Projected Differential Transform Method and Elzaki

World Applied Sciences Journal 32 (9): 1974-1979, 2014 ISSN 1818-4952 © IDOSI Publications, 2014 DOI: 10.5829/idosi.wasj.2014.32.09.1253

Projected Differential Transform Method and Elzaki Transform for Solving System of Nonlinear Partial Differential Equations 1

1

Tarig M. Elzaki and 2Badriah A.S. Alamri

Department of Mathematics, Faculty of Sciences and Arts-Alkamil, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics Department, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

Abstract: In this paper we solve nonlinear system of partial differential equations using the new integral transform "Elzaki transform" and projected differential transform method together. The purpose of the method is obtained analytical or approximate solutions of some nonlinear system of partial differential equations. This method is more efficient and easy to handle such partial differential equations in comparison to other methods. The result showed the efficiency, accuracy and validation of the combined Elzaki transform and projected differential transform method. Key words: Projected differential transform method differential equations INTRODUCTION

∞

∫

= E [ f (t )] v f (t ) e

Several numerical and analytical techniques including the Adomian's decomposition method [1], differential transform method [2-7] and homotopy perturbation method [8-12], have been developed for solving nonlinear partial differential equations. Elzaki transform is a powerful tool for solving some differential equations which cannot solve by Sumudu or Laplace transforms in [13]. In this paper we will present the combined Elzaki transform and projected differential transform method, we apply this method to solve some examples of nonlinear system of partial differential equations, this method has been widely used in solving nonlinear problems to overcome the shortcoming of other methods. Elzaki Transform: The basic definitions of modified of Sumudu transform or Elzaki transform is defined as follows, Elzaki transform of the function f(t) is;

Corresponding Author:

Elzaki transform −

Nonlinear system of partial t v dt

,

t >0

0

Tarig M. Elzaki and Sailh M. Elzaki in [14-17], showed the modified of Sumudu transform or Elzaki transform was applied to partial differential equations, ordinary differential equations, system of ordinary and partial differential equations and integral equations. In this paper, we combined Elzaki transform and projected differential transform methods to solve nonlinear system of partial differential equations. To obtain Elzaki transform of partial derivative we use integration by parts and then we have:  ∂f ( x, t )  1 E = T ( x, v) − vf ( x,0),   ∂t  v  ∂ 2 f ( x, t )  1 ∂f ( x,0) E . = 2 T ( x, v) − f ( x,0) − v 2 ∂t ∂ t v  

Proof: To obtain ELzaki transform of partial derivatives we use integration by parts as follows:

Tarig M. Elzaki, Department of Mathematics, Faculty of Sciences and Arts-Alkamil, King Abdulaziz University, Jeddah-Saudi Arabia.

1974

World Appl. Sci. J., 32 (9): 1974-1979, 2014 p   −t  p −t p −t −t  ∞ ∂f ∂f  v  T ( x, v )  ∂f  v v v  Ε  ( x, t )  = v e dt = − vf ( x,0 ) . dt = lim  ve f ( x, t ) − e f ( x, t )dt  = lim ve  p →∞ p →∞   v ∂t  ∂t  0 ∂t  0 0   0  

∫

∫

∫

We assume that f is piecewise continuous and it is of exponential order.  2 By the same method we find: Ε  ∂ f 2  ∂x

 d2  = 2 T ( x, v )  .  dx

 2  To find: Ε  ∂ f ( x, t )  Let ∂f = g , then we have: 2 ∂t  ∂t   ∂2 f   g ( x, t )  ∂g ( x , t )  Ε  2 ( x, t )  = Ε Ε − vg ( x,0 ) , = v  ∂t   ∂t   ∂2 f  1 ∂f Ε  2 ( x, t )= 2 T ( x , v ) − f ( x , 0 ) − v ( x ,0 ) . ∂t t v ∂  

We can easily extend this result to the nth partial derivative by using mathematical induction. Projected Differential Transform Methods: In this section we introduce the projected differentia transform method [5, 18] which is modified method of the differential transform method. Definition: The basic definition of projected differential transform method of function f(x1, x2,...,xn) is defined as;   1  ∂ k f ( x1, x2 , , xn )  f ( x1, x2 ,, xn −1, k ) =   k! ∂xnk    xn = 0

(1)

Such that f ( x1, x2 ,, xn ) is the original function and f ( x1, x2 ,, xn −1, k ) is projected transform function. And the differential inverse transform of f ( x1, x2 ,, xn −1, k ) is defined as; = f ( x1, x2 ,, xn )

∞

∑ f ( x1, x2,., xn−1, k )( x − x0 )k

k =o

(2)

The fundamental theorems of the projected differential transform are: Theorems:

(1) ( 2)

If z= ( x1, x2 ,......, xn ) u ( x1, x2 ,......, xn ) ± v ( x1, x2 ,......, xn )

Then z= ( x1, x2 ,......, xn −1, k ) u ( x1, x2 ,......, xn −1, k ) ± v ( x1, x2 ,......, xn −1, k ) Ifz ( x1, x2 ,......, xn ) = c u ( x1, x2 ,......, xn )

Then z ( x1, x2 ,......, xn −1, k ) = cu ( x1, x2 ,......, xn −1,k )

1975

World Appl. Sci. J., 32 (9): 1974-1979, 2014

( 3)

Ifz ( x1, x2 ,......, xn ) =

du ( x1, x2 ,......, xn ) dxn

Then z ( x1, x2 ,......, xn −1, k ) = ( k + 1) u ( x1, x2 ,....., xn −1, k + 1)

( 4 ) Ifz ( x1, x2 ,......, xn ) =

d n u ( x1, x2 ,......, xn ) dxnn

Then z ( x1, x2 ,......, xn −1, k ) =

( 5) If

( k + n )! u k!

( x1, x2 ,....., xn −1, k + n )

z ( x1, x2 ,......, xn ) = u ( x1, x2 ,......, xn ) v ( x1, x2 ,......, xn ) k

∑ u ( x1, x2 ,......, xn−1, m ) v ( x1, x2 ,......, xn−1, k − m )

= Then z ( x1, x2 ,......, xn −1, k )

m=0

( 6 ) If z ( x1, x2 ,......, xn ) = u1 ( x1 , x2 ,......, xn ) u2 ( x1, x2 ,......, xn ) .....un ( x1, x2 ,......, xn ) Then z ( x1, x2 ,......, xn −1, k )

k

kn−1

∑ ∑

kn= −1 0 k n= −2 0

......

k3

k2

∑ ∑ u1 ( x1, x2 ,......, xn −1, k1 ) u2 ( x1, x2 ,......, xn−1, k2 − k1 )

= k 2 0= k1 0

×.....un −1 ( x1 , x2 ,......, xn −1, kn −1 − kn − 2 ) un ( x1, x2 ,......, xn −1, k − k n −1 )

(7)

q

q

If z ( x1, x2 ,......, xn ) = x1 1 x2 2 ........xnqn Then

z ( x1= , x2 ,......, xn −1, k )

= xn −1, qn − k ) ( x1, x2 ,......,

Note that is a constant and is a nonnegative integer.

 1 k = qn   0 k ≠ qn

where,

= Am +1 Applications: To illustrate the capability and simplicity of the method, some examples for nonlinear system of partial differential equations will be discussed. = C m +1

Example 2.1: Consider the following system of nonlinear partial differential equations; 1 U t + VU x + U =  Vt + UVx − V = −1

∑ h

∂V ( x, h − m) = U ( x, m) , Dm +1 V ( x, m). ∂x m=0

∑

(5)

From equations (4) and (5), we have:

(3)

A1 == 1, B1 e x , C1 = −1 , D1 = e− x , U ( x,1) = −te x , V ( x,1) = te − x , A2 = 0, B2 = −te x , C2 = 0 , D2 = te − x ,

With the initial conditions; x = U ( x,0) e= , V ( x,0) e− x

h

∂U ( x, h − m) = V ( x, m ) , Bm +1 U ( x, m), ∂x m=0

t2 x t2 −x = e e , , V ( x,2) 2 2 Then the solution of equation (3) is;

= U ( x, 2)

Taking Elzaki transform of equations (3) subject to the initial conditions, we have:  E [U ( x, t )] = v 2e x − vE [VU x + U − 1]  2 −x  E [V ( x, t )] = v e − vE [V xU − V + 1]

