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arXiv · 2312.16426

Spectral approximation of $\psi$-fractional differential equation based on mapped Jacobi functions

Tinggang Zhao·Zhenyu Zhao·Changpin Li·Dongxia Li
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Abstract

Fractional calculus with respect to function $\psi$, also named as $\psi$-fractional calculus, generalizes the Hadamard and the Riemann-Liouville fractional calculi, which causes challenge in numerical treatment. In this paper we study spectral-type methods using mapped Jacobi functions (MJFs) as basis functions and obtain efficient algorithms to solve $\psi$-fractional differential equations. In particular, we setup the Petrov-Galerkin spectral method and spectral collocation method for initial and boundary value problems involving $\psi$-fractional derivatives. We develop basic approximation theory for the MJFs and conduct the error estimates of the derived methods. We also establish a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. Numerical examples confirm the theoretical results and demonstrate the effectiveness of the spectral and collocation methods.

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Tinggang Zhao, Zhenyu Zhao, Changpin Li, Dongxia Li. 2023-12-27. Spectral approximation of $\psi$-fractional differential equation based on mapped Jacobi functions. https://arxiv.org/abs/2312.16426

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