On a Class of Multi-Dimensional Non-linear Time-Fractional Fokker-Planck Equations Capturing Brownian Motion

TL;DR

This paper introduces a semi-analytical Laplace residual power series method for solving multi-dimensional non-linear time-fractional Fokker-Planck equations, effectively capturing anomalous diffusion and nonlocal effects.

Contribution

It develops a novel semi-analytical solution technique combining Laplace transform and residual power series for complex fractional PDEs.

Findings

The method provides accurate solutions for multi-dimensional non-linear problems.

Numerical results show the approach's efficiency and stability.

Error analysis confirms the method's reliability for fractional Fokker-Planck equations.

Abstract

The time-fractional Fokker-Planck equation is a key model for characterizing anomalous diffusion, stochastic transport, and non-equilibrium statistical mechanics with applications in finance, chaotic dynamics, optical physics, and biological systems. In this work, we develop a semi-analytical solution for the multi-dimensional time-fractional Fokker-Planck equation employing the Laplace residual power series method. This method blends the Laplace transform and the traditional residual power series method, guaranteeing efficient solutions incorporating the memory and nonlocal effects. To validate the accuracy and effectiveness of the approach, we address several examples, including non-linear problems in multi-dimensions, and analyze the evolution of errors. The numerical simulations are compared with existing methods to confirm the adopted method's strength. The smooth and stable error evolution promises that the suggested method is a powerful tool for analyzing time-fractional Fokker-Planck equations.