International Journal of Scientific & Engineering Research Volume 2, Issue 10, Oct-2011 1

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Modified Variational Iteration Method for the Solution of nonlinear Partial Differential Equations

Olayiwola , M. O Akinpelu, F. O , Gbolagade, A .W

Abstract-The Variational Iteration Method (VIM) has been shown to solve effectively, easily and accurat ely a large class of linear and nonlinear problem s with approximations converging rapidly to exact solutions.

W e present a new Modified Variational Iteration Method (MVIM) for the solution of some partial differential equations of physical significance.

Index Terms : variational iteration method, lagrange multiplier, Taylor’s series, partial differential equation, m odified variational iteration m ethod, correction functional.

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1.INTRODUCTION

Differential equations play a crucial role in applied mathematics and physics. The results of solving these equations can guide authors to know the described process deeply. But at times it is difficult to obtain the exact solutions to these problems. In recent decades, there has been great development in the numerical analysis and exact method of solving partial differential equations. For instance, Adomian’s decomposition method, Homotopy perturbation method, parameter expanding method etc.
He (1999, 2000, 2006) developed the variational iteration method for solving linear, nonlinear and boundary value problems. The method was first considered by Inokuti, Sekine and Mura (1978) and fully explored by He. J. H. In this method, the solution is given in an infinite series usually converging to an accurate solution. Olayiwola etal (2009) used modified power series method for the solution of systems of differential equations. It is observed that the method solve effectively, easily and accurately a class of linear, nonlinear, ordinary differential equations with approximate solution which converge very rapidly to accurate solution.
In this paper, we present a new modified variational iteration method for the solution of nonlinear partial differential equations.

2. VARIATIONAL ITERATION METHOD

To illustrate the basic concept of the VIM , we consider the following general nonlinear partial differential equation.

Lu x, t Ru x, t Nu x, t

g x, t

(2.1)
where L is a linear time derivative operator, R is a linear operator which has partial derivative with respect to x, N is a nonlinear operator and g is an inhomogeneous term. According to VIM, we can construct a correct fractional as follows:

u n 1

x , t u n x , t   t

Lu n

R u~ n

N u~ n

g d

(2.2)
where is a Lagrange multiplier which can be identified optimally via variational iteration method. The

~ ~

subscript n denote the nth approximation, un is considered as a restricted variation i.e, u n  0 . The

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successive approximation

u n  1 , n  0

of the solution u will be readily obtained upon using the
determined Lagrange multiplier and any selective function u0 , consequently, the solution is given by:

lim u

u n

(2.3)

n  

In Modified VIM, equation (2.2) becomes:

u 0 x , t  

g 0 ( x )  tg 1 ( x )  t 2 g 2 ( x )

(2.4)

u n 1

x , t u n x , t  t Lu n

R u~n

N u~n

g d

(2.5)

where g 2 ( x )

can be found by substituting for u 0 ( x , t ) in (2.1) when t  0 .

3. APPLICATION OF MVIM

3.1 We consider the nonlinear homogeneous gas dynamics equation Hossein etal (2008):

u

t

1 

2  x

( u 2 ) 

u (1  u )  0 ,

0  x

 1,

t  0 .

(3.1)
With the initial conditions

u ( x , 0 ) 

e x

The iteration formula is

u x,t u

x, t

t u 1


 (u 2 )  u

(1  u

d

n1

n 0



2 x n n

n )

(3.2)
Making (3.2) correction functional stationary, the Lagrange multiplier can be identified as:

  1

(3.3)
Using (2.4) and (2.5) in (3.2)

x 1

2x

2x

2 2x

u x,tex ex  e

0

 (2e

2

4e

2e )

d

(3.4)

(ex ex ) e2x  2e2x 2e2x 

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e x

t e x

t e x

2

Similarly
(3.5)

u e x

t e x t

e x

t e x

2

.
.
.

u e x (1  t

2


t t

6

 ......

 ...)

(3.6)

n 2 6

(3.7)

t x

It is obvious that a higher number of iteration makes un converges to exact solution: e .
Figure 1 shows the graphs of exact solution against x for different time (t). Figure 2 shows the graphs of MVIM solution against x for different time (t) while Figure 3 shows the graphs of exact solution and MVIM solution against x for (3.1).

