infinitesimal transformations [13]. Equation (2) is the hyperbolic nonlinear which describes the shock phenomenon inherent in its solution. This equation is handled by using several techniques, such as, differential quadrature method (DQM)[5], Adomian decomposition method(ADM)[3], finite difference method[18]. Equation (3) is one of a class of quasilinear parabolic equations. This equation is solved numerically by finite difference scheme [7] and theoretically by different methods [21]. System of equations (4) couple Burger’s equations. It has been found to describe various kinds of phenomena such as a mathematical model of turbulence, the approximate theory of flow through a shock wave traveling in a viscous fluid [6]. This system is handled by ADM [6] and differential transforms method [20]. System of equations (5) arising in various simplified models of fluid flow and turbulence, and it can be solved by various numerical methods such as ADM [3] and DQM[5]. The aim of this paper is to apply the variational iterations method to solve nonlinear initial value problems that do not apply the method for solving it, by using VIM exact solution and approximate solutions of the problems have been obtained in terms of convergent series with easily computable components. The organization of this paper is as follows; section 2 gives brief ideas of VIM. In section 3, the sufficient conditions are presented to guarantee the convergence of the method. In section, 4; four examples are given to illustrate the effectiveness and the useful of the variational iteration method. In section 5, we presented discussion of our work. Conclusions are presented in the last section.

2. Basic Concepts of Variational Iteration Method: Idea of variational iteration method depends on the general Lagrange's multiplier method [14]. This method has a main feature, which is the solution of a mathematical problem with linearization assumption used as initial approximation or trial function. This approximation converges rapidly to an accurate solution [11].


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To illustrate the basic concepts of the VIM, we consider the following nonlinear differential equation: L(u ) + N (u ) = g ( x)

(6)

where L is a linear operator and N is a nonlinear operator, and g ( x) is an inhomogeneous term. According to the VIM [4], we can construct a correction functional as follows: t u n+1 ( x, t ) = u n ( x, t ) + ∫ λ (τ )[L(u n ( x, t )) + N (u~n ( x,τ )) − g ( x)]dτ 0

(7)

where λ (τ ) is a general Lagrangian multiplier[1,15,16], which can be identified optimally via the variational theory and integration by parts. The subscript n denotes the nth-order approximation, u~n is considered as a restricted variation ( i.e. δ u~n = 0 ) [8]. So, we first determine the Lagrange multiplier λ (τ ) that will be identified optimally via integration by parts. The successive approximations u n+1 , n ≥ 0 of the solution u ( x, t ) will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function u 0 . Consequently, the Solution u ( x, t ) = lim n→∞ u n

(8)

3. Convergence Analysis of the VIM: Here, we will study the convergence analysis as the same manner in [23] of the variational iteration method to the nonlinear equations. Let us consider the Banach space X , with the set of applications, u : Ω → ℜ , with

∫u

2

dx < ∞

Ω

and the associated norm:

u

2

= ∫ u 2 dx Ω

The VIM is convergent if the conditions of the following theorem are satisfied.

Theorem 1[23]: (Banach’s fixed-point theorem) Assume that X be a Banach space and A : X → X is a nonlinear mapping, and suppose that A[u ] − A[u ] ≤ γ u − u for some constant γ < 1 . Then A has a unique fixed point. According to the theorem 1, for the nonlinear mapping

 ∂u ∂u ∂ 2u ∂ 2u ∂ 2u  A[u ] = u ( x, t ) + ∫ λ  F (u, , , 2 , 2 , ) dτ 0 ∂x ∂τ ∂x ∂τ ∂x∂τ   t

A sufficient condition for convergence of the variational iteration method is the strictly contraction of A , such that for u , u ∈ X we have u ≤ M and u ≤ M , for M > 0 .


33

4. Applications of VIM: In this section, we apply the variational iteration method for solving five models of nonlinear initial value problems. Three contains non-linear partial differential equations and the others contain non-linear system of partial differential equations.

