On the Properties of the Semigroup Generated by the RL Fractional Integral

TL;DR

This paper investigates the mathematical properties of semigroups generated by fractional integral operators within Bochner-Lebesgue spaces, which are relevant for solving PDEs across various scientific fields.

Contribution

It provides a detailed analysis of the properties of semigroups generated by fractional integrals, expanding understanding of their role in PDEs.

Findings

Characterization of semigroup properties

Application to PDEs in multiple fields

Extension of classical semigroup theory

Abstract

For operators $A$, it is sometimes possible to define $e^{At}$ as an operator in and of itself provided it meets certain regularity conditions. Like $e^{\lambda x}$ for ODEs, this operator is useful for solving PDEs involving the operator $A$. We call the set of $e^{At}$ a semigroup generated by $A$. In this paper, we discuss the properties of semigroups generated by the fractional integral, an operator appearing in PDEs in increasingly many fields, over Bochner-Lebesgue spaces.