# Operator Ordering and Solution of Pseudo-Evolutionary Equations

## TL;DR

This paper develops a unified analytical framework using umbral and operational techniques to solve pseudo-evolutionary equations, including fractional differential equations, with applications demonstrated through various examples.

## Contribution

It extends existing methods by incorporating operator and time ordering tools to solve a broad class of pseudo-evolutionary equations, especially fractional ones.

## Key findings

- Extended Volterra-Neumann and Feynman-Dyson series to fractional equations
- Demonstrated the effectiveness of umbral and operational methods in solving pseudo-evolutionary problems
- Applied the approach to multiple examples involving fractional calculus

## Abstract

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions of fractional differential equations. We apply the method to a number of examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.