Abstract
Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order (1, 2]. Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.
Highlights
The Fokker-Planck equation FPE has commonly been used to describe the Brownian motion of particles
Normal diffusion in an external force field is often modeled in terms of the following Fokker-Planck equation FPE 1 :
∂2 ∂x2 u x, t, 1.1 where m is the mass of the diffusing test particle, η1 denotes the fraction constant characterising the interaction between the test particle and its embedding, and the force is International Journal of Differential Equations related to the external potential through F x dV x /dx
Summary
The Fokker-Planck equation FPE has commonly been used to describe the Brownian motion of particles. Normal diffusion in an external force field is often modeled in terms of the following Fokker-Planck equation FPE 1 :. Yuste and Acedo proposed an explicit finite difference method and a new von Neumann-type stability analysis for the anomalous subdiffusion equation 1.6 with V x 0 They did not give a convergence analysis and pointed out the difficulty of this task when implicit methods are considered. Effective numerical methods and error analysis for the time-space fractional Fokker-Planck equation with a nonlinear source term are still in their infancy and are open problems. We are unaware of any other published work on numerical methods for the timespace fractional Fokker-Planck type equation with a nonlinear source term. We consider the following time-space fractional Fokker-Planck equation with a nonlinear source term TSFFPE-NST :.
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