Collocation Methods for General Caputo Two-Point Boundary Value Problems

A general class of two-point boundary value problems involving Caputo fractional-order derivatives is considered. Such problems have been solved numerically in recent papers by Pedas and Tamme, and by Kopteva and Stynes, by transforming them to integral equations then solving these by piecewise-polynomial collocation. Here a general theory for this approach is developed, which encompasses the use of a variety of transformations to Volterra integral equations of the second kind. These integral equations have kernels comprising a sum of weakly singular terms; the general structure of solutions to such problems is analysed fully. Then a piecewise-polynomial collocation method for their solution is investigated and its convergence properties are derived, for both the basic collocation method and its iterated variant. From these results, an optimal choice can be made for the transformation to use in any given problem. Numerical results show that our theoretical convergence bounds are often sharp.