Exact solutions to new classes of reaction-diffusion equations containing delay and arbitrary functions

Abstract

The following one-dimensional nonlinear delay reaction-diffusion equations are considered: $$u_t = [G(u)u_x ]_x + F(u,w)$$ , where u=u(x,t), w=u(x,t−τ), and τ is the delay time. New classes of these equations are described that depend on one or two arbitrary functions of one argument and that have exact simple separable, generalized separable, and functional separable solutions. The functional constraints method is used to seek solutions. Exact solutions are also presented for the more complex three-dimensional delay reaction-diffusion equations $$u_t = div[G(u)\nabla u] + F(u,w)$$ . All of the derived solutions are new, contain free parameters, and can be used to solve certain problems and test approximate analytical and numerical methods for solving these or more complex nonlinear delay equations.