Abstract
We propose a new method called the fractional reduced differential transform method (FRDTM) to solve nonlinear fractional partial differential equations such as the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers and Cahn-Allen equations to show the nature of solutions as the fractional derivative parameter is changed. The results prove that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.
Highlights
The space-fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe
The Burgers equations occur in various areas of applied sciences and physical applications, such as modeling of fluid mechanics and financial mathematics, and the equation has still interesting applications in physics and astrophysics
There are many physical applications in science and engineering that can be represented by models using fractional differential equations [ – ], which are quite useful for many physical problems
Summary
The space-fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. Kurulay [ ] found approximate and exact solutions of the space- and time-fractional Burgers equations. We first consider the nonlinear fractional generalized Burgers equation with time- and space-fractional derivatives of the form [ ]. (αk + ) ∂tαk t=t is the reduced transformed function of u(x, t), where α is a parameter which describes the order of time-fractional derivative. 4.3 The time-fractional Cahn-Allen equation Consider the following nonlinear time-fractional Cahn-Allen equation [ , ]: We proceed in this way, and after the th iteration the approximate solution is given by u (x, t) = Uk(x)tαk k=. 5 Tables of numerical calculations we present tables to show the comparison of results of the FRDTM approximate solutions and the exact solution for different values of α and β. This shows that the FRDTM converges faster than the existing methods in the literature
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