The convergence rate for difference approximations to fractional boundary value problems

Abstract

We study fractional differential equations defined in a bounded domain and with Dirichlet boundary conditions. To discretize these equations and in particular the fractional derivative of order α we consider the Grünwald–Letnikov approximation which is first order accurate when we have an open domain but it can be of lower order in bounded domains and sometimes not consistent. Because the fractional derivative is nonlocal, the accuracy of its approximation near the boundary is strongly affected by the cut of the domain. However, as we will prove, the numerical methods are convergent in the discrete L∞ and L2 norms and the first order rate of convergence can be recovered. In some cases, regarding the order of accuracy of the fractional derivative approximation, we can go down an order α, from one, without destroying the convergence of the scheme.