Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method

An approximate analytical solution of fractional Fornberg-Whitham equation was obtained with the help of the two-dimensional di ﬀ erential transformation method (cid:3) DTM (cid:4) . It is indicated that the solutions obtained by the two-dimensional DTM are reliable and present an e ﬀ ective method for strongly nonlinear partial equations. Exact solutions can also be obtained from the known forms of the series solutions.


Basic Definitions
Here are some basic definitions and properties of the fractional calculus theory which can be found in 5, 6, 25, 26 .

Definition 2.1.
A real function f x , x > 0, in the space C μ , μ ∈ R if there exists a real number p > μ, such that f x Definition 2.2. The left-sided Riemann-Liouville fractional integral operator of order α ≥ 0, of a function f ∈ C μ , μ ≥ −1 is defined as The properties of the operator J α can be found in Jang et al. 25 .
The fractional derivative of f x in the Caputo 6 sense is defined as

2.2
The unknown function f f x, t is assumed to be a casual function of fractional derivatives i.e., vanishing for α < 0 taken in Caputo sense as follows.

Definition 2.4.
For m as the smallest integer that exceeds α, the Caputo time-fractional derivative operator of order α > 0 is defined as

Two-Dimensional Differential Transformation Method
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 26 . The proposed method is based on the combination of the classical twodimensional DTM and generalized Taylor's Table 1 formula. Consider a function of two variables u x, y and suppose that it can be represented as a product of two single-variable 4 Abstract and Applied Analysis functions, that is, u x, y f x g y . The basic definitions and fundamental operations of the two-dimensional differential transform of the function are expressed as follows 25-38 . Two-dimensional differential transform of u x, y can be represented as: where 0 < α, β ≤ 1, U α,β k, h F α k G β h is called the spectrum of u x, y . The generalized two-dimensional differential transform of the function u x, y is given by In case of α 1, and β 1, the generalized two-dimensional differential transform 3.2 reduces to the classical two-dimensional differential transform.
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.

The DTM Applied to Fractional Fornberg-Whitham Equation
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 1.

4.5
As α 1, this series has the closed form e x/2−2t/3 , which is an exact solution of the classical gas dynamics equation. The graphs of exact and DTM solutions belonging to examples examined above are shown in Figure 1. It can be deduced that DTM solution corresponds to the exact solutions.
Both the exact results and the approximate solutions obtained for the DTM approximations are plotted in Figure 1. There are no visible differences in the two solutions of each pair of diagrams.

Conclusions
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.