Functional separable solutions of nonlinear convection–diffusion equations with variable coefficients

•Nonlinear convection-diffusion equations are considered.•Arbitrary functions are included in equations.•New functional separable solutions are presented.•Specific equations and their exact solutions are analyzed.•Solutions of delay convection-diffusion equations are given. The paper presents a number of new functional separable solutions to nonlinear convection–diffusion equations of the formc(x)ut=[a(x)ux]x+[b(x)+p(x)f(u)]ux,where f(u) is an arbitrary function. It shows that any three of the four variable coefficients a(x), b(x), c(x), p(x) of such equations can be chosen arbitrarily, and the remaining coefficient can be expressed through the others. Examples of specific equations and their exact solutions are given. The results obtained are generalized to more-complex nonlinear PDEs with variable coefficients. Also some functional separable solutions to nonlinear convection–diffusion equations with delayut=uxx+a(x)f(u,w)ux,w=u(x,t−τ),where τ > 0 is the delay time and f(u, w) is an arbitrary function of two arguments, are obtained.