Local and Non-local Fractional Porous Media Equations

TL;DR

This paper explores fractional porous media equations with local, non-local, and mixed derivatives, showing they admit q-Gaussian solutions and better fit S ext&Ps 500 data than classical models.

Contribution

It introduces fractional extensions of the porous media equation with different derivative types and demonstrates their solutions and improved data fitting capabilities.

Findings

All fractional models admit q-Gaussian solutions.

Local and non-local models fit S ext&Ps 500 data better than classical models.

Differences in free parameters affect fitting quality.

Abstract

Recently it was observed that the probability distribution of the price return in S\&P500 can be modeled by $q$-Gaussian distributions, where various phases (weak, strong super diffusion and normal diffusion) are separated by different fitting parameters (Phys Rev. E 99, 062313, 2019). Here we analyze the fractional extensions of the porous media equation and show that all of them admit solutions in terms of generalized $q$-Gaussian functions. Three kinds of "fractionalization" are considered: \textit{local}, referring to the situation where the fractional derivatives for both space and time are local; \textit{non-local}, where both space and time fractional derivatives are non-local; and \textit{mixed}, where one derivative is local, and another is non-local. Although, for the \textit{local} and \textit{non-local} cases we find $q$-Gaussian solutions , they differ in the number of free parameters. This makes differences to the quality of fitting to the real data. We test the results for the S\&P 500 price return and found that the local and non-local schemes fit the data better than the classic porous media equation.