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

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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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. (2020) doi:10.1007/s11786-019-00448-x

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Keywords

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

Mathematics Subject Classification

  • 45K05
  • 65L20
  • 65M12
  • 65R20