4.1 The nonlinear equations: To uphold our work, we introduced three examples Example 1: Consider the following non-linear partial differential equation; ut − u xx + 2u 3 = 0

(9)

According to the VIM, we can construct the correction functional (7) of equation (9) as follows: t u n+1 ( x, t ) = u n ( x, t ) + ∫ λ (τ )[(u n ( x,τ ))τ − (u~n ( x,τ )) xx − 2(u~n ( x,τ )) 3 ]dτ (10) 0

where λ is a general Lagrange multiplier. The value of λ can be found by considering, (u~n ) xx and (u~ ) 3 as a restricted variations (i.e. (δu~ ) = (δu~ ) 3 = 0 in equation (10), then integrating n

n xx

n

the result by part to obtained λ = −1 . Then the correction functional (10) becomes in the following iteration formula: t

u n+1 ( x, t ) = u n ( x, t ) − ∫ [(u n ( x,τ ))τ − (u n ( x,τ )) xx − 2(u n ( x,τ )) 3 ]dτ 0

(11)

Consequently, the following approximants are obtained by using the above iteration formula 2x + 1 (11) started with the initial approximation u ( x,0) = 2 x + x + 10

2x + 1 x + x + 10 2x + 1 6(2 x + 1) ⋅ t u1 ( x, t ) = 2 − 2 x + x + 10 ( x + x + 10) 2

u 0 ( x ,0 ) =

2

u 2 ( x, t ) =

2x + 1 6(2 x + 1) ⋅ t 36(2 x + 1) ⋅ t 2 − + + small terms x 2 + x + 10 ( x 2 + x + 10) 2 ( x 2 + x + 10) 3

u 3 ( x, t ) =

2x + 1 6(2 x + 1) ⋅ t 36(2 x + 1) ⋅ t 2 216(2 x + 1) ⋅ t 3 − + − + small terms x 2 + x + 10 ( x 2 + x + 10) 2 ( x 2 + x + 10) 3 ( x 2 + x + 10) 4

M (12) and so on. The solution in a closed form is readily found to be [19].


u ( x,t ) =

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2 x +1 x + x + 6 t + 10

(13)

2

Table 1: Comparison between the DQM ,ADM and the VIM solutions at t = 0.1 .

x 0.1 0.3 0.5 0.7 0.9

uexact

u 4 −VIM

u 4 − ADM

0.11204 0.14559 0.17621 0.20356 0.22746

0.11204 0.14559 0.17621 0.20356 0.22746

0.11204 0.14559 0.17621 0.20356 0.22746

DQM 0.11829 0.15318 0.18467 0.21241 0.23628

Example 2: Consider the following non-linear partial differential equation [3,5] u t − uu x = 0

(14)

The correction functional (7) of equation (14) is given as: t u n+1 ( x, t ) = u n ( x, t ) + ∫ λ (τ )[(u n ( x,τ ))τ − u~n ( x,τ )(u~n ( x,τ )) x ]dτ 0

(15)

Proceeding as before we find λ = −1 . Substituting this value into the correction functional (15) gives the following iteration formula: t

u n+1 ( x, t ) = u n ( x, t ) − ∫ [(u n ( x,τ ))τ − u n ( x,τ )(u n ( x,τ )) x ]dτ 0

We start with initial approximation u ( x,0) =

(16)

x , and using the iteration formula (16), we can 10

obtain: x 10 x t u1 ( x, t ) = (1 + ) 10 10 x t t u 2 ( x, t ) = (1 + + ( ) 2 + small terms) 10 10 10 x t t t u3 ( x, t ) = (1 + + ( ) 2 + ( ) 3 + small terms) 10 10 10 10 M u 0 ( x ,0 ) =

(17)

and so on. The solution in a closed form is readily found as; u ( x, t ) =

x t − 10

(18)


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Table 2 : Comparison between the DQM,ADM and the VIM solutions at t = 0.1 .

x 0.1 0.3 0.5 0.7 0.9

uexact

u 4 −VIM

u6 − ADM

0.01010 0.03030 0.05051 0.07071 0.09091

0.01010 0.03030 0.05051 0.07071 0.09091

0.01010 0.03030 0.05051 0.07071 0.09091

DQM 0.01012 0.03025 0.05050 0.07072 0.09093

Example 3: Consider the following nonlinear partial differential equation [7,19,21]; ut +

∂ m u m+1 (u u x ) − −u = 0 ∂x m +1

, m>0

(19)