U ( x= , t ) U ( x,0) + U ( x,1) + U ( x,2) + = ... e x −t . , t ) V ( x,0) + V ( x,1) + V ( x,2) += ... et − x . V ( x=

Example 2.2: Consider the following system of nonlinear partial differential equations;

The inverse Elzaki transform implies that: U ( x, m + 1) = ex − E −1{vE [ Am+1 + Bm+1 − (h)]} , U ( x,0) = ex   −x −1 −x V ( x, m + 1) = e − E {vE [Cm+1 − Dm+1 + (h)]} , V ( x,0) = e

(4) 1976

U t + VxW y − V yWx = −U  V Vt + WxU y + W yU x =  W Wt + U xVy + U yVx =

(6)

World Appl. Sci. J., 32 (9): 1974-1979, 2014

With the initial conditions;

= U ( x, y , t )

+y x− y = U ( x, y,0) e x= , V ( x, y,0) e= , W ( x, y,0)

W ( x, y , t ) =

∑ V ( x, y , m),

∑ W ( x, y , m )

m=0

From equations (7) and (8) we find that:

} } }

A1 = 1 , B1 = 1 , C1 = ex+ y , D1 = −e2 y , E1 = e2 y ,

U (x , y, t ) = e x + y − E −1 vE  −V W + V W + U  x y  y x    x− y −1 − E vE WxU y + WyU x − V  V ( x, y , t ) =e  W ( x, y, t ) = e− x + y − E −1 vE U xV y + U yVx − W    

{

∑

∞

m 0= m 0 = e− x + y ∞

Taking Elzaki transform of equations (6) subject to the initial conditions and then take the inverse Elzaki transform we have:

{ {

∞

= U ( x, y , m), V ( x, y , t )

F1 = ex− y , G1 = −e2x , H1 = e2x , I1 = e− x+ y , U (x, y,1) = −tex+ y , V (x, y,1) = tex− y ,W (x, y,1) = te− x+ y , A2 = 2t , B2 = 2t , C2 = −tex+ y , D2 = 0, E2 = 0,

= F2 te x− y ,= G2 0,= H2 0,= I2 te− x+ y , Applying projected differential transform method to t 2 x+ y t 2 x− y t2 obtain: = U (x, y,2) = e , V ( x, y,2) e , W (x, y,2) = e− x+ y , 2 2 2 . . . U(x, y, m +1) = ex+ y −E−1{vE[ −Am+1 + Bm+1 + Cm+1]} , U(x, y,0) = . . .   ex−y −E−1{vE[ Dm+1 + Em+1 − Fm+1]} , V (x, y,0) = V (x, y, m +1) = . . .  −1 −x+ y e −E {vE[Gm+1 + Hm+1 − Im+1]} , W(x, y,0) = W(x, y, m +1) = Then the solutions U(x, y, t), V(x, y, t), W(x, y, t) are (7) given by: where, U (x= + .... e x + y −t . , y, t ) U (x , y,0) + U ( x, y,1) + U ( x, y, 2)=

h

 ∂V ( x, y , m) ∂W ( x, y , h − m) ,  Am +1 = ∂y ∂x  m=0  h  ∂V ( x, y , m) ∂W ( x, y , h − m) B =  m +1 ∂x ∂y  m=0 C = U ( x, y , m ),  m +1 h  ∂W ( x, y , m) ∂U (x , y , h − m)  Dm +1 = ,  ∂x ∂y m=0  h  ∂W ( x, y , m) ∂U (x , y , h − m) ,  Em +1 = ∂y ∂x  m=0 F  m +1 = V ( x, y , m), h  ∂U (x , y, m) ∂V ( x, y , h − m) Gm +1 = ,  ∂x ∂y m=0  h  ∂U ( x, y , m) ∂V ( x, y , h − m) ,  H m +1 = ∂y ∂x  m =0   I m +1 = W ( x, y , m).  

V ( x= + .... e x − y +t . , y, t ) V ( x, y,0) + V ( x, y,1) + V ( x, y, 2)=

∑

W ( x= + .... e− x + y +t . , y, t ) W ( x, y,0) + W ( x , y,1) + W ( x, y,2)=

∑

Example 2.3: Consider the following Coupled Burger's system,

∑

0 U t − U xx − 2UU x + (UV ) x =  0 Vt − Vxx − 2VVx + (UV ) x =

∑

∑

(9)

With the initial conditions; = U ( x,0) sin = x , V ( x,0) sin x.

