Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 19-25, 2010 Applied Mathematics
Solution of a Kind of Evolution Equation by the Differential Trans-formation and Adomian Decomposition Methods
H. Alpaslan Peker, Onur Karao˘glu
Selçuk University, Faculty of Science, Department of Mathematics, Campus, 42075, Konya Türkiye
e-mail: p ekera@ hotm ail.com,okaraoglu@ yaho o.com
Presented in 2National Workshop of Konya Ere˘gli Kemal Akman College, 13-14 May 2010.
Abstract. In this study, solution of a kind of evolution equation is studied by using both the differential transformation and Adomian decomposition methods and solutions found by both methods are compared.
Key Words: Differential transformation method, Adomian decomposition method, Soliton solution.
2000 Mathematics Subject Classification: 35A22, 35A25, 35C08. 1. Introduction
The evolution equations are the most commonly occurring equations in physics and hydromechanics. The general expression of these equations is given in [1] as follows: (1) = X =0 + X =0 (+1) + +1 + ( )
In general, obtaining the exact solution of this equation is quite hard.
The special cases of this equation are well known equations as well. For instance, RLW equation + = 3 2, Burgers equation + = 2 2, KdV equation + 6 + 3 3 = 0, MKdV equation + 6 2 + 3 3 = 0 and so on [1].
The evolution equation to be considered in this study is the following RLW equation (2) + 1 2 (2) = 3 2 − ∞ +∞ 0
with the initial condition
(3) ( 0) =
This equation is used for modeling the shallow water waves, plasma waves, etc. and it is one of the most important equations among the nonlinear equations. 2. Adomian Decomposition Method
The Adomian decomposition method introduced by G.Adomian [2,3] in the ear-lier of 1980. This method is used for obtaining not only analytical but also ap-proximate solutions of both linear and nonlinear differential and integral equa-tions arising in engineering and applied fields. This method decomposes the solution by a series converging rapidly [4,5] to the exact solution and then every pieces of this series are found by recursion.
The structure of the method is introduced in [2,3] as follows: Consider the following equation
(4) [()] = ()
where () is an unknown function, () is a continuous function and is a nonlinear differential operator consisting of both linear and nonlinear terms. The linear term divided into two parts as + where is the remaining part of the linear operator and is a high order invertible differential operator. Then, the initial equation becomes as follows:
(5) + + =
Since is an invertible differential operator, −1can be applied the both sides
of the above equation as follows:
(6) −1 = −1 − −1 − −1
The decomposition method calculates the solution of () in a series form as = P∞
=0
and decomposes the nonlinear terms Nu as
(7) =
∞
X
=0
where ’s are polynomials, called as Adomian polynomials, depending on 0
1 ’s.
Both ’s and Nu’s can be obtained respectively as
(8) =
∞
X
=0
and (9) () = Ã∞ X =0 ! = ∞ X =0
where is a parameter taking for convenience. From this result, ’s are found
as follows: (10) = 1 ! " ̰ X =0 !# =0
The initial equation can be obtained in the following form:
(11) ∞ X =0 = + −1 − −1 Ã∞ X =0 ! − −1 à ∞ X =0 ! where = (0).
3. Solution by Adomian Decomposition Method
Using Adomian decomposition method, solution of the equation (1) with initial condition (2) is obtained in [1] as follows:
(12) = ∞ X =0 = ∞ X =0 (−) = 1 + 0 1
4. Differential Transformation Method
Differential transform method is a numerical method based on Taylor expansion. This method tries to find coefficients of series expansion of unknown function by using the initial data on the problem. The concept of differential transform method was first proposed by Zhou [6]. It was applied to electric circuit analysis problems by Zhou. Afterwards, it was applied to several systems and differential equations. For instance, initial — value problems [7,8], difference equations [9], integro-differential equations [10], partial differential equations [11— 13], system of ordinary differential equations [14].
Definition 1. The one—dimensional differential transform of a function () at the point = 0 is defined as follows [7—14]:
(13) () = 1 ! ∙ () ¸ =0
Definition 2. The differential inverse transform of () is defined as follows [7—14]: (14) () = ∞ X =0 ()( − 0)
From the definitions 1 and 2, the following is obtained:
(15) () = ∞ X =0 1 ! ∙ () ¸ =0 ( − 0)
The following theorems [7,8] that can be deduced from the definitions 1 and 2 are given below:
Theorem 1. If () = () ± (), then () = () ± ().
Theorem 2. If () = (), then () = (), where is a constant. Theorem 3. If () = () , then () = ( + )! ! ( + ). Theorem 4. If () = ()(), then () = () ⊗ () = X 1=0 (1)( − 1) Theorem 5. If () = () 2() 2 , then () = () ⊗ () = X =0 ( − + 1)( − + 2)()( − + 2)
5. Two Dimensional Differential Transformation Method
The two—dimensional differential transform of a function is defined as follows: Definition 3. The two—dimensional differential transform of a function is de-fined in [11] as follows: (16) ( ) = 1 !! ∙ + ( ) ¸ = 0 = 0
Definition 4. The differential inverse transform of ( ) is defined in [11] as follows: (17) ( ) = ∞ X =0 ∞ X =0 ( )
From the definitions 3 and 4, the following is obtained: (18) ( ) = ∞ X =0 ∞ X =0 1 !! ∙ + ( ) ¸ = 0 = 0
The following theorems are also given in [11,12]:
Theorem 6. If ( ) = ( ) ± ( ), then ( ) = ( ) ± ( ). Theorem 7. If ( ) = ( ), then ( ) = ( ), where is a constant. Theorem 8. If ( ) =( ) , then ( ) = ( + 1) ( + 1 ). Theorem 9. If ( ) =( ) , then ( ) = ( + 1) ( + 1). Theorem 10. If ( ) = ( )( ), then ( ) = X =0 X =0 ( − ) ( − ) Theorem 11. If ( ) = +( ) , then ( ) = ( + 1)( + 2) ( + )( + 1)( + 2) ( + ) ( + + )
6. Solution by the Differential Transformation Method
The two dimensional differential transformation of the RLW equation is as fol-lows: (19) ( + 1) ( + 1) + X =0 X =0 ( − + 1)( − )( − + 1 ) = ( + 1)( + 2)( + 1) ( + 2 + 1)
and the initial condition ( 0) = is transformed as follows:
(20) ( ) =
½
1 = 1 and = 0 0 otherwise
Then, the following series solution is obtained:
(21) ( ) = − + 2− 3+ 4− 5+ = (1 − + 2− 3+ 4− 5+ ) = ∞ X =0 (−) = 1 + 0 1 7. Conclusion
Exact solution of the initial value problem is obtained by both methods and both results are coincides with each other. This study has an importance due to the being an application of differential transformation method to the RLW equation. Adomian decomposition and differential transformation methods are based on integration and differentiation respectively. Since taking integral of sophisticated expressions in some other problems may be difficult, using the differential transformation method would be more rational choice with respect to Adomian decomposition method.
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