Numerical solution of nonlinear fractional differential equations by spline collocation methods

Abstract

An initial value problem for nonlinear fractional differential equations is considered. Using an integral equation reformulation of the initial value problem, some regularity properties of the exact solution are derived. On the basis of these properties, the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed. In particular, the attainable order of convergence of proposed algorithms is studied and a (global) superconvergence effect for a special choice of collocation points is established. Theoretical results are verified by means of numerical examples.