Error Analysis Of A Finite Difference Method On Graded Meshes For A Multiterm Time-fractional Initial-boundary Value Problem

An initial-boundary value problem in one space dimension,whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1,is considered.This problem is a generalisation of the problem considered by Stynes,O'Riordan and Gracia in SIAM J.Numer.Anal.2017,where only one fractional time derivative was present.A priori bounds on the derivatives of the unknown solution are derived.A finite difference method,using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation,is constructed on a mesh that is uniform in space and arbitrarily graded in time.Stability and consistency of the method and a sharp convergence result are proved;hence it is clear how to choose the temporal mesh grading in a optimal way.Numerical results supporting our theoretical results are provided.