Numerical scheme for Erd\'elyi-Kober fractional diffusion equation using Galerkin-Hermite method

TL;DR

This paper develops and analyzes a stable, convergent numerical scheme using Galerkin-Hermite methods for solving Erdélyi-Kober fractional diffusion equations, which model generalized grey Brownian motion.

Contribution

It introduces a novel numerical scheme for Erdélyi-Kober fractional diffusion equations, including stability and convergence analysis, and applies Hermite functions for full discretization.

Findings

The scheme is stable and converges to the exact solution.

Convergence is slower than first order due to singular time terms.

Numerical results validate the theoretical analysis.

Abstract

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erd\'elyi-Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion and grey Brownian motion. To obtain convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erd\'elyi-Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singular in time term appearing in the main equation the proposed method converges slower than first order. Finally, we provide the numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.