High-Order Methods for Systems of Fractional Ordinary Differential Equations and Their Application to Time-Fractional Diffusion Equations

  • Luís L. Ferrás
  • Neville Ford
  • Maria Luísa MorgadoEmail author
  • Magda Rebelo


Taking into account the regularity properties of the solutions of fractional differential equations, we develop a numerical method which is able to deal, with the same accuracy, with both smooth and nonsmooth solutions of systems of fractional ordinary differential equations of the Caputo-type. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples. Finally, we solve the time-fractional diffusion equation using a combination of the method of lines and the newly developed hybrid method.


Fractional diffusion Caputo derivative Nonpolynomial collocation method Polynomial collocation method Method of lines 

Mathematics Subject Classification

45K05 65L20 65M12 65R20 



L.L. Ferrás would like to thank FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) for financial support through the scholarship SFRH/BPD/100353/2014 and Project UID-MAT-00013/2013. M.L. Morgado aknowledges the financial support of FCT, through the Project UID/Multi/04621/2019 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon. This work was also partially supported by FCT through the Project UID/MAT/00297/2019 (Centro de Matemática e Aplicações).


  1. 1.
    Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41, 364–381 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, C.M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, C.M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, New York (2010)CrossRefGoogle Scholar
  7. 7.
    Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electr. Trans. Numer. Anal. 5, 1–6 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Diethelm, K., Ford, J.M., Ford, N.J., Weilbeer, M.: Pitfalls in fast numerical solvers for fractional differential equations. J. Comput. Appl. Math. 186, 482–503 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diethelm, K., Ford, N.J.: Volterra integral equations and fractional calculus: do neighboring solutions intersect? J. Integral Equ. Appl. 1, 25–37 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ford, N.J., Connolly, J.A.: Comparison of numerical methods for fractional differential equations. Commun. Pure Appl. Anal. 5, 289 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ford, N., Morgado, M.: Fractional boundary value problems: analysis and numerical methods. Fract. Calc. Appl. Anal. 14, 554–567 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ford, N.J., Morgado, M.L., Rebelo, M.: Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16, 874–891 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ford, N.J., Yan, Y.: An approach to construct higher order time discretization schemes for time fractional partial differential equations with nonsmooth data. Fract. Calc. Appl. Anal. 20, 1076–1105 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferrás, L.L., Ford, N.J., Morgado, M.L., Rebelo, M.: A numerical method for the solution of the time-fractional diffusion equation. In: Murgante, B. et al. (eds.) Computational Science and Its Applications – ICCSA 2014. ICCSA 2014. Lecture Notes in Computer Science, vol. 8579. Springer, Cham (2014)Google Scholar
  15. 15.
    Gao, G., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129–143 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gu, Y.T., Zhuang, P.: Anomalous sub-diffusion equations by the meshless collocation method. Aust. J. Mech. Eng. 10, 1–8 (2012)CrossRefGoogle Scholar
  18. 18.
    Fenghui, H.: A time-space collocation spectral approximation for a class of time fractional differential equations. Int. J. Differ. Equ. 2012, 1–19 (2012). Article ID 495202MathSciNetCrossRefGoogle Scholar
  19. 19.
    Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205(2), 719–736 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lin, Y., Chuanju, X.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160–176 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Luchko, Y.: Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59, 1766–1772 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Luchko, Y.: Maximum principle and its application for the time-fractional diffusion equations. Fract. Calc. Appl. Anal. 14, 110–124 (2011)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, 141–160 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Murio, D.A.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 5, 1138–1145 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mainardi, F.: Fractional diffusive waves in viscoelastic solids. In: Wegner, J. I., Norwood, F. R. (eds.) Nonlinear Waves in Solids, ASME Book No. AMR 137 93-97, Fairfield (1995)Google Scholar
  27. 27.
    Mainardi, F.: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997)zbMATHGoogle Scholar
  28. 28.
    Mainardi, F., Pagnini, G., Gorenflo, R.: Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187, 295–305 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mainardi, F.: The time fractional diffusion-wave equation. Radiophys. Quant. Electron. 38, 13–24 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nigmatullin, R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi 133, 425–430 (1986)CrossRefGoogle Scholar
  31. 31.
    Rice, J.: On the degree of convergence of nonlinear spline approximation. In: Schoenberg, I.J. (ed.) Approximations with Special Emphasis on Spline Functions, pp. 349–369. Academic Press, New York (1969)Google Scholar
  32. 32.
    Richardson, L.F.: Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Character JSTOR 110, 709–737 (1926)CrossRefGoogle Scholar
  33. 33.
    Schneider, W., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Stynes, M., O’Riordan, E., Gracia, L.J.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sun, H.G., Chen, W., Sze, K.Y.: A semi-discrete finite element method for a class of time-fractional diffusion equations. Philos. Trans. R. Soc. A 371, 20120268 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27, 27–82 (1986)MathSciNetCrossRefGoogle Scholar
  37. 37.
    YingJun, J., JingTangm, M.A.: Moving finite element methods for time fractional partial differential equations. Sci. China Math. 56, 1287–1300 (2013)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Yuste, S.B., Acedo, L.: An explicit finite difference method and a new von Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhao, X., Sun, Z.Z.: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6061–6074 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Luís L. Ferrás
    • 1
  • Neville Ford
    • 2
  • Maria Luísa Morgado
    • 3
    Email author
  • Magda Rebelo
    • 4
  1. 1.Department of Mathematics, Centre of Mathematics (CMAT)University of MinhoGuimarãesPortugal
  2. 2.Department of MathematicsUniversity of ChesterChesterUK
  3. 3.Center for Computational and Stochastic Mathematics, Instituto Superior TécnicoUniversidade de Trás-os-Montes e Alto Douro, Departamento de MatemáticaVila RealPortugal
  4. 4.Departamento de Matemática, Faculdade de Ciências e Tecnologia and Centro de Matemática e Aplicações (CMA)Universidade NOVA de LisboaCaparicaPortugal

Personalised recommendations