Implementation issues of collocation methods for fractional differential equations

The introduction of non-integer derivatives into the models of many processes of science and engineering led to reliable qualitative and quantitative descriptions and overcame the unphysical predictions of classical models, which may happen in certain cases. In this talk, we discuss the numerical solution of fractional differential equations by spline collocation methods. These methods present several advantages: they are continuous methods i.e., they approximate the solution at each point of the time interval; they have high order of convergence, if a suitable mesh of points is set; they have good stability properties, which is a fundamental property to obtain a reliable numerical solution. We focus on the implementation issues, which needs to be addressed to apply one and two-step spline collocation methods, to develop an efficient and accurate mathematical software. They regard the evaluation of fractional integrals of certain polynomial functions, a matrix formulation of the considered method and, in the case of two-step spline collocation methods, the introduction of a suitable starting procedure. We give some details on the proper settings of the parameters of methods, too. Finally, we show some numerical experiments to confirm the theoretical expectation on the order of convergence and to compare one and two step collocation methods.