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

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.