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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
Article
  • 96 Downloads

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.

Keywords

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

Mathematics Subject Classification

34A08 65L12 

Notes

References

  1. 1.
    Baeumer, B., Kovács, M., Meerschaert, M.M., Sankaranarayanan, H.: Boundary conditions for fractional diffusion. J. Comput. Appl. Math. 336, 408–424 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    del Castillo-Negrete, D.: Fractional diffusion models of nonlocal transport. Phys. Plasmas 13(8), 082308 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Diethelm, K.: An application-oriented exposition using differential operators of Caputo type. In: Morel, J.-M., Takens, F., Teissier, B. (eds.) The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)Google Scholar
  4. 4.
    Eliazar, I.I., Shlesinger, M.F.: Fractional motions. Phys. Rep. 527(2), 101–129 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ervin, V.J., Heuer, N., Roop, J.P.: Regularity of the solution to 1-D fractional order diffusion equations. Math. Comput. 87(313), 2273–2294 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Applied Mathematics, vol. 16. Chapman & Hall/CRC, Boca Raton (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gracia, J.L., O’Riordan, E., Stynes, M.: Convergence in positive time for a finite difference method applied to a fractional convection–diffusion problem. Comput. Methods Appl. Math. 18(1), 33–42 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gracia, J.L., O’Riordan, E., Stynes, M.: A fitted scheme for a Caputo initial-boundary value problem. J. Sci. Comput. 76(1), 583–609 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gracia, J.L., Stynes, M.: Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. J. Comput. Appl. Math. 273, 103–115 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jia, L., Huanzhen, C., Ervin, V.J.: Existence and regularity of solutions to 1-D fractional order diffusion equations. Electron. J. Differ. Equ. 2019(93), 1–21 (2019)zbMATHGoogle Scholar
  11. 11.
    Kelly, J.F., Sankaranarayanan, H., Meerschaert, M.M.: Boundary conditions for two-sided fractional diffusion. J. Comput. Phys. 376, 1089–1107 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liang, H., Stynes, M.: Collocation methods for general Caputo two-point boundary value problems. J. Sci. Comput. 76(1), 390–425 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Meng, X., Stynes, M.: The Green’s function and a maximum principle for a Caputo two-point boundary value problem with a convection term. J. Math. Anal. Appl. 461(1), 198–218 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Patie, P., Simon, T.: Intertwining certain fractional derivatives. Potential Anal. 36(4), 569–587 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pedas, A., Tamme, E.: Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math. 236(13), 3349–3359 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Podlubny, I.: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. In: Ames, W.F. (ed.) Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, Inc., San Diego (1999)Google Scholar
  18. 18.
    Stynes, M., Gracia, J.L.: A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA J. Numer. Anal. 35(2), 698–721 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vainikko, G.: Multidimensional Weakly Singular Integral Equations. Lecture Notes in Mathematics, vol. 1549. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  20. 20.
    Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics, vol. 27, expanded edn. Springer, Berlin (2000)CrossRefGoogle Scholar
  21. 21.
    Wang, H., Yang, D.: Wellposedness of Neumann boundary-value problems of space-fractional differential equations. Fract. Calc. Appl. Anal. 20(6), 1356–1381 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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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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