Nonpolynomial collocation approximation of solutions to fractional differential equations

Fractional Calculus and Applied Analysis, 16(4) (2013), 874-891
http://dx.doi.org/10.2478/s13540-013-0054-3

We propose a non-polynomial collocation method for solving fractional differential equations. The construction of such a scheme is based on the classical equivalence between certain fractional differential equations and corresponding Volterra integral equations. Usually, we cannot expect the solution of a fractional differential equation to be smooth and this poses a challenge to the convergence analysis of numerical schemes. In this paper, the approach we present takes into account the potential non-regularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. An error analysis is provided for the linear case and several examples are presented and discussed.