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: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2019/6159024 |
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| _version_ | 1832551312145776640 |
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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 |