An efficient collocation method for a Caputo two-point boundary value problem

A two-point boundary value problem is considered on the interval [0,1], where the leading term in the differential operator is a Caputo fractional-order derivative of order 2-δ with 0<δ<1. The problem is reformulated as a Volterra integral equation of the second kind in terms of the quantity u′(x)-u′(0), where u is the solution of the original problem. A collocation method that uses piecewise polynomials of arbitrary order is developed and analysed for this Volterra problem; then by postprocessing an approximate solution uh of u is computed. Error bounds in the maximum norm are proved for u-uh and u′-u_h′. Numerical results are presented to demonstrate the sharpness of these bounds.