The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.