Separation of variables and exact solutions to nonlinear diffusion equations with x-dependent convection and absorption

This paper considers nonlinear diffusion equations with x-dependent convection and source terms: ut=(D(u)ux)x+Q(x,u)ux+P(x,u). The functional separation of variables of the equations is studied by using the generalized conditional symmetry approach. We formulate conditions for such equations which admit the functionally separable solutions. As a consequence, some exact solutions to the resulting equations are constructed. Finally, we consider a special case for the equations which admit the functionally separable solutions when the convection and source terms are independent of x.