Spline collocation for nonlinear fractional boundary value problems

Abstract

We consider a class of boundary value problems for nonlinear fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties and spline collocation techniques, the numerical solution of boundary value problems by suitable non-polynomial approximations is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. Theoretical results are tested by two numerical examples.