Double Laplace Variational Iteration Method for

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these problems such as Backlund transformation [3], Hirota's bilinear method [4, 5], ... transform to solve a new type of equations called nonlinear convolution ...
Archives Des Sciences

Vol 65, No. 12;Dec 2012

Double Laplace Variational Iteration Method for Solution of Nonlinear Convolution Partial Differential Equations Tarig. M. Elzaki 1

Mathematics Department, Faculty of Sciences and Arts-Alkamil, King Abdulaziz University, JeddahSaudi Arabia. Mathematics Department, Faculty of Sciences, Sudan University of Sciences and Technology-Sudan. E-mail: [email protected]

Abstract In this paper we combine double Laplace transform and modified variational iteration method to solve new type of nonlinear partial differential equation called nonlinear convolution partial differential equations, it is possible to find the exact solutions or better approximate solutions of these equations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. This method is used for solving nonlinear convolution partial differential equations. The solutions obtained by this method show the accuracy and efficiency of the method. Keywords Modified Variational Iteration Method, Nonlinear Convolution Partial Differential Equations, Lagrange Multiplier, Double Laplace Transform. 1. Introduction It is well known that there are many nonlinear partial differential equations which are used in the study of several fields for example physics, mechanics, etc. The solutions of these equations can give more understanding of the described process. But because of the complexity of the nonlinear partial differential equations and the limitations of mathematical methods, it is difficult to obtain the exact solutions for these problems. Thus, this complexity hinders further applications of nonlinear partial differential equations [1, 2]. A broad class of analytical and numerical methods were used to handle these problems such as Backlund transformation [3], Hirota’s bilinear method [4, 5], Darboux transformation [6], Symmetry method [7], the inverse scattering transformation [8], the tanh method [9, 10], the Adomian decomposition method [11, 12], the improved Adomian decomposition method [13], the exp-function method [14] and other asymptotic methods [15] for strongly nonlinear equations. In 1978, Inokuti et al. [16] proposed a general use of Lagrange multiplier to solve nonlinear problems, which was intended to solve problems in quantum mechanics. Subsequently, in 1999, the variational iteration method (VIM) was first proposed by Ji-Huan He [17, 18]. The idea of the VIM is to construct an iteration method based on a correction functional that includes a generalized Lagrange multiplier. The value of the multiplier is chosen using variational theory so that each iteration improves the accuracy of the solution. In this paper, we have applied the modified variational iteration method (VIM) and double Laplace transform to solve a new type of equations called nonlinear convolution partial differential equation, The VIM reduced the size of calculations without any need to transform the nonlinear terms. And double Laplace transform reduce double convolution terms, It is obvious that the method gives rapidly convergent successive approximations through determining the Lagrange multiplier. He’s variational iteration method gives several successive approximations by using the iteration of the correction functional. This method uses the initial values for selecting the zeroth approximation. The results obtained confirm the accuracy and efficiency of the method.

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Vol 65, No. 12;Dec 2012

Definition Let F1 (x , y ) defined as

, F2 (x , y ) be integrable functions, then the convolution of F1 (x , y ) , F2 (x , y ) is y x

F1 (x , y )  F2 (x , y )    F1 (x   , y   ) F2 ( , )d  d  0 0

and known as double convolution with respect to The Laplace transform is defined as

x and y see [25, 26]. 

L 2  f (x , t )   F ( p , s )   e

 px

0



e

 st

f (x , t )dtdx

0

Where x , t  0 and p , s are complex values. And further the double Laplace transform of first and second partial derivatives are given by

 f (x , t )   f (x , t )  (i ) L 2   pF ( p , s )  F (0, s ) (ii ) L 2   sF ( p , s )  F ( p , 0)   x   t    2 f (x , t )  F (0, s ) (iii ) L 2   p 2 F ( p , s )  pF (0, s )   2 x  x    2 f (x , t )  F ( p , 0) (vi ) L 2   s 2 F ( p , s )  sF ( p , 0)   2 t  t 

2-Variational Iteration Method (VIM) Consider the partial differential equation

L u (x , t )   R u (x , t )   N u (x , t )   0

(1)

With the initial conditions

u (x , 0) (2)  g (x , t ) t Where L is a linear operator, R is a linear operator less than L and N is the nonlinear operator. And N u (x , t )  , R u (x , t )  don't have partial derivatives with respect to t . u (x , 0)  h (x )

,

According to variational iteration method, we can construct a correction functional ac follows t

u n 1 (x , t )  u n (x , t )    ( )  Lu n (x ,  )  Run (x ,  )  Nun (x ,  ) d 

(3)

0

 is a Lagrange multiplier, Run (x ,  ) , Nun (x ,  )  Run  0 ,  Nun  0 and   1 .

