Mittag-Leffler function and fractional differential equations

TL;DR

This paper introduces a novel operational-umbral method to solve fractional Fokker-Planck equations with Riemann-Liouville derivatives, enabling effective handling of operator ordering and advancing fractional differential equation solutions.

Contribution

It presents a new combined operational-umbral approach for solving fractional PDEs, specifically the fractional Fokker-Planck equation, with improved operator handling capabilities.

Findings

Successfully applied the method to fractional Fokker-Planck equations

Enhanced ability to manage operator ordering in fractional derivatives

Demonstrated the method's effectiveness through theoretical analysis

Abstract

We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $\alpha$ ($0 < \alpha < 1$) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering.