Analysis of Univariate Stable Distributions using Fractional Calculus and Real-World Applications

Abstract

In this study we see, how are apply fractional calculus to demonstrate a relationship between the stable distributions? To do so, it is necessary to first formulate the appropriate fractional diffusion equations. Standard diffusion equations can be extended into what are called fractional diffusion equations. This enlargement can be accomplished by considering either a time or a space derivative on a fractional scale. The fractional derivative is used to extend the reach of standard diffusion equations in this article. Obtain some analytic-numerical approximations for the PDF of the univariate stable distributions by employing some analytic-numerical approaches, such as the homotopy perturbation method, the Adomian decomposition method, and the variational iteration method, which are employed to solve partial differential equations (PDEs) and perform stability analysis. By using fractional calculus, one can precisely manage a wide variety of mathematical models. It is applied to a wide variety of problems, including those involving turbulence, pollution, population growth and spread, landscape development, medical imaging, and complex systems.