A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

Abstract

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations

“…Spectral method, which is well known as a global high accurate numerical approach, has received great attention and has been extensively applied to solving fractional differential equations (FDEs), see, e.g., [4,6,7,11,13,14] and the references therein. Diethelm [5] proved that the Caputo fractional boundary value problem can be reformulated as a Volterra or Fredholm integral equation.…”

On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems

“…Spectral methods have been extensively used in numerical solutions of fractional differential equations (Zaky 2019b;, function approximations (Liu et al 2019), and other variational problems (Zaky et al 2018a;Ezz-Eldien 2016). The most attractive property of spectral methods may be that they are capable of providing highly accurate solutions to smooth problems with significantly less unknowns than using finite-element or finite difference methods.…”