According to the VIM, we can construct the correction functional (7) of equation (19) as follows: t (u~ (x,τ ))m+1 ~ ∂ un+1 ( x, t ) = un (x, t ) + ∫ λ(τ )[(un (x,τ ))τ + ((u~n (x,τ ))m (u~n ( x,τ ))x ) − n − un (x,τ )]dτ (20) 0 m +1 ∂x Proceeding as before we find λ = −1 . Substituting this value into the correction functional (20) gives the following iteration formula: t

un+1 ( x, t ) = un ( x, t ) − ∫ [(un ( x,τ ))τ + 0

(u ( x,τ ))m+1 ∂ ((un ( x,τ ))m (un ( x,τ )) x ) − n − un ( x,τ )]dτ m +1 ∂x

(21) 1

Using the above iteration formula (21) and the initial approximations u ( x,0) = (e x + e − x ) m+1 , we obtain the following successive approximations: 1

u 0 ( x,0) = (e x + e − x ) m+1 1

u1 ( x, t ) = (e x + e − x ) m+1 ⋅ (1 + t ) 1

u 2 ( x, t ) = (e x + e − x ) m+1 ⋅ (1 + t + 1 − x m +1

u 3 ( x, t ) = (e + e ) x

⋅ (1 + t +

t2 ) 2!

(22)

t2 t3 + ) 2! 3!

M and so on. The solution in a closed form is given by: 1

u ( x, t ) = e t (e x + e − x ) m+1

(23)


36

Table 3: Comparison between the DQM and the VIM solutions m = 0.5 at t = 0.1 .

x 0.1 0.3 0.5 0.7 0.9

uexact

u5 −VIM

u5 − ADM

1.7602 1.807 1.9006 2.0414 2.23

1.7602 1.807 1.9006 2.0414 2.23

1.7602 1.807 1.9006 2.0414 2.23

DQM 1.7601 1.8071 1.9005 2.0413 2.2296

4.2 The non-linear systems equations We now introduce two examples for non-linear system equations

Example 4: Consider the following non-linear system equations [20, 6] ut − u xx − 2uu x + (uv) x = 0

(24)

vt − v xx − 2vv x + (uv) x = 0 Following the analysis presented above, we construct a correction functional for (24) as: t u (x, t) = u (x, t) + λ(τ )[(u (x,τ )) − (u~ (x,τ )) − 2u~ (x,τ )(u~ (x,τ )) + (u~ (x,τ )v~ (x,τ )) ]dτ τ ∫ ( x, t) = v (x, t) + ∫ λ(τ )[(v (x,τ ))τ − (v~ (x,τ ))

n+1

vn+1

n

n

0

n

t

n

n

0

n

xx

n

n

x

n

n

x

~ ~ ~ ~ xx − 2vn ( x,τ )(vn ( x,τ ))x + (un ( x,τ )vn ( x,τ ))x ]dτ

(25)

Proceeding as before we find λ = −1 . The correction functional (25) becomes in the following iteration formula: t

un+1 (x, t) = un ( x, t) − ∫ [(un ( x,τ ))τ − (un (x,τ ))xx − 2un (x,τ )(un (x,τ ))x + (un ( x,τ )vn ( x,τ ))x ]dτ 0 t

vn+1 ( x, t) = vn (x, t) − ∫ [(vn ( x,τ ))τ − (vn ( x,τ ))xx − 2vn (x,τ )(vn ( x,τ ))x + (un (x,τ )vn ( x,τ ))x ]dτ

(26)

0

Consequently, the following approximations are obtained by using the above iteration formulas (26) with the initial approximations u ( x,0) = v( x,0) = sin( x) .

u 0 ( x,0) = v0 ( x,0) = sin( x) u1 ( x, t ) = v1 ( x, t ) = sin( x) ⋅ (1 − t ) t2 ) 2! t2 t3 u 3 ( x, t ) = v3 ( x, t ) = sin( x) ⋅ (1 − t + − ) 2! 3! M u 2 ( x, t ) = v2 ( x, t ) = sin( x) ⋅ (1 − t +

(27)

So on. According to the equation (8), the solutions are given as: u ( x, t ) = v( x, t ) = sin( x) ⋅ e t

(28)


