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

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...

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Main Authors: Jianke Zhang, Zhirou Wei, Lifeng Li, Chang Zhou
Format: Article
Language:English
Published: Wiley 2019-01-01
Series:Complexity
Online Access:http://dx.doi.org/10.1155/2019/6159024
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author Jianke Zhang
Zhirou Wei
Lifeng Li
Chang Zhou
author_facet Jianke Zhang
Zhirou Wei
Lifeng Li
Chang Zhou
author_sort Jianke Zhang
collection DOAJ
description 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.
format Article
id doaj-art-45d7a33b97244adb8eb6741ef8ee39a7
institution Kabale University
issn 1076-2787
1099-0526
language English
publishDate 2019-01-01
publisher Wiley
record_format Article
series Complexity
spelling doaj-art-45d7a33b97244adb8eb6741ef8ee39a72025-02-03T06:01:47ZengWileyComplexity1076-27871099-05262019-01-01201910.1155/2019/61590246159024Least-Squares Residual Power Series Method for the Time-Fractional Differential EquationsJianke Zhang0Zhirou Wei1Lifeng Li2Chang Zhou3School of Science, Xi’an University of Post and Telecommunications, Xi’an 710121, ChinaSchool of Science, Xi’an University of Post and Telecommunications, Xi’an 710121, ChinaSchool of Science, Xi’an University of Post and Telecommunications, Xi’an 710121, ChinaSchool of Science, Xi’an University of Post and Telecommunications, Xi’an 710121, ChinaIn 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.http://dx.doi.org/10.1155/2019/6159024
spellingShingle Jianke Zhang
Zhirou Wei
Lifeng Li
Chang Zhou
Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations
Complexity
title Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations
title_full Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations
title_fullStr Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations
title_full_unstemmed Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations
title_short Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations
title_sort least squares residual power series method for the time fractional differential equations
url http://dx.doi.org/10.1155/2019/6159024
work_keys_str_mv AT jiankezhang leastsquaresresidualpowerseriesmethodforthetimefractionaldifferentialequations
AT zhirouwei leastsquaresresidualpowerseriesmethodforthetimefractionaldifferentialequations
AT lifengli leastsquaresresidualpowerseriesmethodforthetimefractionaldifferentialequations
AT changzhou leastsquaresresidualpowerseriesmethodforthetimefractionaldifferentialequations