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

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. This opens the door for future research on more complex scenarios, including those with variable coefficients and other types of initial conditions. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/ACMathX/FFPEDDIC and in the supplementary materials ( M168290_Supplementary_Materials.pdf [675KB], FFPEDDIC-main.zip [3.68MB]). [Formula: see text]