37

Table 4: Comparison between the DQM , ADM and the VIM solutions at t = 0.1 .

x 0.1 0.3 0.5 0.7 0.9

uexact

u5 −VIM

u5 − ADM

0.09033 0.26740 0.43380 0.58291 0.70878

0.09033 0.26740 0.43380 0.58291 0.70878

0.09033 0.26740 0.43380 0.58291 0.70878

DQM 0.09035 0.26742 0.43384 0.58292 0.70881

Example 5: Consider the following two-dimension non-linear system equations [3]

ut − uu x + vu y = 0 vt − uv x + vv y = 0

(29)

To solve the system (29) by means of variational iteration method, we construct a correction functional (7) as: t un+1 (x, y, t) = un (x, y,t) + ∫ λ(τ )[(un (x, y,τ ))τ − u~n (x, y,τ )(u~n (x, y,τ ))x + (u~n (x, y,τ )v~n (x, y,τ ))y ]dτ 0 t

vn+1 (x, y,t) = vn (x, y,t) + ∫ λ(τ )[(vn (x, y,τ ))τ − v~n (x, y,τ )(v~n (x, y,τ ))x + (v~n (x, y,τ )v~n (x, y,τ ))x ]dτ

(30)

0

Proceeding as before we find λ = −1 . The correction functional (30) becomes in the following iteration formula: t

un+1 (x, y, t) = un (x, y, t) − ∫ [(un (x, y,τ ))τ − un (x, y,τ )(un (x, y,τ ))x + (un (x, y,τ )vn (x, y,τ ))y ]dτ 0 t

vn+1 (x, y, t) = vn (x, y, t) − ∫ [(vn (x, y,τ ))τ − vn (x, y,τ )(vn (x, y,τ ))x + (vn (x, y,τ )vn (x, y,τ ))x ]dτ

(31)

0

Consequently, the following approximants are obtained by using the above iteration formulas (31) with the initial approximations u ( x, y,0) = v( x, y,0) = x + y

u 0 ( x , y ,0 ) = v 0 ( x , y ,0 ) = x + y u1 ( x, y, t ) = v1 ( x, y, t ) = ( x + y )(1 + 2t ) u 2 ( x, y, t ) = v2 ( x, y, t ) = ( x + y )(1 + 2t + (2t ) 2 + small terms )

(32)

u 3 ( x, y, t ) = v3 ( x, y, t ) = ( x + y )(1 + 2t + (2t ) 2 + (2t ) 3 + small terms ) M So on. The solution in a closed form is readily found as; u ( x, y, t ) = v( x, y, t ) = ( x + y ) /(1 − 2t )

(33)


38

Table 5: Comparison between the DQM, ADM and the VIM solutions at t = 0.1 .

x=y

uexact

u8 −VIM

u12 − ADM

0.1 0.3 0.5 0.7 0.9

0.2500 0.7500 1.2500 1.7500 2.2500

0.2500 0.7500 1.2500 1.7500 2.2500

0.2500 0.7500 1.2500 1.7500 2.2500

DQM 0.24842 0.74551 1.24566 1.74983 2.24678

5. Discussion In this study examples are solved by the VIM on the bound region 0 ≤ x ≤ 1 (or 0 ≤ y ≤ 1 ), t ≥ 0 . The obtained results from the variational iteration method are listed in the Tables (1-5) at t = 0.1 . Tables show the comparison between our solution using VIM and the solution of the same problem by using ADM and DQM. The results were obtained show that VIM solutions with lees iterations converge rapidly to the exact solution u ( x, t ) or v( x, t ) at t = 0.1 . In examples 1, 2,3 the fourth iterations are needed when considering for arriving at the accurate solutions, and in examples 4,5 we need fifth and eighth iterations, respectively. From the comparison between the results of VIM and exact solution, we notice that the approximate solutions obtained by using VIM are very well accurate when the number of iterations is increasing with smallest value of time. Consequently, we concluded that the errors should be smaller with smaller time. Moreover, the problems in this study are tested by the fixed point theorem 1. The results confirm that A has a unique fixed point, and satisfying the convergence condition in the theorem 1, with the value of γ for; t

Example 1: γ = 1 − ∫ (δ 1 − δ 2 + 6 M 2 ) dτ 0

t

Example 2: γ = 1 − ∫ (δ 1 − δ 2 M ) dτ 0

t

Example 3: γ = ∫ (δ 1 + M m (δ 22 − 1) − 1) dτ − 1 0

t

Example 4: γ = ∫ (δ 1 − δ 2 − M (δ 3 − δ 4 )) dτ − 1 0 t

Example 5: γ = ∫ (δ 1 − M (δ 2 − δ 3 )) dτ − 1 0

where, δ ' s are the absolute values of differential operators that appear in partial differential equations.

