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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
Article
  • 2 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

45K05 65L20 65M12 65R20 

Notes

Acknowledgements

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).

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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

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