# Generalized substantial fractional operators and well-posedness of Cauchy problem

## TL;DR

This paper introduces generalized substantial fractional operators, analyzes their fundamental properties, and studies the well-posedness of related fractional differential equations, advancing the mathematical modeling of anomalous diffusion.

## Contribution

It presents new generalized substantial fractional integral and derivatives, and establishes existence, uniqueness, and stability results for associated differential equations.

## Key findings

- New generalized substantial fractional operators introduced
- Fundamental properties of these operators analyzed
- Existence and uniqueness of solutions established

## Abstract

In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.