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

and Applied Analysis 3 Table 1: Operations of the two-dimensional differential transform. Original function Transformed function u x, y f x, y ∓ g x, y Uα,β k, h Fα,β k, h ∓G k, h u x, y ξf x, y Uα,β k, h ξFα,β k, h u x, y ∂f x, y /∂x Uα,β k, h k 1 F k 1, h u x, y D ∗x0f x, y , 0 < α ≤ 1 Uα,β k, h Γ α k 1 1 /Γ αk 1 Fα,β k 1, h u x, y D ∗y0f x, y , 0 < α ≤ 1 Uα,β k, h Γ α h 1 1 /Γ αh 1 Fα,β k, h 1 u x, y x − x0 mα y − y0 nβ Uα,β k, h δ k −m,h − n { 1, k r, h s 0, otherwise u x, y f x, y g x, y Uα,β k, h ∑k m 0 ∑h n 0 Fα,β m,h − n Gα,β k −m,n u x, y f x, y g x, y h x, y Uα,β k, h ∑k k4 0 ∑k−k4 k3 0 ∑h k2 0 ∑h−k2 k1 0 Fα,β k4, h − k2 − k1 Gα,β k3, k2 Hα,β k − k4 − k3, k1 Definition 2.3. The fractional derivative of f x in the Caputo 6 sense is defined as D ∗f x J m−α Df x 1 Γ m − α ∫x 0 x − t m−α−1 f m t dt, for m − 1 < α < m, m ∈ N, x > 0, f ∈ C −1. 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 D ∗tf x, t ∂f x, t ∂tα ⎪⎪⎨ ⎪⎩ 1 Γ m − α ∫ t 0 t − τ m−α−1 ∂ f x, τ ∂tm dτ, m − 1 < α < m, ∂f x, t ∂tm , α m ∈ N. 2.3 3. 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: