Computational analysis of a normalized time-fractional Fokker–Planck equation

Abstract

We propose a normalized time-fractional Fokker–Planck equation (TFFPE). A finite difference method is used to develop a computational method for solving the equation, and the system’s dynamics are investigated through computational simulations. The proposed model demonstrates accuracy and efficiency in approximating analytical solutions. Numerical tests validate the method’s effectiveness and highlight the impact of various fractional orders on the dynamics of the normalized time-fractional Fokker–Planck equation. The numerical tests emphasize the significant impact of different fractional orders on the temporal evolution of the system. Specifically, the computational results demonstrate how varying the fractional order influences the diffusion process, with lower orders exhibiting stronger memory effects and slower diffusion, while higher orders lead to faster propagation and a transition towards classical diffusion behavior. This work contributes to the understanding of fractional dynamics and provides a robust tool for simulating time-fractional systems.