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

Neetu Garg, Varsha R

TL;DR

The paper addresses multi-dimensional time-fractional Fokker-Planck equations with Caputo derivatives, proposing a Laplace residual power series (LRPS) method to obtain semi-analytical solutions without discretization or linearization. By transforming the problem via the Laplace transform, representing the solution as a fractional power series in time, and enforcing a robust residual condition, the authors derive accurate generalized solutions for linear and nonlinear problems in $d=1,2,3$ dimensions, with error analyses and comparisons to existing methods. Key contributions include a systematic LRPS construction, demonstration on various benchmark problems with exact or reference solutions, and discussion of the influence of the fractional order $\gamma$ and the presence of control terms. The work offers a memory-informed, high-accuracy tool for fractional diffusion and stochastic transport models with potential extensions to space-time fractional FPEs, impacting physics, biology, and finance where anomalous diffusion is important.

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.

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

TL;DR

The paper addresses multi-dimensional time-fractional Fokker-Planck equations with Caputo derivatives, proposing a Laplace residual power series (LRPS) method to obtain semi-analytical solutions without discretization or linearization. By transforming the problem via the Laplace transform, representing the solution as a fractional power series in time, and enforcing a robust residual condition, the authors derive accurate generalized solutions for linear and nonlinear problems in dimensions, with error analyses and comparisons to existing methods. Key contributions include a systematic LRPS construction, demonstration on various benchmark problems with exact or reference solutions, and discussion of the influence of the fractional order and the presence of control terms. The work offers a memory-informed, high-accuracy tool for fractional diffusion and stochastic transport models with potential extensions to space-time fractional FPEs, impacting physics, biology, and finance where anomalous diffusion is important.

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.
Paper Structure (7 sections, 4 theorems, 71 equations, 16 figures, 21 tables)

This paper contains 7 sections, 4 theorems, 71 equations, 16 figures, 21 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of LRPS solution for TFFPE
  4. Applications
  5. Discussions
  6. Consequence of the control term in TFFPE
  7. Conclusion

Key Result

Lemma 2.3

IP99 Consider a piecewise continuous function $\nu(\zeta, \tau)$ defined in the domain $I \times[0 \times \infty)$ of the exponential order $\delta$. Assume $\mathcal{L}\{\nu(\zeta, \tau)\}=\mathcal{V}(\zeta, s)$, then

Figures (16)

  • Figure 1: The variation of solution profile for various fractional orders for Example \ref{['ex1']}.
  • Figure 2: The variation of $\nu_8(\zeta,\tau)$ for different time $\tau$ for Example \ref{['ex1']}.
  • Figure 3: The variation of solution profile for various fractional orders for Example \ref{['ex2']}.
  • Figure 4: The variation of $\nu_8(\zeta,\tau)$ for different time $\tau$ for Example \ref{['ex2']}.
  • Figure 5: Error plots for Example \ref{['ex2']}.
  • ...and 11 more figures

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 1
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2
  • Example 4.1
  • ...and 6 more