A finite difference method for an initial–boundary value problem with a Riemann–Liouville–Caputo spatial fractional derivative

Abstract

An initial–boundary value problem with a Riemann–Liouville–Caputo space fractional derivative of order α∈(1,2) is considered, where the boundary conditions are reflecting. A fractional Friedrichs’ inequality is derived and is used to prove that the problem approaches a steady-state solution when the source term is zero. The solution of the general problem is approximated using a finite difference scheme defined on a uniform mesh and the error analysis is given in detail for typical solutions which have a weak singularity near the spatial boundary x=0. It is proved that the scheme converges with first order in the maximum norm. Numerical results are given that corroborate our theoretical results for the order of convergence of the difference scheme, the approach of the solution to steady state, and mass conservation.