Vol 20, No. 4;Apr 2013
Solution of Nonlinear Partial Differential Equations by the Combined Laplace Transform and the new Modified Variational Iteration Method 1Elsayed
A. Elnour , and 2Tarig M. Elzaki 1Math. Dept., Faculty of Sciences and Arts, P.O.Box 80200,Code 21589 Khulais, King Abdulaziz University, Jeddah, Saudi Arabia 1Math. Dept.,Faculty of Education, Alzaeim Alazhari University, Sudan 2Math. Dept., Faculty of Sciences and Arts, Alkamil, King Abdulaziz University, Jeddah, Saudi Arabia 2Math. Dept.,Faculty of Science, Sudan University of Science and Technology, Sudan E-mail:
[email protected],
[email protected],
Abstract In this paper, we present a reliable combined Laplace transform and the new modified variational iteration method to solve some nonlinear partial differential equations. The analytical results of these equations have been obtained in terms of convergent series with easily computable components. The nonlinear terms in these equations can be handled by using the new modified variational iteration method. This method is more efficient and easy to handle such nonlinear partial differential equations. Keywords: Laplace transforms, variational iteration method, nonlinear partial differential equations. 1. Introduction The nonlinear equations represent the most important phenomena occurring in the world. Manipulation of nonlinear phenomena is of great importance in applied mathematics, physics, and issues related to engineering. The importance of obtaining the exact solution of nonlinear partial differential equations in physics and applied mathematics is still a big problem that needs new methods to discover new exact or approximate solutions. In the resent years, many authors mainly had paid attention to study solution of nonlinear partial differential equations by using various methods. Among these are the Adomian decomposition method, [20] homotopy perturbation method [14-17, 27], variational iteration method, differential transform method [2, 3, 18, 19], and projected differential transform method [25, 26, 28]. In this work, we will use the new modified form of Laplace variational iteration method which we will call the new modified variational iteration Laplace transform method and it will be employed in a straight forward manner. This method provides an effective and efficient way of solving a wide range of nonlinear operator equations. The advantage of this method is its capability of combining two powerful methods for obtaining exact solutions for linear and nonlinear partial differential equations. This article considers the effectiveness of the new modified variational iteration Laplace transform method in solving nonlinear partial differential equations. 2. Laplace Transforms The basic definition of Laplace transform is given as follows: Laplace transform of the function f (t ) is
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∞
L [ f (t )] = F (s ) = ∫ f (t ) e − st dt ,
Re s > 0
0
(1) And the inverse Laplace transform is given by
L
−1
[ F (s )] = f
(t ) =
1
α +i ∞
2π i α ∫
e st F (s ) ds , α > 0
−i ∞
−1
Obviously L and L are linear integral operators. In this paper, we combined Laplace transform and variational iteration method to solve nonlinear partial differential equations. To obtain Laplace transform of partial derivative we use integration by parts, and then we have:
∂f (x , t ) L = sF (x , s ) − f (x , 0) , ∂t ∂f (x , t ) d L = [ F (x , s ) ] ∂x dx
,
∂ 2f (x , t ) ∂f (x , 0) 2 L = s F (x , s ) − sf (x , 0) − 2 ∂t ∂t ∂ 2f (x , t ) d 2 L = 2 [ F (x , s )] . 2 ∂x dx
Where F (x , s ) is the Laplace transform of f ( x , t ) . We can easily extend this result to the nth partial derivative by using mathematical induction. 3. Variational Iteration Method To illustrate the basic concept of the He's VIM, we consider the following general differential equations
L [u (x , t )] + N [u (x , t )] = g (x , t ) With the initial condition
(2)
u (x , 0) = h (x )
(3)
Where L is a linear operator of the first order, N is nonlinear operator and g ( x , t ) is inhomogeneous term. According to variational iteration method [29] we can construct a correction functional as follows. t
u n +1 = u n + ∫ λ [ Lu n (x , s ) + Nuɶn (x , s ) − g (x , s )] ds
(4)
0
Where
λ
is a Lagrange multiplier ( λ = −1 ), the subscripts n denote the nth approximation, uɶn
is considered as a restricted variation, i.e.
δ uɶn = 0 .
