In the present paper, the hybrid method and the RDTM will be applied for solving Rosenau-Hyman equation that is given as
Special case of Gilson-Pickering equation is Rosenau-Hyman equation for ɛ = 0,κ = 0,α = 1,β = 3. Gilson-Pickering equation is defined as follows:
In the target of this paper, by using the hybrid method and the RDTM, the approximate solution of the Rosenau-Hyman equation is obtained and determined the accuracy of this method for solving this equation. The remainder of this study is continued as follows: In Section 2, a brief introduction for the Rosenau-Hyman equation, the hybrid method and the RDTM are presented. In Section 3, the hybrid method and the RDTM are applied to the Rosenau-Hyman equation in a few steps. And then, the numerical results are obtained for the Rosenau-Hyman equation. The tables and graphs are given that the results are presented. Finally, this study ends with conclusion section.
2 Description of the Methods
Here, the hybrid method and the RDTM which are analyzed the numerical solution of the Rosenau-Hyman equation are introduced, respectively. The differential transform method was studied in the solution of linear and nonlinear problems by Zhou in 1986 . Two-dimensional differential transform method was applied to two dimensional partial differential equations by Chen and Ho in 1999 . The definition and properties of the differential transform method are introduced as follows:
The differential transform of u(x,t) based on t – time variable is defined as
The reverse of the U(i,k) differential function based on t – time variable is defined as follows:
The differential transform and finite difference method apply to t – and x – variables. Linear and nonlinear partial differential equations are solved by using hybrid method with a few iteration and with obtaining rapid convergence. Some properties of differential transform and finite difference method, which using to solve the Rosenau-Hyman equation, are presented the Table 1 and 2 as follows:
Some properties of differential transform based on t – variable.
|W (i,k) = (k + 1 )W (i,k + 1 )|
|w(x,t) = αw(x,t)||W (i,k) = αW (i,k)|
Some properties of central difference and differential transform based on x–variable.
|w(x,t) = sinh(x)||W (x,t) = sinh(xi)|
|w(x,t) = cosh(x)||W (x,t) = cosh(xi)|
Some properties of the RDTM.
|w(x,t) = u(x,t) ± v(x,t)||Wk(x) = Uk(x) ±Vk(x)|
|w(x,t) = αu(x,t),α is a constant.||Wk(x) = αUk(x)|
|w(x,t) = u(x,t)v(x,t)|
|Wk(x) = (k + 1 )Uk+1(x)|
u(x,t) and Uk(x) are defined as original function and the transformed function in the RDTM, respectively. The transformed function is defined as
The differential inverse transform of Uk(x) is defined as follows:
3 Implementation and Comparison of the Hybrid Method and the RDTM to the Rosenau-Hyman Equation
In the present study, the Rosenau-Hyman equation is analyzed by the hybrid method and the RDTM. The obtained numerical solutions are compared with the exact solution and each other.
To demonstrate the basic idea of the hybrid method and the RDTM, we consider the following Rosenau-Hyman equation:
Now, let us solve the Rosenau-Hyman equation using the hybrid method and the RDTM respectively as follows:
The solution with the hybrid method:
According to the hybrid method procedure, the presented differential transforms in the above can be obtained the following recurrence relation by putting in its place on equation (1).
The solution with the RDTM:
According to t – time variable and Table 3, we take differential transform of the equation (6) and the initial condition (7), respectively. If these differential transformations are written in the Eq. (6), the following recurrence relation is obtained:
Here, the following differential transform coefficients are found by using above iteration procedure and two iterations for k = 0,1,2,....
These differential transform coefficients are written in the equation (7), the approximate solution is found as
One can see that the obtained approximate solutions by the hybrid method and the RDTM are quite close to their exact solutions.
The exact, the approximate solutions and error values for different values for t = 0.0001, x are presented in Table 4 as.
Comparison of the approximate solutions (hybrid method and RDTM) with exact solution u(x,t) and the error values for t = 0.0001.
|x(xi)||Hybrid Method||RDTM||Exact||Error of RDTM||Error of Hybrid Method|
In this study, we have successfully acquired approximate solution of the Rosenau-Hyman equation using the hybrid method and the RDTM. The obtained numerical solutions by these method were compared with the exact solution and with each other. These solutions are of high accuracy and also the results show that the proposed methods were strongly and efficient for solving the Rosenau-Hyman equation. In conclusion, It was revealed these methods are as good as the other methods in terms of fast convergence. As suggestion, these methods are a potential important approximate method for further works in strongly nonlinear partial differential equations.
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