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

By Natalia Kopteva, Martin Stynes. 2015

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A two-point boundary value problem is considered on the interval $$[0,1]$$[0,1], where the leading term in the differential operator is a Caputo fractional-order derivative of order $$2-\delta $$2-δ with $$0<\delta <1$$0<δ<1. The problem is reformulated as a Volterra integral equation of the second kind in terms of the quantity $$u'(x)-u'(0)$$u′(x)-u′(0), where $$u$$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 $$u_h$$uh of $$u$$u is computed. Error bounds in the maximum norm are proved for $$u-u_h$$u-uh and $$u'-u_h'$$u′-uh′. Numerical results are presented to demonstrate the sharpness of these bounds.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b034208786d44c71e79d4c37f65fabe6a1e43ea9