Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann–Liouville–Caputo fractional derivative

  • José Luis Gracia
  • Eugene O’Riordan
  • Martin StynesEmail author


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.


Fractional differential equation Riemann–Liouville–Caputo fractional derivative Weak singularity Maximum principle Finite difference scheme 

Mathematics Subject Classification

34A08 65L12 



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.IUMA and Department of Applied Mathematics, Torres Quevedo Building, Campus Rio EbroUniversity of ZaragozaZaragozaSpain
  2. 2.School of Mathematical SciencesDublin City UniversityDublin 9Ireland
  3. 3.Applied Mathematics DivisionBeijing Computational Science Research CenterHaidian District, BeijingChina

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