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An approximate analytical solution of fractional Fornberg-Whitham equation was obtained with the help of the two-dimensional differential transformation method (DTM). It is indicated that the solutions obtained by the two-dimensional DTM are reliable and present an effective method for strongly nonlinear partial equations. Exact solutions can also be obtained from the known forms of the series solutions.

A homogeneous nonlinear fractional Fornberg-Whitham equation [

Subscripts denote the partial differentiation unless stated otherwise. Fornberg and Whitham obtained a peaked solution of the form

See fractional diffusion equation with absorbent term and external force by Das and Gupta [

The goal of this paper is to extend the two-dimensional differential transform method to solve fractional Fornberg-Whitham equation.

This paper is organized as follows.

In Section

Here are some basic definitions and properties of the fractional calculus theory which can be found in [

A real function

The left-sided Riemann-Liouville fractional integral operator of order

The fractional derivative of

For

DTM is an analytic method based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional high order Taylor series method requires symbolic computation. However, the DTM obtains a polynomial series solution by means of an iterative procedure. The method is well addressed by Odibat and Momani [

Operations of the two-dimensional differential transform.

Original function | Transformed function |
---|---|

In case of

From the above definitions, it can be found that the concept of two-dimensional differential transform is derived from two-dimensional differential transform which is obtained from two-dimensional Taylor series expansion.

In this section, we will research the solution of fractional Fornberg-Whitham equation, which has been widely examined in the literature. We described the implementation of the DTM for the fractional Fornberg-Whitham equation in detail. To solve (

The graphs of exact and DTM solutions belonging to examples examined above are shown in Figure

The surface shows the solution

Both the exact results and the approximate solutions obtained for the DTM approximations are plotted in Figure

In this paper, the applicability of the fractional differential transformation method to the solution of fractional Fornberg-Whitham equation with a number of initial and boundary values has been proved. DTM can be applied to many complicated linear and strongly nonlinear partial differential equations and does not require linearization, discretization, or perturbation. The obtained results indicate that this method is powerful and meaningful for solving the nonlinear fractional Fornberg-Whitham type differential equations.