A MATLAB Implementation of Spline Collocation Methods for Fractional Differential Equations

The present paper illustrates a MATLAB program for the solution of fractional differential equations. It is based on a spline collocation method on a graded mesh, introduced by Pedas and Tamme in [J. Comput. Appl. Math. 255, 216–230 (2014)]. This is the first program proposed to implement spline collocation methods for fractional differential equations, and it is one of the few algorithms available in the literature for these functional equations. An explicit formulation of the method is derived, and the computational kernel is a nonlinear system to be solved at each time step. Such system involves some fractional integrals, whose analytical expression is given; their computation requires the knowledge of the coefficients of some polynomials and the evaluation of some special functions. The method is written in a compact matrix form, to improve the efficiency of the MATLAB implementation. The overall algorithm is outlined and then the attention is focused on some routines, which are given. In particular, some MATLAB native routines are used to evaluate special functions and to compute the coefficients of some polynomials. The complete list of the input and output parameters is available. Finally, an example of usage of the MATLAB program on a test problem is provided and some numerical experiments are shown.