An Alternative Finite Difference Stability Analysis for a Multiterm Time-Fractional Initial-Boundary Value Problem

Abstract

A fractional initial-boundary value problem is considered, where the differential operator includes a sum of Caputo temporal derivatives, and the solution has a weak singularity at the initial time $t$ = 0. The problem is solved numerically by a finite difference method based on applying the L1 method to discretise each temporal derivative on a graded mesh. Stability of this method is proved by generalising the analysis of Stynes $et$ $al$., SIAM J. Numer. Anal. 55 (2017), where the case of a single temporal derivative was investigated. This stability result is used to prove a sharp error estimate for the finite difference method.