  By using Jackson's q-exponential function we introduce the generating
function, the recursive formulas and the second order q-differential equation
for the q-Hermite polynomials. This allows us to solve the q-heat equation in
terms of q-Kampe de Feriet polynomials with arbitrary N moving zeroes, and to
find operator solution for the Initial Value Problem for the q-heat equation.
By the q-analog of the Cole-Hopf transformation we construct the q-Burgers type
nonlinear heat equation with quadratic dispersion and the cubic nonlinearity.
In q -> 1 limit it reduces to the standard Burgers equation. Exact solutions
for the q-Burgers equation in the form of moving poles, singular and regular
q-shock soliton solutions are found.
