The Explicit Formula for Solution of Anomalous Diffusion Equation in the Multi-Dimensional Space
D. K. DurdievBukhara Branch of the Institute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan, 200100, Bukhara, UzbekistanElina ShishkinaVoronezh State University, 394018, Voronezh, RussiaС. М. СитникBelgorod State National Research University (BelGU), 308015, Belgorod, Russia
ABI
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.