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

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, Centre of Mathematics (CMAT), University of Minho, Campus de Azurém, 4800-058, Guimarães, Portugal

    Luís L. Ferrás

  2. Department of Mathematics, University of Chester, Chester, CH1 4BJ, UK

    Neville Ford

  3. Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, Universidade de Trás-os-Montes e Alto Douro, Departamento de Matemática, UTAD Quinta de Prados, 5001-801, Vila Real, Portugal

    Maria Luísa Morgado

  4. Departamento de Matemática, Faculdade de Ciências e Tecnologia and Centro de Matemática e Aplicações (CMA), Universidade NOVA de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    Magda Rebelo

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  1. Luís L. Ferrás
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  2. Neville Ford
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  3. Maria Luísa Morgado
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  4. Magda Rebelo
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Correspondence to Maria Luísa Morgado.

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Ferrás, L.L., Ford, N., Morgado, M.L. et al. High-Order Methods for Systems of Fractional Ordinary Differential Equations and Their Application to Time-Fractional Diffusion Equations. Math.Comput.Sci. 15, 535–551 (2021). https://doi.org/10.1007/s11786-019-00448-x

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  • DOI: https://doi.org/10.1007/s11786-019-00448-x

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