Direct and inverse source problems for heat equation in quantum calculus

TL;DR

This paper investigates the existence, uniqueness, and stability of solutions for heat equations in quantum calculus, including inverse problems, using Fourier series in Hilbert spaces, with applications to various q-operators.

Contribution

It introduces a framework for solving weak and inverse heat problems in quantum calculus within Hilbert spaces, extending to several q-operator examples.

Findings

Proved existence and uniqueness of solutions.

Established stability of inverse problem solutions.

Applied analysis to multiple q-operator examples.

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.