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 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.
Highlights
During the last decade, fractional calculus has emerged as a model for a broad range of nonclassical phenomena in the applied sciences and engineering [1,2,3,4,5]
Later, generalized fractional operator was used successfully in papers of Mura and Mainardi [12] to model a class of self-similar stochastic processes with stationary increments, which provided models for both slowand fast-anomalous diffusion
We propose a spectral collocation method for differential and integral equations with generalized fractional operators
Summary
Fractional calculus has emerged as a model for a broad range of nonclassical phenomena in the applied sciences and engineering [1,2,3,4,5]. Numerical methods with high order convergence have been developed for fractional differential equations, e.g., spectral method [22], discontinuous Galerkin method [23], and wavelet method [24]. They have not yet been applied to fractional differential equations with generalized fractional operator. Through a suitable variable transform technique, spectral approximation formulas are proposed for generalized fractional operators based on Jacobi polynomials. Operational matrices are constructed and efficient spectral collocation methods are proposed for the generalized fractional differential and integral equations appearing in the references and applications.
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