Construction of exact solutions in implicit form for PDEs: New functional separable solutions of non-linear reaction-diffusion equations with variable coefficients

The paper deals with different classes of non-linear reaction-diffusion equations with variable coefficients c(x)u(t) = [a(x)f(u)u(x)](x) + b(x)g(u), that admit exact solutions. The direct method for constructing functional separable solutions to these and more complex non-linear equations of mathematical physics is described. The method is based on the representation of solutions in implicit form integral h(u) du = xi(x)omega(t) + eta(x), where the functions h(u), xi(x), eta(x), and omega(t) are determined further by analyzing the resulting functional-differential equations. Examples of specific reaction-diffusion type equations and their exact solutions are given. The main attention is paid to non-linear equations of a fairly general form, which contain several arbitrary functions dependent on the unknown u and /or the spatial variable x (it is important to note that exact solutions of non-linear PDEs, that contain arbitrary functions and therefore have significant generality, are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial-boundary value problems). Many new generalized traveling-wave solutions and functional separable solutions are described.