# Generalised Fractional Evolution Equations of Caputo Type

## TL;DR

This paper investigates generalized Caputo-type fractional evolution equations, establishing well-posedness and stochastic representations, and introduces a probabilistic extension of Mittag-Leffler functions to encompass classical and extended fractional PDEs.

## Contribution

It provides new analytical and probabilistic methods for solving generalized fractional evolution equations, including a probabilistic interpretation of Mittag-Leffler functions.

## Key findings

- Established well-posedness of generalized fractional equations.
- Derived stochastic representations for solutions.
- Extended classical fractional PDEs with new probabilistic insights.

## Abstract

This paper is devoted to the study of generalised time-fractional evolution equations involving Caputo type derivatives. Using analytical methods and probabilistic arguments we obtain well-posedness results and stochastic representations for the solutions. These results encompass known linear and non-linear equations from classical fractional partial differential equations such as the time-space-fractional diffusion equation, as well as their far reaching extensions. \\ Meaning is given to a probabilistic generalisation of Mittag-Leffler functions.