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

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. Wellposedness 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.