Well-posedness of heat and wave equations generated by Rubin's q-difference operator in Sobolev spaces

TL;DR

This paper establishes the well-posedness and explicit solution formulas for heat and wave equations involving Rubin's q-difference operator within Sobolev spaces, extending classical PDE theory to difference-differential operators.

Contribution

It introduces the analysis of parabolic and hyperbolic equations generated by Rubin's q-difference operator and proves their well-posedness with explicit solution representations.

Findings

Unique solutions exist for the heat and wave equations with Rubin's operator.

Solutions can be explicitly represented in Sobolev spaces.

The results extend classical PDE theory to q-difference operators.

Abstract

In this paper, we investigate difference-differential operators of parabolic and hyperbolic types. Namely, we consider non-homogenous heat and wave equations for Rubin's difference operator. Well-posedness results are obtained in appropriate Sobolev type spaces. In particular, we prove that the heat and wave equations generated by Rubin's difference operator have unique solutions. We even show that these solutions can be represented by explicit formulas