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

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 $k\ge 1$ generated by Jackson's $q$-difference operator.