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

Abstract

A radial basis functions (RBFs) method for solving nonlinear time- and space-fractional Fokker-Planck equation is presented. The time-fractional derivative is Caputo type and the space-fractional derivative is Caputo or Riemann-Liouville type. The Caputo and Riemann-Liouville fractional derivatives of RBFs are 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 cite{CaoLiChen} and, 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. Two numerical examples are included to demonstrate the applicability, accuracy and stability of the method. Numerical experiments show that the experimental order of convergence is $4-alpha$, $alpha$ is the order of time derivative.