  By generating function based on the Jackson's q-exponential function and
standard exponential function, we introduce a new q-analogue of Hermite and
Kampe-de Feriet polynomials. In contrast to standard Hermite polynomials, with
triple recurrence relation, our polynomials satisfy multiple term recurrence
relation, derived by the q-logarithmic function. It allow us to introduce the
q-Heat equation with standard time evolution and the q-deformed space
derivative. We found solution of this equation in terms of q-Kampe-de Feriet
polynomials with arbitrary number of moving zeros, and solved the initial value
problem in operator form. By q-analog of the Cole-Hopf transformation we find a
new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular
everywhere single and multiple q-Shock soliton solutions and their time
evolution are studied. A novel, self-similarity property of these q-shock
solitons is found. The results are extended to the time dependent
q-Schr\"{o}dinger equation and the q-Madelung fluid type representation is
derived.
