A Fast and Accurate Solver for the Fractional Fokker-Planck Equation with Dirac-Delta Initial Conditions

TL;DR

This work develops a fast, high-precision solver for the free-space fractional Fokker-Planck equation with Dirac-delta initial data by leveraging an integral representation based on Fourier analysis and a one-dimensional Bessel formulation. It introduces robust numerical techniques to handle singular near-origin integrals and slow-decaying far-field tails via expansion, re-weighting, and windowing, together with a scaling law that enables efficient evaluation across scales and parameters. Crucially, the method avoids time stepping and achieves machine-precision accuracy for moderate times while remaining applicable to high dimensions and to initial data given by sums of Gaussians through linear superposition. The solver demonstrates accurate solution displays, rigorous error analyses, and scalable running times, paving the way for extensions to variable coefficients and more complex initial conditions in nonlocal diffusion problems.

Abstract

The classical Fokker-Planck equation (FPE) is a key tool in physics for describing systems influenced by drag forces and Gaussian noise, with applications spanning multiple fields. We consider the fractional Fokker-Planck equation (FFPE), which models the time evolution of probability densities for systems driven by Lévy processes, relevant in scenarios where Gaussian assumptions fail. The paper presents an efficient and accurate numerical approach for the free-space FFPE with constant coefficients and Dirac-delta initial conditions. This method utilizes the integral representation of the solutions and enables the efficient handling of very high-dimensional problems using fast algorithms. Our work is the first to present a high-precision numerical solver for the free-space FFPE with Dirac-delta initial conditions. In addition to Dirac-delta initial data, we demonstrate the effectiveness of our method for initial conditions given by sums of Gaussians. This opens the door for future research on more complex scenarios, including those with variable coefficients and other types of initial conditions.

Figures (10)

• Figure 1: Relative error comparison with and without scaling for $\alpha = 1 / 2$, $D_{\textnormal{o}} = 0$, $D_{\textnormal{f}} = 8$, and $d = 7$. The left plot shows the relative error with scaling, while the right plot shows the ratio of the error with scaling to the error without scaling. In the right plot, areas where the ratio is smaller than 1 (indicating that scaling performs better) are shown in red, and areas where the ratio is larger than 1 (indicating that scaling performs worse) are shown in blue. The color bars indicate the magnitude of the errors and their ratios on a logarithmic scale. These plots illustrate the effectiveness of scaling in reducing the error across different values of $y$ and $t$ for large $y$.
• Figure 2: A graphical representation of the function $\exp(- \tau |k|^{2\alpha})$, where $\tau = 0.01$ and $\alpha = 0.4$. The plot highlights the near-origin singularity and the far-field slow decay.
• Figure 3: An example of the windowing function $w_{M,\gamma}(z)$ with parameters $M = 10$ and $\gamma = 0.5$. The key points $(\gamma M, 1)$ and $(M, 0)$ are marked in red. This illustrates how the function smoothly transitions from $1$ to $0$ around the interval $[\gamma M, M]$, demonstrating the behavior of the windowing function in truncating values outside the specified range.
• Figure 4: Solution representations in one dimension with $x_{0} = 0$ and $b = 0$ for $d = 1$ and $t = 0.05$. The plot shows $p(x, t)$ for different parameter settings: $D_{\textnormal{o}} = 4, D_{\textnormal{f}} = 0$ (pure ordinary diffusion, blue), $D_{\textnormal{o}} = 2, D_{\textnormal{f}} = 2, \alpha = 0.7$ (mixed diffusion, green), and $D_{\textnormal{o}} = 0, D_{\textnormal{f}} = 4, \alpha = 0.3$ (pure fractional diffusion, red). The graph illustrates how the density function $p(x, t)$ varies with $x$ under different diffusion conditions.
• Figure 5: Special case examples for various values of $d$ and $y$ with $t = 0.025$. The left plot illustrates the behavior of $\tilde{p}(y, 0.025)$ with parameter settings $\alpha = 1 / 2$, $D_{\textnormal{o}} = 1$, and $D_{\textnormal{f}} = 8$ (mixed diffusion). The right plot shows $\tilde{p}(y, 0.025)$ with $\alpha = 1 / 2$, $D_{\textnormal{o}} = 0$, and $D_{\textnormal{f}} = 8$ (pure fractional diffusion). Different colors represent different values of $d$. The plots depict how $\tilde{p}(y, t)$ varies with $y$ for different dimensions $d$ under the given diffusion conditions.

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Theorem 3.1
• Remark C.1