The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space

TL;DR

This paper derives an explicit multi-dimensional solution for the anomalous diffusion equation using Laplace and Fourier transforms, involving Mittag-Leffler functions and Fox H-functions, applicable in infinite domains with specific initial and boundary conditions.

Contribution

It provides the first explicit formula for the solution of the n-dimensional anomalous diffusion equation with memory effects, expanding analytical tools for complex diffusion processes.

Findings

Explicit solution expressed via Fox H-functions

Solution applicable to infinite domains with specific initial conditions

Demonstrates the connection between anomalous diffusion and integro-differential equations

Abstract

This paper intends on obtaining the explicit solution of $n$-dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is $t^{-\alpha}E_{1-\alpha, 1-\alpha}(-t^{1-\alpha}),\alpha\in(0, 1),$ where $E_{\alpha, \beta}$ is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.