Exact solutions to the nonlinear diffusion-convection equation with variable coefficients and source term

Abstract

The nonlinear diffusion-convection equation f(x)ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u) with variable coefficients and source term has been studied. This equation is symmetrically reduced by the generalized conditional symmetry method. Some exact solutions to the resulting equations are constructed, with the diffusion terms D(u)=um (m≠-1,0,1) and D(u)=eu. These exact solutions are also the generalized functional separable solutions. Solutions to the equation with constant coefficients are covered by those exact solutions to the equation with variable coefficients.