Applications of Separation Variables Approach in Solving Time-Fractional PDEs

Yinghui He,Yunmei Zhao

doi:10.1155/2018/3892691

Abstract

Based on the homogenous balanced principle and subequation method, an improved separation variables function-expansion method is proposed to seek exact solutions of time-fractional nonlinear PDEs. This method is novel and meaningful without using Leibniz rule and chain rule of fractional derivative which have been proved to be incorrect. By using this method, we studied a nonlinear time-fractional PDE with diffusion term. Some general solutions are obtained which contain many arbitrary parameters. Solutions given in related reference are just our especial case. And we also obtained some new type of solutions.

Highlights

It is well known that fractional-order models are more adequate than previously used integer-order models due to an exact description of nonlinear phenomena
FPDEs are increasingly used in mathematical modeling of fluid mechanics, biological and chemical processes, signal processing, and control systems, and they are used in fractal and differential geometry, and so on(see [1,2,3,4,5,6,7,8] and their references cited)
Many natural phenomena associated with realtime problems which depend on both time instant and the previous time history, especially, can be successfully modeled by time-fractional nonlinear partial differential equations

Summary

Introduction

It is well known that fractional-order models are more adequate than previously used integer-order models due to an exact description of nonlinear phenomena. In this paper we will adopt Caputo fractional derivative definition to investigate exact solution of the following type of nonlinear time-fractional partial differential equation: CDtαu. Feng [19] introduced a fractional DαG/G method for seeking traveling wave solutions of space-timefractional partial differential equations under the following modified Riemann-Liouville derivative definition [20]: Dtαf (t). Which can be converted into the following fractional ordinary differential equation with respect to the variable ξ: They suppose that the solution of (7) can be expressed by a polynomial in DαG/G as follows: U (ξ) m. Substituting (8) into (7), equating each coefficient of this polynomial on DξαG/G to zero, they can obtain a large number of exact solutions of space-time fractional partial differential equations (6).

Introduction of Improved Separation Variable Function-Expansion Method

Exact Solutions of a Nonlinear Time-Fractional PDE with Diffusion Term

Conclusions