Numerical approximation of a class of time-fractional differential equations

Abstract

We consider a class of linear fractional partial differential equations containing two time-fractional derivatives of orders α, β ∈ (0, 2) and elliptic operator on space variable. Three main types of such equations with α and β in the corresponding subintervals were determined. The existence of weak solutions of the corresponding initial-boundary value problems has been proved. Some finite difference schemes approximating these problems are proposed and their stability is proved. Estimates of their convergence rates, in special discrete energetic Sobolev’s norms, are obtained. The theoretical results are confirmed by numerical examples.