are considered as restricted variations, i.e

Then the variational iteration formula can be obtained as t

u n 1 (x , t )  u n (x , t )    Lu n (x ,  )  Ru n (x ,  )  Nu n (x ,  ) d 

(4)

0

The second term on the right is called the correction term. Eq (4) can be solved iteratively using u 0 (x , t ) as the initial approximation, u n 1 (x , t ) , n  0 . Then the solution is

u (x , t )  lim u n (x , t ) n 

3- Modified Variational Iteration Double Laplace Transform Method (MVIDLTM) Consider the nonlinear convolution partial differential equation 589

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Vol 65, No. 12;Dec 2012

L u (x , t )   R u (x , t )   N



u (x ,t )  0

With the initial conditions

u (x , 0)  h (x ) Where

L

nonlinear

N



u (x , 0)  g (x , t ) t

,

is a linear operator, convolution

(5) (6)

R is a linear operator less than L , and N  u (x , t )  is the

term,

and

in

this

paper

we

assume

that

k

  m u (x , t )   , where a , n , m , k are positive integers. m  x 

u (x , t )  au n (x , t ) ** 

Also in this work we assume that L



2 . t 2

Take double Laplace transform on both sides of eq (5), to find k    m u (x , t )   n L 2  Lu (x , t )   L 2  Ru (x , t )   L 2 au (x , t ) **   0 m   x   k    m u (x , t )   F ( p , 0) 2 n s F ( p , s )  sF ( p , 0)   L 2 Ru (x , t )  au (x , t ) **    m t  x   

(7)

(8)

And the single Laplace transform of initial conditions given by

F ( p , 0)  H ( p ) , Then equation (8) becomes

F ( p , 0) G (p) t

k    m u (x , t )   H (p) G (p) 1 n F (p,s )   2  2 L 2 Ru (x , t )  au (x , t ) **    m s2 s s  x   

(9)

Appling the double inverse Laplace transform to eq (9) to obtain

k 1    m u (x , t )   n (10) u (x , t )  k (x , t )  L  2 L 2 Ru (x , t )  au (x , t ) **    m  x  s      Where k ( x , t ) represents the terms arising from the source term and the prescribed initial conditions. Now the first derivative with respect to t of eq (10) is given by k    mu (x , t )      1  1 n u (x , t )  k (x , t )  L 2  2 L 2 Ru (x , t )  au (x , t )**     0 (11) m t t t  s  x    1 2

By the correction function of the irrational method we have

u n 1 (x , t )  u n (x , t ) k    m u (x ,  )      1  1   n   (u n ) (x ,  )  k (x ,  )  L 2  2 L 2 Ru n (x ,  )  au (x ,  ) **    d    x m     s  0      t

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Then the new correction function (new modified VIM) is given by k 1    m u (x , t )   n u n 1 (x , t )  u n (x , t )  L  2 L 2 Ru n (x , t )  au (x , t ) **    , n  0 (12) m  x  s     In the last we find the solution in the form u ( x , t )  lim u n ( x , t ) 1 2

n 

Where double inverse Laplace transform exist. First, in particular consider the second order nonlinear partial differential equation with double convolution term. 2

 2u u (x , 0)  u   u 2  (x  4)u  4u   0   0 , u (x , 0)  x , 2 t t  x 

(13)

Take double Laplace transform of eq (13), and single Laplace transform of initial conditions, we have 2  2 1 2 1  u     L u  4 u  xu  4 u  2     sp 2 s 2 p 2 s 2   x   1  2 Where U ( p , 0)  2 , U ( p , 0)  2 p t p

U (p,s ) 

Take double inverse Laplace transform to find that

u (x , t )  x  2xt  L 21

1 s2

2   u   2 L u  4 u  xu  L 4 u L  2   2  x    2  

By using eq (12), we have the new correction function

u n 1 (x , t )  u n (x , t )  L 21

1 s2

2   u n   2 L u  4 u  xu  4 L u L  2 n n n 2 n 2    x   

(14)

Then we have

u 0 (x , t )  x  2xt L 2  x  2xt 2  4L 2  x  2xt       2 2 2 L 2  x  2x t   L 2  4x  8xt  L 2 1  2t   8 8 2 4 64 2 5 64 2 6 u1 (x , t )  x  2xt  2xt 2  xt 3  x t  x t  x t 3! 2.4! 2.5! 2.6! 1 u1 (x , t )  x  2xt  L 21 2 s

And so on, then the exact solution of eq (13) is

u (x , t )  x  2xt  2xt 2 

8 3 xt  ....  xe 2t 3!

