This talk deals with the numerical solution, by two-step spline collocation methods, of nonlinear fractional differential equations. Fractional differential equations are widely applied in various fields, including physics, engineering, finance, and biomedical sciences, where they model complex phenomena such as anomalous diffusion, viscoelastic materials, signal processing, and neural dynamics with greater accuracy than classical differential equations. Two-step spline collocation methods are highly accurate when a suitably graded mesh is adopted and exhibit strong stability properties, as demonstrated in [1, 2]. Notably, they surpass one-step collocation methods proposed in [5], in terms of accuracy while maintaining the same computational cost. However, for an efficient implementation, several key aspects must be addressed: selecting the optimal grading exponent for the mesh, devising an appropriate starting procedure, and ensuring a cost-effective computation of the fractional integrals involved in the method’s formulation. This talk aims to explore these aspects in depth and outline a general algorithm for applying two-step collocation methods [3, 4]. REFERENCES [1] Cardone, A., Conte, D., Paternoster, B., Two-step collocation methods for fractional differential equations, Discrete Contin. Dyn. Syst. Ser. B (2018) 23(7):2709–2725. https://doi.org/10.3934/dcdsb.2018088. [2] Cardone, A., Conte, D., Paternoster, B., Stability of two-step spline collocation methods for fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. (2022) 115:106726. https://doi.org/10.1016/j.cnsns.2022.106726. [3] Cardone, A., Conte, D., Paternoster, B., A MATLAB code for fractional differential equations based on two-step spline collocation methods, Springer INdAM Series (2023) 50:121–146. https://doi.org/10.1007/978-981-19-7716-9 8. [4] Cardone, A., Conte, D., Paternoster, B., tsfcoll, GitHub repository (2024), https://github.com/cardange/tsfcoll. [5] Pedas, A., Tamme, E., Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math. (2014) 255:216–230. https://doi.org/10.1016/j.cam.2013.04.049