Analytical-numeric formulas for the probability density function of multivariate stable and geo-stable distributions

The main problem in applying the multivariate stable and geo-stable distributions is the lack of a closed-form formula for the probability density function. In this article, we provide a solution by using the connection between these two distributions and fractional calculus. The analytical-numeric formulas for the probability density function for these two probability distributions are obtained via three analytic approximation methods - homotopy perturbation method, Adomian decomposition method, and variational iteration method. Finally, the convergent analysis of the applied methods is discussed.