Direct and inverse source problems for heat equation in quantum calculus

Abstract

In this paper, we explore the  weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces.  For this, we use the Fourier series expansions.  Moreover,  we prove  the existence, uniqueness and stability of the weak solution of the inverse problem with   a final determination condition. We give some examples such as  the $q$-Sturm–Liouville problem, the $q$-Bessel operator,  the $q$-deformed Hamiltonian, the fractional Sturm-Liouville operator, and  the restricted fractional Laplacian, covered by our analysis.