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We propose new type of $q$-diffusive heat equation with nonsymmetric$q$-extension of the diffusion term. Written in relative gradient variablesthis system appears as the $q$- viscous Burgers' equation. Exact solutions ofthis equation in polynomial form as generalized Kampe de Feriet polynomials,corresponding dynamical symmetry and description in terms of Bell polynomialsare derived. We found the generating function for these polynomials byapplication of dynamical symmetry and the Zassenhaus formula. We haveconstructed and analyzed shock solitons and their interactions with different$q$. We obtain modification of the soliton relative speeds depending on valueof $q$.For $q< 1$ the soliton speed becomes bounded from above and as a resultin addition to usual Burgers soliton process of fusion, we found a newphenomena, when soliton with higher amplitude but smaller velocity is fissingto two solitons. q-Semiclassical expansion of these equations are found interms of Bernoulli polynomials in power of $\ln q$.

q-Viscous Burgers' Equation: Dynamical Symmetry, Shock Solitons and q-Semiclassical Expansion