Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation

Abstract

In this study, a radial basis functions (RBFs) method for solving nonlinear timeand space-fractional Fokker-Planck equation is presented. The time-fractional derivative is of the Caputo type, and the space-fractional derivatives are considered in the sense of Caputo or Riemann-Liouville. The Caputo and Riemann-Liouville fractional derivatives of RBFs are computed and utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step, the time-fractional derivative is approximated by the high order formulas introduced in [6], and then a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy, and stability of the method. Numerical experiments show that the experimental order of convergence is 4 − α where α is the order of time derivative.