6. Conclusions In this paper, the variation iteration method has been successfully employed to obtain the approximate analytical solutions of non-linear boundary and initial value problems. The method has been applied directly without using linearization or any restrictive assumptions. The comparison of the numerical results of VIM with other solutions by using other methods show that the variational iteration method is a powerful mathematical tool to solving nonlinear partial differential equations and faster in convergence to exact solution.


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References [1] M.A. Abdou, A.A. Soliman,Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math., 181 (2005)245–251. [2] W.F. Ames, Nonlinear Partial Differential Equations in Engineering. I, Academic Press, New York, 1967. [3] A.H. Ali and A. S. J. Al-Saif, Adomian decomposition method for solving some models of nonlinear partial differential equations, Basrah J. Sci., 26(2008), 1-11. [4] B. Batiha, M. S. M. Noorani and I. Hashim, Application of variational iteration method to a general Riccati equation, Inter. Math. Forum, 2 (2007), 2759 – 2770. [5] R. Bellman, B.G. Kashef and J. Casti, Differential quadrature: A Technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., 10 (1972) 40-52. [6] S. M. El-Sayed, D. Kaya, On the numerical solution of the system of two dimensional Burger's equations by the decomposition method, Appl. Math. Comput., 158 (2004), 101–109. [7] V.A. Galaktionov and A.A. Samarskii, Difference solutions of a class of quasilinear parabolic Equations I, USSR Comput. Math. and Math. Phys., 23(3), (1983), 83-91. [8] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167(1998), 57–68. [9] J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Math. Appl. Mech. Eng., 167(1998), 69–73. [10] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114 (2000), 115–123. [11] J. He, Approximate Analytical Solution of Blasius’ Equation, Comm. in Nonlinear Sci. and Numer. Simu., 4(1998), 75-78. [12] J. He, Variational iteration method, a kind of non-linear analytical technique: some examples, Inter. J. of Non-Linear Mech., 34(1999), 699-708. [13] J.M. Hill, Differential Equations and Group Methods for Scientists and Engineers, CRC Press, Boca Raton, 1992. [14] M. Inokuti, H. Sekine and T. Mura, General use of the Lagrange multiplier in Nonlinear Mathematical Physics, In: Nemat-Nassed S, Ed., Variational Method in the Mechanics of Solids, 156-162, Pergamon Press,N.Y., U.S.A., 1978. [15] M. Matinfar and M. Ghanbari, Solving the Fisher’s equation by means of variational iteration method, Int. J. Contemp. Math. Sci., 4 (2009), 343 – 348. [16] M. Matinfar, H. seinzadeh M. Ganbari, Numerical implementation of the variational iteration method for the Lienard equation, World J. Mode. Simu., 4(2008), 205-210.


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[17] M.O. Miansari, M.E. Miansari, A. Barari and D.D. Gangi, Application of He’s variational iteration method to nonlinear Helmholtz and fifth-order Kdv Equation, J. Appl. Math., Stat. Inf., 5(2009), 5-20. [18] B. J. Noye, Numerical Solution of Partial Differential Equations, North Holland, South Australia, 1981. [19] A.D. Polyanin and V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC, U.S.A, 2004. [20] A.S.V. Ravi Kanth and K. Aruna, Differential transform method for solving linear and nonlinear systems of partial differential equations, Phys. Let. A, 372 (2008), 6896–6898. [21] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow- up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. [22] H. Stephani, Differential Equations, their Solution using Symmetries, Ed. M. MacCallum, C.U.P., Cambridge, 1989. [23] M. Tatari, M. Dehghan, On the convergence of He’s variational iteration method, J. Comput. and Appl. Math., 207(2007), 121 – 128.