2023/12/27 by Tinggang Zhao, Zhao, Tinggang, Zhenyu Zhao +5
Mathematics · #35K55 #65D32 #65F60 #65M12 #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #G.1.2 #G.1.9 #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2312.16426
openalex publication_date 2023/12/27 · openalex created_date 2023/12/30 · openalex updated_date 2026/07/28
Fractional calculus with respect to function ψ, also named as ψ-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 ψ-fractional differential equations. In particular, we setup the Petrov-Galerkin spectral method and spectral collocation method for initial and boundary value problems involving ψ-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.