Numerical analysis of linear and nonlinear time-fractional subdiffusion equations

TL;DR

This paper develops a new discrete fractional Grönwall inequality and applies it to analyze the stability, convergence, and unconditional convergence of Galerkin spectral methods for linear and nonlinear time-fractional subdiffusion equations.

Contribution

It introduces a novel discrete fractional Grönwall inequality and demonstrates its application in analyzing spectral methods for fractional subdiffusion equations.

Findings

Proves stability and convergence of spectral methods for linear equations.

Establishes unconditional convergence for nonlinear equations.

Provides a new analytical tool for fractional PDEs.

Abstract

In this paper, a new type of the discrete fractional Gr{\"o}nwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Gr{\"o}nwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.