An α-robust error analysis of the [formula omitted] scheme for time-fractional integro-differential initial-boundary value problems

Abstract

An initial-boundary value problem governed by the time-fractional integro-differential equation Dtαu−κItβΔu−λΔu=f is considered, where (x,t)∈Ω×(0,T]; here Dtα is a Caputo temporal derivative of order α∈(0,1), Itβ is a Riemann-Liouville temporal integral of order β∈(0,1), and Ω is a rectangular-type domain in Rd for some d≥1. The positive constants κ and λ and the function f are given data. A numerical scheme is constructed by employing the L1‾ scheme for the Caputo derivative, an averaged product integration rule for the Riemann-Liouville integral, and a standard discretisation (3 points in each coordinate direction) of Δu. The scheme satisfies a discrete comparison principle, which is used to establish an error bound for the computed solution that is O(N−2) in time on graded temporal meshes with N points. This result is α-robust: it does not blow up as α→1−. Numerical results show the sharpness of our theoretical results.