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.