Modified Variational Iteration Method for Solving Fourth Order ...

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1587-1592 © Research India Publications http://www.ripublication.com

Modified Variational Iteration Method for Solving Fourth Order Parabolic PDEs With Variable Coefficients Alla M. Elsheikh and Tarig M. Elzaki Mathematics Department, Faculty of Science, Sudan University of Science and Technology, Mathematics Department, Faculty of Science and Arts, Alkamil, University of Jeddah, Jeddah, Saudi Arabia.

Abstract In this paper, we apply a new Modified Variational Iteration Method (MVIM) to solve one dimensional fourth order parabolic linear partial differential equations with variable coefficients. This method is a combination of the two initial conditions. Keywords: Modified Variational Iteration Method, linear partial differential equation, exact solution.

Introduction Many problems of physical interest are described by linear partial differential equations with initial and boundary conditions. One of them is fourth order parabolic partial differential equations with variable coefficients; these equations emerge on the transverse vibration issue [1]. In recent years, many research workers have paid attention to find the solution of these equations by using various methods. Among these are the variational iteration method [Biazar and Ghazvini (2007)], Adomian decomposition method [Wazwaz (2001) and Biazar etal (2007)], homotopy perturbation method [Mehdi Dehghan and Jalil Manafian (2008)], homotopy analysis method [Najeeb Alam Khan (2010)] and Laplace decomposition algorithm [Majid Khan, Muhammad Asif Gondal and Yasir Khan (2011)]. In this paper, we use the Modified Variational Iteration Method. This method is a useful technique for solving linear and nonlinear differential equations. The principle point of this paper is to integrate initial conditions for solving higher order linear partial differential equations with variable coefficients. This method gives the solution as convergent series leads to the exact solution.

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Alla M. Elsheikh and Tarig M. Elzaki

Modified Variational Iteration Method: Consider a one dimensional linear, non-homogeneous fourth order parabolic partial differential equation with variable coefficients of the form,  2u  4u (1)   ( x )   ( x ,t ), t 2 x 4 Where,  ( x ) is a variable coefficient, with the following initial conditions, u ( x, 0)  h( x), (2) u (x ,0)  f (x ) , and t and the boundary conditions are,  4u  4u (3) u (a,t )  1 (t ),u (b ,t )  2 (t ), ( a , t )   ( t ), (b ,t )   4 (t ), 3 x 4 x 4 Apply modified variational iteration method of Eq. (1), t   2u ( x ,  )   4u u n 1 ( x ,t )  u n ( x ,t )    ( )  n 2   ( x ) 4n ( x ,  )   ( x ,  )  d  , (4) x    0

Where  is a Lagrange multiplier (    t ), the subscripts n denote the n th approximation, u n is considered as a restricted variation, i.e. u n  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 

Applications To show that the method is effected, we have solved homogeneous and nonhomogeneous one dimensional fourth order linear partial differential equations with the initial and boundary condition. Example 1. Consider the fourth order homogenous partial differential equations, 1  2u  1 x 4   4u    4  0,  x  1,t  0 2 2 t  x 120  t With the following initial conditions, u x5 u ( x ,0)  0, ( x ,0)  1  , t 120 and the boundary conditions,

(5)

(6)

Modified Variational Iteration Method for Solving Fourth Order Parabolic

 (0.5)5  121 u (0.5, t )   1  sin t ,  sin t ,u (1,t )  120  120   2u  2u 1 (0.5, t )  0.02084sin t , (1, t )  sin t . 2 2 t t 6 Appling Modified Variational Iteration Method to Eq.(5), we get: t   2u ( x ,  )  1 x 4   4u n  u n 1 ( x ,t )  u n ( x ,t )    ( )  n 2    4 (x ,  ) d  ,  x 120  x 0    take     t , then:

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(7)

(8)

t   2u ( x ,  )  1 x 4   4u n  u n 1 ( x ,t )  u n ( x ,t )   (  t )  n 2    4 (x ,  ) d  ,  x 120  x 0    5  x  take, u 0 ( x ,t )  u ( x ,0)   1   t , then:  120  t  1 x 4    x5  u1 ( x , t )   1  t  (   t )      x  d   120  0  x 120  

 x5   x 5 t 3  1  t  1      120   120  3! t    x5   x 5 t 3 x 5   1 x 4   3  u 2 ( x ,t )  1  t  1     (  t )    1  t     x   x  d  3!    120   120  3! 0   120   x 120 

 x5   x 5 t 3  x 5 t 5  1  t  1   1        120   120  3!  120  5!  x5   x 5 t 3  x 5 t 5 u 3 ( x ,t )  1  t  1    1     120   120  3!  120  5!

 x 5  t 3   1 x 4  3  5  (   t ) 1   t    x   x  x       d  0 120 6 x 120 3! 5!       t

 x5   x 5 t 3  x 5 t 5  x 5 t 7  1  t  1   1   1          120   120  3!  120  5!  120  7! :  x5   x 5 t 3  x 5 t 5  x 5 t 7 u n (x ,t )  1  t  1   1   1          ...  120   120  3!  120  5!  120  7!

  x 5  t 3 t 5 t 7  1    t     ...   120   3! 5! 7! 

