Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

Jian Zu,Bochao Chen,Fei Xu,Li Qin

doi:10.1155/2018/2394735

Abstract

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

Highlights

In the field of science and engineering, many physical phenomena can be described by differential equations with variable coefficients
In [14], power series solutions of higher-order ordinary differential equations are obtained by residual power series method (RPS)
We present a general residual power series method (GRPS) for constructing power series solutions of time-space fractional differential equations with variable coefficients: Dtmαu (x, t) + P (x) G (u) = F (x, t), Discrete Dynamics in Nature and Society

Summary

Introduction

In the field of science and engineering, many physical phenomena can be described by differential equations with variable coefficients. In [14], power series solutions of higher-order ordinary differential equations are obtained by RPS Inspired by this approach, we present a general residual power series method (GRPS) for constructing power series solutions of time-space fractional differential equations with variable coefficients: Dtmαu (x, t) + P (x) G (u) = F (x, t) , Discrete Dynamics in Nature and Society. Dβpj xj mean the Caputo fractional derivative with respect to t of order iα and xj of order βpj, respectively Such type of differential equation provides an exact description of some physical phenomena in fluid dynamics, electrodynamics, and elastic mechanics. We will apply GRPS to a series of PDE with variable coefficients, including fourth-order parabolic equations, fractional heat equation, and fractional wave equation.

Concepts on Fractional Calculus Theory

Algorithm of GRPS

Applications of GRPS to PDEs with Variable Coefficients

Conclusions