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Erschienen in:
01.12.2015
An efficient collocation method for a Caputo two-point boundary value problem
Erschienen in: BIT Numerical Mathematics | Ausgabe 4/2015
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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.