On the $q$-analogue of Duhamel's principle

TL;DR

This paper revisits Duhamel's principle, providing a self-contained proof for classical cases and establishing a new $q$-analogue for $q$-evolution equations using Jackson's $q$-difference operator.

Contribution

It introduces a novel $q$-analogue of Duhamel's principle for $q$-evolution equations, extending the classical theory to the $q$-calculus setting.

Findings

Self-contained proof of classical Duhamel's principle

First establishment of a $q$-analogue for $q$-evolution equations

Extension of Duhamel's principle to $q$-difference operators

Abstract

In this paper, we revisit the classical Duhamel's principle and provide a self-contained proof of this fundamental tool for linear evolution equations and systems of coupled equations. Moreover, we establish a $q$-analogue of Duhamel's principle for $q$-evolution equations of order $k\geq 1$ generated by Jackson's $q$-difference operator.