The numerical treatment of fractional differential equations raises a challenging issue regarding the accuracy. As a matter of fact, due to the lack of smoothness of the solution near the origin, several numerical methods exhibit low order of convergence. Spline collocation methods are able to overcome this problem, when applied on a suitable graded mesh [1, 2, 3, 4]. Within this class, we pay special attention to two-step collocation methods, which doubles the order of convergence at the same computational cost of one-step collocation methods. In this talk, we illustrate both one and two-step spline collocation methods, analyze their convergence and stability properties. In addition, we discuss their efficient MATLAB implementation and show some numerical examples. The presented results have been obtained in collaboration with Dajana Conte and Beatrice Paternoster, from University of Salerno. References  Cardone, A., Conte, D., Paternoster, B., Two-step collocation methods for fractional differential equations, Discrete Contin. Dyn. Syst. Ser. B 22 (2018) 1–17.  Cardone, A., Conte, D., Stability analysis of spline collocation methods for fractional differential equations, Math. Comput. Simulation 178 (2020) 501-514.  Cardone, A., Conte, D., Paternoster, B., A MATLAB implementation of spline collocation methods for fractional differential equations. Lect. Notes Comput. Sci., 12949 LNCS:387–401, 2021.  Pedas, A., Tamme, E., Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math. 255 (2014) 216–230
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