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

Abstract

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 $$2-\delta $$ with $$0<\delta <1$$ . 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 $$u_h$$ of $$u$$ is computed. Error bounds in the maximum norm are proved for $$u-u_h$$ and $$u'-u_h'$$ . Numerical results are presented to demonstrate the sharpness of these bounds.