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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2019-01-01
|
| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2019/3734617 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 1687-9643 1687-9651 |