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The exact solution is:  x5  u (x ,t )  1   sin t  120  Example 2. Consider fourth order homogenous partial differential equation, 4  2u  x u   1 4  0, 0  x  1,t  0 t 2  sin x  x with the following initial conditions, u u ( x ,0)  x  sin x , ( x ,0)  x  sin x t And the boundary conditions, u (0, t )  0,u (1, t )  e t (1  sin1),  2u  2u (0, t )  0, (1, t )  e t sin1. 2 2 t t Solution: Appling Modified Variational Iteration Method to Eq.(10), we get: t 4   2u ( x ,  )  x   u n (x ,  )  u n 1 ( x ,t )  u n ( x ,t )    ( )  n 2   1 d  4  sin x  x    0 take,     t , then: t 4   2u ( x ,  )  x   u n (x ,  )  u n 1 ( x ,t )  u n ( x ,t )   (  t )  n 2   1 d  4   sin x  x     0 Using initial conditions from Eq. (11), we get: u 0 ( x , t )  ( x  sin x )  (  x  sin x )t

4   2u ( x ,  )  x   u 0 (x ,  )  u1 ( x , t )  u 0 ( x , t )   (  t )  0 2   1 d  4  sin x  x    0 t

t 2 t 3   ( x  sin x )  (  x  sin x )t  ( x  sin x )     2! 3!  4   2u ( x ,  )  x   u (x ,  )  u 2 ( x , t )  u1 ( x , t )   (  t )  1 2   1  1 4 d   sin x  x    0 t

t 2 t 3  t 4 t5   ( x  sin x )  (  x  sin x )t  ( x  sin x )     ( x  sin x )     2! 3!   4! 5!  : t 2 t 3  t 4 t 5  u n ( x ,t )  ( x  sin x )  (  x  sin x )t  ( x  sin x )     ( x  sin x )     ...  2! 3!   4! 5!    t2 t3 t4 t5  ( x  sin x ) 1  t      ... 2! 3! 4! 5!  

(9)

(10)

(11)

(12)

Modified Variational Iteration Method for Solving Fourth Order Parabolic

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The exact solution: u (x ,t )  (x  sin x )e t Example 3. Consider fourth order homogenous partial differential equation,  2u  4u  4 6  (13)  (1  x ) 4   x  x 3  x 7  cos t , 0  x  1,t  0 2 t x 7!   with the following initial conditions, 6 u u ( x ,0)  x 7 , ( x ,0)  0. (14) 7! t And the boundary conditions, 6 u (0, t )  0,u (1, t )  cos t , 7! (15) 2 2 u u 1 (0, t )  0, 2 (1, t )  cos t . t 2 t 20 Solution: Appling Modified Variational Iteration Method to Eq. (13), to find: t   2u ( x ,  )   4u ( x ,  )  4 6  u n 1 ( x ,t )  u n ( x ,t )    ( )  n 2  1  x  n 4   x  x 3  x 7  cos  d  x 7!      0 take,     t , then:   2u n ( x ,  )   4u n ( x ,  )  4 6  u n 1 ( x ,t )  u n ( x ,t )   (  t )   1  x    x  x 3  x 7  cos  d  2 4 x 7!      0 Using initial conditions from Eq. (14), to get: 6 u 0 ( x ,t )  x 7 , 7! t   2u 0 ( x ,  )   4u 0 ( x ,  )  4 6  u1 ( x ,t )  u 0 ( x ,t )   (  t )   (1  x )   x  x 3  x 7  cos  d  2 4 x 7!      0 t

6 7 t2 x cos t  ( x 4  x 3 )   x 4  x 3  (1  cos t ) 7! 2! 6  x 7 cos t   x 4  x 3  (1  cos t )  noiseterms 7! t   2u ( x ,  )   4u ( x ,  )  6  u 2 ( x ,t )  u1 ( x ,t )   (  t )  1 2  (1  x ) 1 4   x 4  x 3  x 7  cos  d  x 7!      0 

6 7 t2 x cos t  (1  x ) 7! 2! 6 7  x cos t  noiseterms 7! : 

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6 7 x cos t 7! This is the approximate solution. u n ( x ,t ) 

Conclusion In this paper, we applied modified variational iteration method for solving one dimensional fourth order homogenous and non-homogenous linear partial differential equations with variable coefficients. The modified variational iteration method is successfully implemented by using the initial conditions only. And give the exact or approximate solution to the equation.

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Majid Khan, Muhammad Asif Gondal and Yasir Khan, An Efficient Laplace Decomposition algorithm for Fourth order Parabolic Partial Differential Equations with variable coefficients, World Applied Sciences Journal 13 (12), 2011, pp2463-2466. Tarig M. Elzaki and Eman M. A. Hilal, Homotopy Perturbation and ELzaki Transform for solving Nonlinear Partial Differential equations, Mathematical Theory and Modeling, 2(3), 2012, pp33-42. Tarig M. Elzaki and Salih M. Elzaki, Applications of New Transform “ELzaki Transform” to Partial Differential Equations, Global Journal of Pure and Applied Mathematics, (7)1, 2011, pp65-70. Hradyesh Kumar Mishra and Atulya K Nagar, He-Laplace Method for Linear and Nonlinear Partial Differential Equations, Journal of Applied Mathematics, Vol.2012, Article ID 180315, 16 pages. Mehdi Dehghan and Jalil Manafian, The Solution of the Variable Coefficients Fourth-Order Parabolic Partial Differential Equations by the Homotopy Perturbation Method, Verlag der Zeitschrift fur Naturforschung, Tu¨bingen, 64a, 2009, pp420-430. M. Hussain and Majid Khan, Modified Laplace Decomposition Method, Journal of Applied Mathematical Sciences, 4(36), 2010, pp 1769-1783. Najeeb Alam Khan, Asmat Ara, Muhammad Afzal and Azam Khan, Analytical Aspect of Fourth order Parabolic Partial differential Equations with variable coefficients, Mathematical and Computational Applications, 15(3), 2010, pp. 481-489 Prem Kiran G. Bhadane and V. H. Pradhan, Elzaki Transform Homotopy Perturbation Method for Solving Fourth Order Parabolic PDE with Variable Coefficients, Journal of Engineering Research and Applications, 6(24), 2013, pp.2170-2173.