Figure 1

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Figure 2

Figure 3

3.2 We consider the nonlinear equation:

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u u

3 u

6 u


u ( ) 2

u 2   0 .

t x

x 3

x 6

(3.8)

u ( x , 0 )  e x

Similarly

u e x

t e x

0 (3.9)

2

u 1 e

u e x

t e x

t e x

t e x

2


t e x t e x

(3.10)

2

.
.

u e x (1  t

2 6



t t t

 ....

 ...)

(3.11)

n 2 6 24

(3.12)
This converges to:

x t

u n e

Figure 4 shows the graphs of exact solution against x for different time (t). Figure 5 shows the graphs of MVIM solution against x for different time (t) while Figure 6 shows the graphs of exact solution and MVIM solution against x for (3.8).

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Figure 4

Figure 5

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Figure 6

3.3 : Consider a nonlinear non-homogenous partial differential equation ; Dogan (2002)

u

e u

u

(  u ) 2

2 u

e u (1 

x t ) .

t x x

x 2

(3.13)

u ( x , 0 )  ln

x , x  0

Appling MVIM to (3.13)

u 0  ln

x t x

…………………………………………………………(3.14)

   2

u x,tu x,t

t un eun

un (

un )2

un eun

(1 xt) d

n1

n 0

 


x x

x2

(3.15)

In the same way, we compute other components for n

 1 , 2 . .

Figure 7 shows the graphs of exact solution and MVIM solution against x for (3.13).
.

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Figure 7

4. CONCLUSIONS

In this work, the MVIM was applied to the solution of nonlinear partial differential equations. The numerical results demonstrated that the method is accurate, reliable and converges faster with less computation when compared with other methods in the literature.

REFRENCES

[1] Abulwafa E.M, Abdou M.A, Mahmoud A.A, “ Nonlinear fluid flows in pipe-like domain problems using VIM class”, Soliton And Fractals 2007, 32(4) 1384-1397.
[2] Abulwafa E.M, Abdou M.A, Mahmoud A.A. “ The solution of nonlinear coagulation problem with mass class”, Soliton and Fractals. 2006,29 (2) 313-330.
[3] Awodola T. O.” Variable Velocity Influence on the Vibration of Simply Supported Bernoulli - Euler Beam Under Exponentially Varying Magnitude Moving Load”. Journal of Mathematics and Statistics 3 (4): 228-232, 2007
[4] Barari A “ An approximate solution for boundary value problems in structural Engineering and
Fluid Mechanics”. Mathematical problems in Engineering. 2008
[5] Batiha B. “Application of Variational Iterational method to a General Riccat Equation”. Int.

Mathematical Forum, 2, 2007, no 56, 2759-2770.

[6] Biazar J., Ghazuini H, “He’s Variatonal iteration method for solving linear and non-linear system of ordinary differential equations”. Applied Maths and Computation 2007, 191, 281-287.
[7] Bildik N., Konuralp A. “The use of variation iteration method, differential transform method and Adomain decomposition method for solving different types of nonlinear pole”. Int. J. nonlinear Science; Numerical Simulation 2006,7(1) 65-70.
[8] Celik E “ Numerical method to solve chemical differential Allgerai equation”. Int. J. of Quantum

chemistry, 2002,89(5), 447-451,.

[9] Diethelin K. and Ford N.J, “Analysis of fractional differential equation”. J. math and Appl.2002, 265
229-248.
[10] Djidjeli K. “Numerical methods for special nonlinear boundary value problems of order 2m”. J. of

Computational and Appl. Mathematics Vol. 47, 2006,no 1, 161-169,.

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research Volume 2, Issue 10, Oct-2011 9

ISSN 2229-5518

[11] Dogan Kaya. “The use of Adomian decomposition method for solving a specific nonlinear partial differential equations”. Bull. Beg. Maths. Soc. 9(2002), 343-349.
[12] Draganescu G E, capalnasan V. “Nonlinear relaxation phenomenon in polycrystalline solid” int. J

nonlinear Numerical Simulation 4(3) 219-225.

