A Fully Discrete Discontinuous Galerkin Method for Nonlinear Fractional Fokker-Planck Equation

The fractional Fokker-Planck equation is often used to characterize anomalous diffusion. In this paper, a fully discrete approximation for the nonlinear spatial fractional Fokker-Planck equation is given, where the discontinuous Galerkin finite element approach is utilized in time domain and the Galerkin finite element approach is utilized in spatial domain. The priori error estimate is derived in detail. Numerical examples are presented which are inline with the theoretical convergence rate. 1. Introduction Many models in physics, chemistry are successfully described by the Langevin equation, which has been introduced almost 100 years before. And for some particular cases, say diffusion, the original Langevin equation can be transformed into the Fokker-Planck equation. H？nggi and Thomas [1] associated a Gaussian distribution of the increments of the noise generating process with the classical Fokker-Planck equation. Sun et al. [2] discussed the fractional model for anomalous diffusion. Metzler et al. [3] and Dubkov and Spagnolo [4] derived the fractional Fokker-Planck equation from different anomalous diffusion procedures. Metzler and Klafter [5] discussed fractional kinetic equation and its relation to the fractional Fokker-Planck equation. Dubkov et al. [6] introduced Fokker-Planck equation for Lévy flights. Now the Fokker-Planck equation is one of the best tools for characterizing anomalous diffusion, especially sub-/super-diffusion. Meanwhile the fractional Fokker-Planck equation has been found to be used in relatively wide field of applied sciences, such as plasma physics, population dynamics, biophysics, engineering, neuroscience, nonlinear hydrodynamics, and marking; see [7–13]. The Fokker-Planck equation describes the changes of a random function in space and in time. So different assumptions on probability density function lead to a variety of space-time equations. In this paper, we mainly study the model described by the following fractional Fokker-Planck equation, which is a special case in [12]: where denotes left fractional derivative with order in the sense of Caputo. There are some numerical methods to find the approximate solutions of the fractional differential equations [14–21]. But the discontinuous Galerkin finite element method is a very attractive method for partial differential equations because of its flexibility and efficiency in terms of mesh and shape functions. And the higher order of convergence can be achieved without over many iterations. Such a method was first proposed and analyzed in the early 1970s as a technique to