4 - Fractional Diffusion and Heavy Tail Distributions: Stable Distribution

Abstract

In this chapter, our objective is twofold. First, we establish a connection between the stable distributions with fractional calculus. This is accomplished by defining appropriate fractional diffusion equations, the fundamental solution of which provides the PDF for the univariate and multivariate stable distributions.1 Second, by using some analytic-numerical approaches such as the homotopy perturbation method, the Adomian decomposition method, and the variational iteration method, which are used for solving partial differential equations (PDEs), we obtain some analytic-numerical approximations for the PDF of the univariate and multivariate stable distributions.