Orthogonal-polynomial-based spectral methods for fractional differential equations: theories and applications
File(s)
Author(s)
Pu, Tianyi
Type
Thesis
Abstract
Fractional calculus is a well-established field whose practical applications have gained significant momentum in recent years, though solving fractional equations remains a substantial challenge. This project advances orthogonal-polynomial-related spectral methods for fractional integro-differential equations (FIE/FDEs) through several contributions. The Jacobi-fractional-polynomial spectral method \cite{zayernouri2014fractional} for one-sided FIE/FDEs is extended with additional operators, improved flexibility, and a stable procedure for generating operational matrices, demonstrating superior performance in challenging examples. For the half-order fractional Laplacian $(-\Delta)^{1/2}$, three new approaches are developed: (1) a recurrence-based computation of half-order Riesz potentials of Legendre polynomials, potentially enabling approximation in $H^{1/2}$; (2) the discovery of an orthogonal basis related to the finite Hilbert transform, with associated operational matrices and a spectral method for $(-\Delta)^{1/2}$ on $[-1,1]$; and (3) an enhancement of a sum-space spectral method via a novel complete basis, addressing limitations in earlier work. These developments collectively broaden the toolkit for fractional spectral methods and lay the groundwork for continued research in this area.
Version
Open Access
Date Issued
2024-10-02
Date Awarded
01/12/2025
License URL
Advisor
Olver, Sheehan
Sponsor
Departmental Roth Scholarship
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)