High-order methods for systems of fractional ordinary differential equations and their application to time-fractional diffusion equations

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.L.L. Ferras would like to thank FCT - Fundacao para a Ciencia 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 Tecnico, University of Lisbon. This work was also partially supported by FCT through the Project UID/MAT/00297/2019 (Centro de Matematica e Aplicacoes)