Discrete Dynamics in Nature and Society

Volume 2018, Article ID 2394735, 9 pages

https://doi.org/10.1155/2018/2394735

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

^{1}School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China^{2}College of Mathematics, Jilin University, Changchun 130012, China^{3}Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China

Correspondence should be addressed to Jian Zu; nc.ude.unen@001juz

Received 21 March 2018; Revised 24 May 2018; Accepted 11 June 2018; Published 9 July 2018

Academic Editor: Jorge E. Macias-Diaz

Copyright © 2018 Bochao Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

In the field of science and engineering, many physical phenomena can be described by differential equations with variable coefficients. For example, some physical problems in inhomogeneous media [1–3]. In the past, many assumptions on integral order differential equations were applied artificially to describe the systems with memory properties and hereditary properties. Some significant information will be lost by such assumptions. Generally, fractional calculus provides an effective tool to describe memory properties and hereditary properties of different materials and processes without extra assumptions. Now the fractional differential equation has attracted a great deal of interest in several areas including chemistry, physics, engineering, and even finance and social sciences [4, 5]. Some recent progress in fractional calculus can be found in [6–8].

The analytical series solutions of differential equations are of fundamental importance in applied science. Various numerical and analytical methods are proposed such as Adomian decomposition method [9, 10], Fractional complex transform method [11], and Laplace transform method [12]. Although lots of methods are put forward, scientists are still looking for more effective ways to solve specific problems, especially for the fractional equations with variable coefficients.

The residual power series method (RPS), proposed by Abu Arqud in [13], is an efficient and easy method for constructing power series solutions of differential equations without linearization, perturbation, or discretization. Different from the classical power series method, RPS does not need to compare the coefficients of the corresponding terms. This method computes the coefficients of the power series by a chain of equations with one or more variables. One advantage is that RPS is not affected by computational round-off errors and also does not require large computer memory and extensive time. 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:where , , , and with , ; . Here and mean the Caputo fractional derivative with respect to of order and of order , respectively. Such type of differential equation provides an exact description of some physical phenomena in fluid dynamics, electrodynamics, and elastic mechanics.

RPS has been extended to many partial differential equations (PDE), especially to fractional partial differential equations (FPDE), such as time-fractional dispersive PDE [15, 16], time-fractional KdV-Burgers equations [17], homogeneous time-fractional wave equation [18], and time-space fractional Boussinesq equations [19]. In the present paper, we will apply GRPS to a series of PDE with variable coefficients, including fourth-order parabolic equations, fractional heat equation, and fractional wave equation. For other approximation and numerical techniques for FPDE, we refer to finite difference methods [20, 21], differential transform method [22, 23], wavelet method [24], Adomian’s decomposition method [25], variational iteration method [26, 27], homotopy analysis method [28], homotopy perturbation method [29], tau method [30, 31], and so on.

The paper is organized as follows: some necessary definitions and theorems will be presented in Section 2. In Section 3, we propose the main steps of GRPS for the general time-space fractional equations with variable coefficients. In Section 4, the applications of GRPS to some different equations with variable coefficients are given, including fourth-order parabolic equations, fractional heat equations, and fractional wave equations. Finally, conclusions are presented in Section 5.

#### 2. Concepts on Fractional Calculus Theory

There are several definitions of the fractional integration with order , and they are not necessarily equivalent to each other. The two most common ones are Riemann-Liouville’s definition and Caputo’s definition; see [32, 33].

*Definition 1. *The Mittag-Leffler function is defined as follows:

*Definition 2. *A real function , , is said to be in the space , , if there exists a real number such that , where , and it is said to be in the space , if , .

*Definition 3. *Let , . The Riemann-Liouville fractional integral operator of order of is defined as follows:

*Definition 4. *The Caputo time-fractional derivative operator of order of is defined as follows:

*Definition 5. *The Caputo space fractional derivative operator of order is defined as follows:

*Definition 6. *A power series representation of the form is called a fractional power series (FPS) about , where is a variable and are the coefficients of the series.

Theorem 7 (see [34]). *Suppose that has a FPS representation at of the form where is the radius of convergence of the FPS. If for , then the coefficients will take the form of where (-times).*

#### 3. Algorithm of GRPS

In this section, we give a general RPS to obtain fractional power series solutions for any-order time-space fractional differential equations with variable coefficients (1). The analytic function can be expanded as follows:where is the radius of convergence of above series. Substitute the initial conditions in (1); we have which implies So we have the initial guess approximation of in the following form:Define the approximate solution of (1) by the th truncated series:Before applying GRPS to solve (1), we give some notationsSubstituting the th truncated approximate solutions into (1), we obtain the th residual functionwhere Then we have the following facts: (1);(2);(3).

Assume thatSince we haveIn fact, this relation is a fundamental rule in GRPS. So the FPS solution of (1) iswhere

#### 4. Applications of GRPS to PDEs with Variable Coefficients

##### 4.1. Fourth-Order Parabolic Equation with Variable Coefficients in

Let us consider the fourth-order parabolic differential equationwhere is the ratio of flexural rigidity of the beam to its mass per unit length; see [35]. In [35], the initial conditions and the boundary conditions of (23) areandrespectively. According to (10), can be written in the following form: The initial approximation is Now by (15), denote By (16) and (18), we assume thatLetting in (29), it showswhich impliesSo the 2nd truncated approximate solution of (23) isSimilarly, can be constructed as follows: So the th truncated approximate solution of (23) is Finally, if we defineit is easy to verify that in (35) is the exact solution of (23) with boundary value condition (25).

Numerical comparisons are studied next. Figure 1 shows the exact solution of (23) with . In Figure 2, , , , and represent the 9th-, 10th-, 11th-, and 12th-order truncated approximate solution of with . It shows that these GRPS approximate solutions are convergent to the exact solution .