Equation (4) is called a correction functional. The successive approximation u n +1 , of the solution
u will be readily obtained by using the determined Lagrange multiplier and any selective function u 0 , consequently, the solution is given by u = lim u n n →∞
4. The new Modified Variational Iteration Laplace Transform Method (VILTM) In this work we assume that
L is an operator of the first order
∂ in equation (2). ∂t
Taking Laplace transform on both sides of equation (2), to get
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L [ Lu (x , t ) ] + L [ Nu (x , t )] = L [ g (x , t ) ]
(5)
Using the differentiation property of Laplace transform and initial condition (3), we have:
sL [u (x , t )] − h (x ) = L [ g (x , t ) ] − L [ Nu (x , t )]
(6)
Appling the inverse Laplace transform on both sides of equation (6) to find:
1 u (x , t ) = G (x , t ) − L −1 L [ Nu (x , t ) ] , s
(7)
where G ( x , t ) represents the terms arising from the source term and the prescribed initial condition. Take the first partial derivative with respect to t of equation (7), to obtain:
∂ ∂ ∂ 1 u (x , t ) − G (x , t ) + L −1 L [ Nu (x , t ) ] = 0 ∂t ∂t ∂t s
(8)
By the correction function of the variational iteration method
∂ ∂ −1 1 u n +1 (x , t ) = u n (x , t ) − ∫ (u n )ς (x , ς ) − G (x , ς ) + L L [ Nu n (x , ς ) ]d ς ∂ς ∂ς ς 0 t
Or
1 u n +1 (x , t ) = G (x , t ) − L−1 L [ Nu n (x , t ) ] s
(9)
Equation (9) is the new modified correction functional of Laplace transform and the variational iteration method, and the solution u is given by
u (x , t ) = lim u n (x , t ) . n →∞
Illustrative Examples In this section, we solve some nonlinear partial differential equations by using the new modified variational iteration Laplace transform method, therefore we have: Example 1 Consider the following nonlinear partial differential equation u t + uu x = 0 , u (x , 0) = − x (10) Taking Laplace transform of equation (10), subject to the initial condition, we have:
L [u (x , t )] = −
x 1 − L [uu x ] s s
The inverse Laplace transform implies that:
1 u (x , t ) = − x − L −1 L [uu x ] s by the new correction function we find:
1 u n +1 (x , t ) = − x − L−1 L [u n (u n )x ] s Now we apply the new modified variational iteration Laplace transform method,
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u 0 (x , t ) = −x 1 x u 1 (x , t ) = − x − L−1 L [ x ] = −x − L −1 2 = − x − xt s s 1 1 2 2 u 2 (x , t ) = − x − L −1 x 2 + 3 + 4 = − x − xt − xt 2 − xt 3 s s 3 s . . .
. . .
. . .
So we deduce the series solution to be
(
)
u ( x , t ) = − x 1 + t + t 2 + t 3 + ... =
x , t −1
which is the exact solution. Example 2 Consider the following nonlinear partial differential equation
∂u ∂u ∂ 2u = , +u ∂t ∂x ∂x 2 2
u (x , 0) = x 2
(11)
Taking Laplace transform of equation (11), subject to the initial condition, we have: 2 x 2 1 ∂u ∂ 2u L [u (x , t )] = + L + u s s ∂x ∂x 2
Take the inverse Laplace transform to find that:
1 ∂u 2 ∂ 2u u ( x , t ) = x + L L + u ∂x 2 s ∂x 2
−1
The new correction functional is given as
1 ∂u 2 ∂ 2u n u n +1 (x , t ) = x 2 + L−1 L n + u n ∂x 2 s ∂x This is the new modified variational iteration Laplace transform method. The solution in series form is given by:
u 0 (x , t ) = x 2 6x 2 u 1 (x , t ) = x 2 + L −1 2 = x 2 + 6x 2t s u 2 (x , t ) = x 2 (1 + 6t + 36t 2 + 72t 3 ) .
.
.
.
.
.
.
.
.