Using the same method in example 1 to obtain:

∑

U ( x, t ) = sin x + E −1 {vE [U xx + 2UU x − UVx − U xV ]}   −1 V ( x, t ) = sin x + E {vE [Vxx + 2VVx − UVx − U xV ]}

(8)

The standard Elzaki transform defines the solutions U(x, y, t), V(x, y, t), W(x, y, t) by the series;

Applying projected differential transform method to obtain: 1977

World Appl. Sci. J., 32 (9): 1974-1979, 2014

U(x= , m +1) E−1{vE[ Am+1 + Bm+1 − Cm+1 − Dm+1]} , U = (x,0) sin x

3.

(x, m +1) E−1{vE[ Em+1 + Fm+1 − Cm+1 − Dm+1]= V= } , V (x,0) sin x

(10)

4.

h  ∂2U(x, m) ∂U(x, h − m) = , B 2 U(x, m) , Am+1 = 1 + m 2 ∂x ∂x  m=0  h  ∂V (x, h − m) ∂2V (x, m) U(x, m) , Em+1 , Cm+1 = ∂x ∂x2  m=0  h h ∂U(x, h − m) ∂V(x, h − m)  V (x, m) , , Fm+1 2 V (x, m) Dm+1 = x ∂ ∂x = m 0= m 0  (11) From equations (10) and (11), we have:

5.

where,

∑

∑

∑

6.

∑

7.

2sin x cos x , C1 = sin x cos x, A1 = B1 = − sin x , D1 = sin x cos x, F1 = 2sin x cos x , E1 = − sin x, , V ( x,1) = U ( x,1) = −t sin x −t sin x, A2 = t sin x , B2 = −4t sin x cos x , C2 = − 2sin x cos x ,

8.

D2 = t sin x, −2t sin x cos x, F2 = −4t sin x cos x , E2 = t2 t2 sin x , V ( x,2) sin x, = 2 2 .

.

. . . . equations (9) are; Then the solutions of

. .

U ( x, 2) = .

U ( x= , t ) U ( x,0) + U ( x,1) + U ( x, 2) += .... e −t sin x,

9.

10.

V ( x= , t ) V ( x,0) + V ( x,1) + V ( x, 2) + = .... e −t sin x,

CONCLUSION In this paper, the authors presented the combined Elzaki transform and projected differential transform method. This method considerably capable of solving a wide range class of nonlinear system of partial differential equations. REFERENCES 1.

2.

Hashim, I.M., SM. Noorani, R. Ahmed, S.A. Bakar, E.S.I. Ismail and A.M. Zakaria, 2006. Accuracy of the Adomian decomposition method applied to the Lorenz System Chaos, 2005.08.135. Abdel - Hassan, I.H., 2004. Differential transformation technique for solving higher-order initial value problem. Applied Math. Comput, 154-299-311.

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13.

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World Appl. Sci. J., 32 (9): 1974-1979, 2014

15. Tarig M. Elzaki and Salih M. Elzaki, 2011. Application of New Transform “Elzaki Transform” to Partial Differential Equations, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768, 1: 65-70. 16. Tarig M. Elzaki and Salih M. Elzaki, 2011. On the Connections Between Laplace and Elzaki transforms, Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 6(1): 1-11. 17. Tarig M. Elzaki and Salih M. Elzaki, 2011. On the Elzaki Transform and Ordinary Differential Equation With Variable Coefficients, Advances in Theoretical and Applied Mathematics. ISSN 09734554, 6(1): 13-18. 18. Salih M. Elzaki, M.A. Bashir and L. Lajimi, 2011. Solution of Linear and Nonlinear Partial Differential Equation by Using Projected Differential Transform Method, Applied Mathematics, Elixir Appl. Math., 40: 5234-5236.

Appendix: Elzaki transform of some Functions f(t) 1 t tn eat

sinat

cosat

E[f(t)[ = T(u) v2 v3 n ! vn+2

v2 1 − av av3

1 + a2v2 v2

1 + a2v2

1979