He’s Homotopy Perturbation Method and Fractional Complex Transform for Analysis Time Fractional Fornberg-Whitham Equation

In this article, time fractional Fornberg-Whitham equation of He’s fractional derivative is studied. To transform the fractional model into its equivalent differential equation, the fractional complex transform is used and He’s homotopy perturbation method is implemented to get the approximate analytical solutions of the fractional-order problems. The graphs are plotted to analysis the fractional-order mathematical modeling.


Introduction
Over the last few years, the study of the fractional calculus and applications in the area of life science, physics and the engineering has been paid a great attention. The fractional calculus are also used in many other fields, such as optics, solitary waves, control theory of dynamical systems, and so on, which can be derived by linear or nonlinear fractional order differential equations. Recently, the studies of nonlinear problems and their effects are of widely significance. Many analytical and approximation methods have been presented to solve nonlinear fractional differential equations such as [1][2][3][4][5][6][7][8][9][10][11]. The homotopy perturbation method (HPM) [12][13][14] is widely applied to various science and engineering problems. This method was first proposed by He [12]. Fractional complex transform was suggested also by He et al. [15][16][17][18][19], which converts the fractional differential equation into its equivalent differential equation, so that the HPM can be effectively used. Now it is considered as a powerful method to find the approximation solutions of nonlinear fractional order differential equations.
In this paper, we study the time fractional Fornberg-Whitham (FW) equation [20] as follows: with the initial condition: here b is the fractal dimensions of the fractal medium and @ b @t b is He's fractional derivative defined [21][22][23]: ðs À tÞ nÀbÀ1 ½u 0 ðsÞ À uðsÞds where u 0 ðx; tÞ is the solution of its starting point of the nonlinear fractional order model. When b ¼ 1, Eq. (1) becomes to be the original Fornberg-Whitham (FW) model which is a significant mathematical equation in mathematical physics. This equation was presented by Parkes et al. [24][25][26] in 1978 which describes nonlinear water waves with peakon solutions. The peakon solution is a special solitary wave solution which is peaked in the limiting case. The FW equation has been found to require peakon results as a simulation for limiting wave heights as well as the frequency of wave breaks. Now many scholars have researched the FW model for the fractional-order derivative because of its fractional calculus applications.

HPM Procedure
To illustrate the basic ideas of this method [27][28][29][30][31][32], we consider the following nonlinear functional equation: with the following boundary conditions: where A is a general functional operator, B is a boundary operator, f ðrÞ is a known analytical function, and Ã is the boundary of the domain . The operator A can be decomposed into two operators L and N , where L is linear, and N is nonlinear operator. Eq. (4) can be written as follows: LðuÞ þ NðuÞ À f ðrÞ ¼ 0 (6) Using the homotopy technique, we construct a following homotopy: where the homotopy parameter p is considered as a small parameter.

Numerical Results and Discussion
The exact result [20] of Eq. (1): In Figs

Conclusion
In this manuscript, a mathematical technology is used to find the solution of time fractional Fornberg-Whitham equation. The fractional-derivatives are discussed within He's fractional derivative. The solutions are determined for fractional-order problems which shows the high accuracy and efficiency. This is easy and can be extended to other nonlinear differential equations with fractal derivatives in science and engineering.

Conflicts of Interest:
The authors declare that we have no conflicts of interest to report regarding the present study.