Symbolic computation on generalized Hopf–Cole transformation for a forced Burgers model with variable coefficients from fluid dynamics

Considering the inhomogeneities of media, nonuniformities of boundaries and external forces, a forced Burgers model with space- and time-dependent coefficients is hereby investigated. In this paper, we perform symbolic computation and construct the generalized Hopf–Cole transformation from such a model to the standard heat equation with the relevant constraint conditions on the variable-coefficient and external-force functions. Physically speaking, this transformation provides the feasibility of linearizing many forced and/or variable-coefficient Burgers models from various branches of physics. Specially, we present the N-shock-wave-like solution, based on which the coalescence structures of shock waves with inhomogeneous and forcing effects are discussed, and possible applications in some fields are also pointed out. In like manner, we can also generalize the Hopf–Cole transformation to bilinearize many other variable-coefficient nonlinear evolution equations.