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Published in:
12-09-2019
Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann–Liouville–Caputo fractional derivative
Published in: BIT Numerical Mathematics | Issue 2/2020
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Abstract
The Riemann–Liouville–Caputo (RLC) derivative is a new class of derivative that is motivated by modelling considerations; it lies between the more familiar Riemann–Liouville and Caputo derivatives. The present paper studies a two-point boundary value problem on the interval [0, L] whose highest-order derivative is an RLC derivative of order \(\alpha \in (1,2)\). It is shown that the choice of boundary condition at \(x=0\) strongly influences the regularity of the solution. For the case where the solution lies in \(C^1[0,L]\cap C^{q+1}(0,L]\) for some positive integer q, a finite difference scheme is used to solve the problem numerically on a uniform mesh. In the error analysis of the scheme, the weakly singular behaviour of the solution at \(x=0\) is taken into account and it is shown that the method is almost first-order convergent. Numerical results are presented to illustrate the performance of the method.