## International Journal of Differential Equations

### Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator

#### Abstract

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 $\alpha -th (\alpha >\mathrm{0})$ 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.

#### Article information

Source
Int. J. Differ. Equ., Volume 2019 (2019), Article ID 3734617, 14 pages.

Dates
Accepted: 13 November 2018
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.ijde/1551150374

Digital Object Identifier
doi:10.1155/2019/3734617

#### Citation

Xu, Qinwu; Zheng, Zhoushun. Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator. Int. J. Differ. Equ. 2019 (2019), Article ID 3734617, 14 pages. doi:10.1155/2019/3734617. https://projecteuclid.org/euclid.ijde/1551150374

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