[13] Ganji D.D. A Comparison of Variational iteration method with Adomain decomposition method
in some highly nonlinear equations.” International Journal of Science and Technology Vol. 2, No 2,
2007,179-188.
[14] Gbadeyan, J A& S.T Oni , “Dynamics behavior of beam and rectangular plates under moving loads.” Journal of Sound and Vibration 1995,199,33-50.
[15] He. J.H “Approximate analytical solution for seepage flow with fractional derivatives in porous media”. Compt. Math. App. Mech. Eng. 167. 1998 57-68.
[16] He. J.H “Approximate solution of nonlinear differential equations with convolution product nonlinearities” Comp. Math. Apply Mech. Eng. 167: 199869-73.
[17] He. J.H “Variational iteration method for autonomous ordinary differential system.” App. Math and Computation, 2000; 114:115-123.
[18] Hossein Jafari etal “Application of Homotopy Perturbation Method for Solving Gas Dynamic
Equation”. Applied Mathematical Sciences, Vol.2,2008. No 48,2393-2396.
[19] Jemal Peradze “ A Numerical Algorithm for a Kirchhoff-Type Nonlinear Static Beam”. Journal of

Applied MathematicsVolume 2009,

[20] Inokuti M. et al. (1978): General use of the Lagrange multiplier in nonlinear mathematical physics
in: S. Nemat Nasser (Ed), Vanational method in the mechanics of solid, Pergamon Press, 156-162. [21] Ji-Huan He (2007): Variational iteration methods some recent results and new interpretation. J. of
competition and Applied martherates 207 3-17.
[22] Ji-Huan He. (1999): Variational Iteration method: a kind of non-linear analytical technique: Some examples. Int. Journal of Non-linear mechanics 3494) 699-708.
[23] Levant Yilmaz “Some Considerations on the series solution of Differential equations and its engineering Applications.” RMZ Materials and Geo-environment, Vol. 53,2006, No 1, 247-259.
[24] Lu J.F. “Variational Iteration method for solving two-point boundary value problems.” Journal of

computational and Applied Mathematics 2007(1) 92-95.

[25] Ma T.F and Silva J. Da , “Iterative solution for a beam equation with nonlinear boundary conditions of third order”, Applied mathematics and Computation, Vol. 159 ,2004,no 1. 11-18.
[26] Muhammed A.N, Syed T.M ,”Variational Iteration Decomposition method for solving Eight-Order
Boundary Value Problem.”Diff. Equation and Nonlinear mechanics. 2007 Vol. Id 19529.
[27] Nuran Guzek, Bayram M. , “Power Series solution of nonlinear first order differential equations systems” Trakya Univ. J.Se, 2005,6(1). 107-111.
[28] Odibat Z.M, Momani S. “Application of Variation iteration method to nonlinear differential equation of fractional order.” Int. J to Nonlinear Science. Numerical Simulation 2006,7(1) 27-34.
[29] Olayiwola M.O et al,”Application of modified power series method for the solution of system of
differential equations”. Journal of Nigerian Association of Mathematical Physics Vol. 15, 2009,pp 91-94.
[30] Olayiwola M.O, Gbolagade A.W. “ On the Chaotic behavoiour of Hamonically driven Oscillator” Far East J. of Dynamical Systems Vol. 10, 2008, issue 1, 47-52.
[31] Podhibny I. “ Fractional Differential equations” Academic Press, 1999, San Diego,.
[32] Saharay S.. Bera R.K. “ Solution of an extraordinary differential equation by Adomain decomposition method”. J. of Applied Maths 4,2004, 331-338.
[33] Siddiqi S.S. Twizell. E.H. “ Spline solution of linear eight-order boundary value problems”

Computer methods in Apply Mechanics and engineering Vol. 131, 1996,no 3-4, 309-325.

[34] Tatari M, Delighan M. “On the Convergence of He’s Variational iteration method” Journal of

computational and applied Mathematics 2007 (1) 121-128.

[35] Wang . L and Ni. Q “ Vibration of Slender Structures Subjected to Axial Flow or Axially Towed in
Quiescent Fluid” Advances in Acoustics and Vibration Volume 2009, Article ID 432340
[36] Wazwaz A M. “ A reliable algorithm for obtaining positive solution for nonlinear boundary value problems.” Compute Math App 41(10-11) 2001, 1237-1244.
[37] Wazwaz, A M. “Analytic treatment for variable cefficient fourth-order parabolic partial differential equations.” Appl math. Comput. 123: 2001, pp. 219-227
[38] Wazwaz, A. M. “ Partial differential equations, methods and applications.” A. A. Balkema Publishers
[39] Wei-Xia Qian “He’s Iteration Formulation for solving Nonlinear Algebraic equations.” Journal of

Physics: Conference series 2008, 96 1-6.

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ISSN 2229-5518

[40] Yujun Cui, Yumei Zou " Existence and Uniqueness of solutions for Fourth-Order Boundary-Value

Problems in Banach Spaces." Electronic Journal of Differential Equations, Vol. 2009(2009), No. 33, pp.

1-8

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