Second, consider the first order convolution partial differential equation

u u u 3  u    x  0 , u (x , 0)  0 t x 12x

(15)

Take double Laplace transform of eq (13), and single Laplace transform of initial condition, we have

sU ( p , s )  591

 u3 1 u   L  u   2  2 sp x  12x ISSN 1661-464X

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Take double inverse Laplace transform to obtain

1   u 3   u u (x , t )  xt  L 21 L 2    L 2 u  L 2  s   12x   x

    

Using eq (12) to find the new correction function in the form

1   u 3 u n 1 (x , t )  u n (x , t )  L 21 L 2  n s   12x Then we have

  u n   L 2 u n  L 2   x 

    

(16)

u 0 (x , t )  xt 1   x 2t 3   u1 (x , t )  xt  L 21 L 2    L 2  xt  L 2 t    s   12  

 1   1  1 1 xt  L 21  3 4   2 2   2    xt s  p s  p s   ps   u 0 (x , t )  u 1 (x , t )  u 2 (x , t )........  u n (x , t )  xt Then the exact solution of eq (13) is u ( x , t )  xt Conclusion In this paper we introduce new equations called convolution nonlinear partial differential equations, and solve it by using combine double Laplace transform and new modified variatrional iteration method. The method is applied in a direct way without using linearization. This method is successfully implemented by using the initial conditions only and solves nonlinear convolution partial differential equations without using Adomian's polynomials. References [1] J. Hu, H. Zhang, A new method for finding exact traveling wave solutions to nonlinear partial differential equations, Phys. Lett. A 286 (2001) 175–179. [2] L. Debnath, Nonlinear Water Waves, Academic Press, Boston, 1994. [3] M.R. Miura, Backlund Transformation, Springer-Verlag, Berlin, 1978. [4] R. Hirota, in: RK Bullogh, PJ Caudrey (Eds.), Direct Methods in Soliton Theory, in: Solitons, Springer, Berlin, 1980. [5] R. Hirota, Exact envelope–soliton solutions of a nonlinear wave, J. Math. Phys. 14 (7) (1973) 805– 809. [6] C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux Transformation in Solitons Theory and Geometry Applications, Shangai Science Technology Press, Shanghai, 1999. [7] P.J. Olver, Application of Lie Group to Differential Equation, Springer, New York, 1986. [8] M.J. Ablowitz, P.A. Clarkson, Solitons: Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [9] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, J. Comput. Appl. Math. 164 (2004) 529–541. [10] A.M. Wazwaz, The tanh method for traveling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (3) (2004) 713–723. [11] G. Adomian, Solving Frontier Problem of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994. [12] G. Adomian, R. Rach, Explicit solutions for differential equations, Appl. Math. Comput. 23 (1) (1987) 49–59. [13] T.A. Abassy, Improved Adomian decomposition method, Comput. Math. Appl. 59 (1) (2010) 42– 54.

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[14] S. Zhang, Exp-function method for constructing explicit and exact solutions of a lattice equation, Appl. Math. Comput. 199 (1) (2008) 242–249. [15] M.A. Helal, M.S. Mehanna, A comparison between two different methods for solving KdV– Burgers equation, Chaos Solitons Fractals 28 (2) (2006) 320–326. [16] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, 1978, 156–162. [17] J.H. He, Variational iteration method kind of non-linear analytical technique: Some examples, Int. J. Non-Linear Mech. 34 (1999) 699–708. [18] J. H. He, Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 235–236. [19] A. M. Wazwaz, A study on linear and nonlinear Schrodinger equations by the variational iteration method, Chaos, Solitons and Fractals 37 (2008) 1136–1142. [20] S. Duangpithak and M. Torvattanabun, Variational Iteration Method for Solving Nonlinear Reaction-Diffusion-Convection Problems, Appl. Math. Sci., 6 (17) (2012) 843-849. [21] Abdul Majid Wazwaz, A comparison between the variational iteration method and Adomian decomposition method, Journal of Computational and Applied Mathematics 207 (2007) 129 – 136. [22] H. Jafari, C.M. Khalique, M. Khan and M. Ghasemi, A two-step Laplace decomposition method for solving nonlinear partial differential equations, International Journal of the Physical Sciences 6 (2011) 4102-4109. [23] Kilicman, A. and H. E. Gadain, 2009. An Application of Double Laplace Transform and Double Sumudu Transform, Lobachevskii Journal of Mathematics, 2009, Vol. 30, No. 3, pp. 214--223. [24] Kiliçman, A. and Eltayeb, H. 2009. On the Second Order Linear Partial Differential Equations with Variable Coefficients and Double Volume 34(2), pp. 257--270. [25] Kiliçman, A. and Eltayeb, H. 2007. A note on the classifications of hyperbolic and elliptic equations with polynomial coefficients, Appl. Math. Lett, 21:1124-1128. Lamb, G. L. Jr. 1995. Introductory Applications

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