Spectral collocation method for a class of fractional diffusion differential equations with nonsmooth solutions

Abstract

In this paper, we develop spectral collocation method for a class of fractional diffusion differential equations. Since the solutions of these fractional differential equations usually exhibit singularities at the end‐points, it can not be well approximated by classical polynomial basis functions. We first give nonclassical interpolants based on the associated Jacobi‐Gauss points and obtain the corresponding fractional differentiation matrices. Second, we prove the convergence for the developed spectral collocation method. Finally, several numerical examples are considered to demonstrate the validity and applicability of the developed basis functions.

Fast spectral solver for the inversion of boundary data problem of Poisson equation in a doubly connected domain

“…Spectral method is an important tool in scientific computing and engineering for solving differential equations 11–14 . Compared with finite difference and finite element method, the method has convergence speed of infinite order, but the traditional view holds that it will lead to dense and ill‐conditioned matrix.…”

“…9,10 Spectral method is an important tool in scientific computing and engineering for solving differential equations. [11][12][13][14] Compared with finite difference and finite element method, the method has convergence speed of infinite order, but the traditional view holds that it will lead to dense and ill-conditioned matrix. In this paper, an ultraspherical spectral method is introduced, which can converge to the solution by superalgebra, and only needs to solve an almost banded and well-posed linear system.…”