The series solution is given by
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x2 u (x , t ) = x (1 + 6t + 36t + 72t + ...) = 1 − 6t 2
2
3
Example 3 Consider the following nonlinear partial differential equation 2 ∂u ∂u 2 ∂ u , = 2u + u ∂t ∂x 2 ∂x 2
u (x , 0) =
x +1 2
(12)
Using the same method in above examples to find the new correction functional in the form:
u n +1 (x , t ) =
2 x + 1 −1 1 ∂ 2u n ∂u + L L 2u n n + u n2 2 ∂x 2 ∂x s
Then we have:
x +1 2 x + 1 −1 x + 1 1 x + 1 t u1 (x , t ) = + L 1+ 2= 2 2 2 4 s
u 0 (x , t ) =
u 2 (x , t ) = .
x +1 t 3 2 1 3 1 4 1+ + t + t + t 2 2 8 8 64 . .
.
.
.
.
.
.
Then the series solution is given by:
x +1 t u (x , t ) = 1 + + 2 2
1 3 . 1 2 2 t 2 + .... = x + 1 (1 − t )− 2 , 2! 2
which is the exact solution of equation (12). Example 4 Consider the following nonlinear partial differential equation
∂ 2u ∂u 2 −x + + u − u = te , ∂t 2 ∂x 2
u (x , 0) = 0 ,
∂u (x , 0) = e − x ∂t
(13)
Taking Laplace transform of equation (13), subject to the initial conditions, we have:
s L [u (x , t )] − e 2
−x
2 −x ∂u 2 = L te + u − −u ∂x
Take the inverse Laplace transform to find that:
u (x , t ) = te
−x
2 1 − x ∂u 2 + L 2 L te + u − − u ∂x s −1
The new correct functional is given as
u n +1 (x , t ) = te
−x
2 1 − x ∂u n 2 + L 2 L te + u n − −un ∂x s −1
This is the new modified variational iteration Laplace transform method. 178
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The solution in series form is given by:
u 0 (x , t ) = te − x u 1 (x , t ) = te − x u 2 (x , t ) = te − x .
.
.
.
.
.
The series solution is given by
u (x , t ) = te − x Which is an exactly the same solution obtained by Wazwaz (2006) using the Adomian decomposition method. Conclusion In this paper we applied new method for solving nonlinear partial differential equations. The method is applied in a direct way without using linearization. In fact the proposed new modified variatrional iteration Laplace transform method is successfully implemented by using the initial conditions only and solves nonlinear partial differential equations without using Adomian's polynomials. Acknowledgement This work was funded by the Deanship of Scientific Research (DSR) , King Abdulaziz University, Jeddah, under grant No. (857-006-D1433). The authors, therefore, acknowledge with thanks DSR technical and financial support. References [1] S. lslam, Yasir Khan, Naeem Faraz and Francis Austin, (2010), Numerical Solution of Logistic Differential Equations by using the Laplace Decomposition Method, World Applied Sciences Journal 8 (9):1100-1105. [2] Nuran Guzel and Muhammet Nurulay, (2008), Solution of Shiff Systems By using Differential Transform Method, Dunlupinar universities Fen Bilimleri Enstitusu Dergisi, ISSN 1302-3055, PP. 49-59. [3] Shin- Hsiang Chang, I-Ling Chang, (2008), a new algorithm for calculating one – dimensional differential transform of nonlinear functions, Applied Mathematics and Computation 195, 799808. [4] Tarig M. Elzaki, (2011), The New Integral Transform “Elzaki Transform” Global Journal of Pure and Applied Mathematics, ISSN 0973-1768,Number 1, pp. 57-64. [5] Tarig M. Elzaki & Salih M. Elzaki, (2011), Application of New Transform “Elzaki Transform” to Partial Differential Equations, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768,Number 1, pp. 65-70. [6] Tarig M. Elzaki & Salih M. Elzaki, (2011), On the Connections between Laplace and Elzaki transforms, Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 Volume 6, Number 1, pp. 1-11. [7] Tarig M. Elzaki & Salih M. Elzaki, (2011), On the Elzaki Transform and Ordinary Differential Equation With Variable Coefficients, Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 6, Number 1,pp. 13-18. [8] Lokenath Debnath and D. Bhatta. (2006), Integral transform and their Application second Edition, Chapman & Hall /CRC. [9] A.Kilicman and H.E.Gadain. (2009), An application of double Laplace transform and Sumudu transform, Lobachevskii J. Math.30 (3), pp.214-223.
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