Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations

Chang Zhou,Zhirou Wei,Jianke Zhang,Lifeng Li

doi:10.1155/2019/6159024

Abstract

In this study, an applicable and effective method, which is based on a least-squares residual power series method (LSRPSM), is proposed to solve the time-fractional differential equations. The least-squares residual power series method combines the residual power series method with the least-squares method. These calculations depend on the sense of Caputo. Firstly, using the classic residual power series method, the analytical solution can be solved. Secondly, the concept of fractional Wronskian is introduced, which is applied to validate the linear independence of the functions. Thirdly, a linear combination of the first few terms as an approximate solution is used, which contains unknown coefficients. Finally, the least-squares method is proposed to obtain the unknown coefficients. The approximate solutions are solved by the least-squares residual power series method with the fewer expansion terms than the classic residual power series method. The examples are shown in datum and images.The examples show that the new method has an accelerate convergence than the classic residual power series method.

Highlights

Some existing methods have been modi ed by the least-squares method so that the approximate solution achieves higher accuracy
Kumar and Koundal [18] proposed an approach in which the system of nonlinear fractional partial di erential equations was gured out by Complexity generalized least-squares homotopy perturbations. e approximate analytical solutions for nonlinear differential equations were solved by the least-squares homotopy perturbation method [19]. e linear and nonlinear fractional partial differential equations were figured out by the leastsquares homotopy perturbation method [20]
The least-squares method is combined with the residual power series method, which is called the leastsquares residual power series method (LSRPSM)

Summary

Introduction

Some existing methods have been modi ed by the least-squares method so that the approximate solution achieves higher accuracy. Kumar and Koundal [18] proposed an approach in which the system of nonlinear fractional partial di erential equations was gured out by Complexity generalized least-squares homotopy perturbations. E approximate analytical solutions for nonlinear differential equations were solved by the least-squares homotopy perturbation method [19]. E linear and nonlinear fractional partial differential equations were figured out by the leastsquares homotopy perturbation method [20]. Compared with the classic residual power series method, a more accurate approximate solution with fewer expansion terms can be obtained by the new method. Is section presents the definition of fractional partial Wronskian The definition of the Caputo fractional is introduced systematically. is section presents the definition of fractional partial Wronskian

Methods

Conclusion