Using the q²-Laplace transform and its inverse transform introduced early on by Hahn and deeply studied by Abdi we have prove that the q-analogues of the heat and wave equations are linked as in the classical case of Bragg and Dettman. As an application, we proved first, through the q-wave polynomials, that the q-Hermite and the q-little Jacobi polynomials are related. Second, we have given a q-analogue of the Poisson kernel studied by Fitouhi and Annabi.