Time discretization of fractional subdiffusion equations via fractional resolvent operators

Abstract

In this work, we study time discretization of subdiffusion equations, that is, fractional differential equations of order α∈(0,1). Assuming that A is the generator of a fractional resolvent family {Sα,α(t)}t≥0, which allows to write the solution to the subdiffusion equation ∂tαu(t)=Au(t)+f(t) as a variation of constants formula, we find an interesting connection between {Sα,α(t)}t≥0 and a discrete resolvent family {Sα,αn}n∈N and then, by using the properties of {Sα,α(t)}t≥0, we study the existence of solutions to the discrete subdiffusion equation C∇αun=Aun+fn,n∈N, where, based on the backward Euler method for a τ>0 given, C∇αun is an approximation of ∂tαu(t) at time tn≔τn. We study simultaneously the fractional derivative in the Caputo and Riemann–Liouville sense. We also provide error estimates and some experiments to illustrate the results.