Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator
Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design hig...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2019/3734617 |
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| _version_ | 1832551851314118656 |
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| author | Qinwu Xu Zhoushun Zheng |
| author_facet | Qinwu Xu Zhoushun Zheng |
| author_sort | Qinwu Xu |
| collection | DOAJ |
| description | Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for α-th (α>0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a variable transform technique. Then, operational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differential and integral equations with different fractional operators. At last, the method is applied to generalized fractional ordinary differential equation and Hadamard-type integral equations, and exponential convergence of the method is confirmed. Further, based on the proposed method, a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed. |
| format | Article |
| id | doaj-art-fece21278ac54a47abc1c71c408ad475 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-fece21278ac54a47abc1c71c408ad4752025-02-03T06:00:22ZengWileyInternational Journal of Differential Equations1687-96431687-96512019-01-01201910.1155/2019/37346173734617Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional OperatorQinwu Xu0Zhoushun Zheng1Department of Mathematical Science, Nanjing University, 210093 Nanjing, ChinaSchool of Mathematics and Statistics, Central South University, 410083 Changsha, ChinaGeneralized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for α-th (α>0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a variable transform technique. Then, operational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differential and integral equations with different fractional operators. At last, the method is applied to generalized fractional ordinary differential equation and Hadamard-type integral equations, and exponential convergence of the method is confirmed. Further, based on the proposed method, a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed.http://dx.doi.org/10.1155/2019/3734617 |
| spellingShingle | Qinwu Xu Zhoushun Zheng Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator International Journal of Differential Equations |
| title | Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator |
| title_full | Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator |
| title_fullStr | Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator |
| title_full_unstemmed | Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator |
| title_short | Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator |
| title_sort | spectral collocation method for fractional differential integral equations with generalized fractional operator |
| url | http://dx.doi.org/10.1155/2019/3734617 |
| work_keys_str_mv | AT qinwuxu spectralcollocationmethodforfractionaldifferentialintegralequationswithgeneralizedfractionaloperator AT zhoushunzheng spectralcollocationmethodforfractionaldifferentialintegralequationswithgeneralizedfractionaloperator |