On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus

Abstract

In this paper, we explore a generalised solution of the Cauchy problems for the $q$ -heat and $q$ -wave equations which are generated by Jackson’s and the $q$ -Sturm-Liouville operators with respect to $t$ and $x$ , respectively. For this, we use a new method, where a crucial tool is used to represent functions in the Fourier series expansions in a Hilbert space on quantum calculus. We show that these solutions can be represented by explicit formulas generated by the $q$ -Mittag-Leffler function. Moreover, we prove the unique existence and stability of the weak solutions.