q-Heat Operator and q-Poisson’s Operator

2000 Mathematics Subject Classification: 33D15, 33D90, 39A13

In this paper we study the q-heat and q-Poisson’s operators associated with the q-operator ∆q (see[5]). We begin by summarizing some statements concerning the q-even translation operator Tx,q, defined by Fitouhi and Bouzeffour in [5]. Then, we establish some basic properties of the q-heat semi-group such as boundedness and positivity. In the second part, we introduce the q-Poisson operator P^t, and address its main properties. We show in particular how these operators can be used to solve the initial and boundary value problems related to the q-heat and q-Laplace equation respectively.