Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion

Abstract

We present a numerical method to solve a time-space fractional Fokker–Planck equation with a space-time dependent force field F(x, t), and diffusion d(x, t). When the problem being modelled includes time dependent coefficients, the time fractional operator, that typically appears on the right hand side of the fractional equation, should not act on those coefficients and consequently the differential equation can not be simplified using the standard technique of transferring the time fractional operator to the left hand side of the equation. We take this into account when deriving the numerical method. Discussions on the unconditional stability and accuracy of the method are presented, including results that show the order of convergence is affected by the regularity of solutions. The numerical experiments confirm that the convergence of the method is second order in time and space for sufficiently regular solutions and they also illustrate how the order of convergence can depend on the regularity of the solutions. In this case, the rate of convergence can be improved by considering a non-uniform mesh.