Numerical treatment of fractional differential models

Several fracture and fatigue problems are modelled by fractional differential equations and the number of such models is rapidly increasing. This communication deals with the numerical solution of fractional differential equations by a class of one and two step spline collocation methods, recently introduced and studied in (Cardone, Conte and Paternoster, 2018), in (Cardone and Conte, 2020) and in (Pedas and Tamme, 2014). These methods have higher order of convergence with respect to most numerical methods for fractional differential equations. We illustrate the construction of these methods, with some details on the computation of certain fractional integrals. Then we describe the convergence analysis, which takes place in a suitable Banach space, which is related to the smoothness of the analytical solution. We discuss the optimal choice of the method parameters, to obtain the maximum order of convergence. In particular, we show how to set the collocation parameters and the grading exponent. Last part of communication is devoted to some numerical experiments on significant test problems from the literature, which confirm theoretical results and make comparison between one and two step spline collocation methods with respect accuracy and efficiency. Finally, some concluding remarks and future development of the work are reported.