New generalized and functional separable solutions to non-linear delay reaction–diffusion equations

• Non-linear delay reaction–diffusion equations are considered.
• New generalized and functional separable solutions are presented.
• Most equations contain arbitrary functions of one argument.
• Some equations contain an arbitrary function of two arguments.
• Some solutions involve infinitely many free parameters.

We present a number of new generalized separable, functional separable, periodic and antiperiodic exact solutions to non-linear delay reaction–diffusion equations of the formut=kuxx+F(u,w),ut=kuxx+F(u,w),where u=u(x,t)u=u(x,t) and w=u(x,t−τ)w=u(x,t−τ), with τ   being the delay time. The generalized separable solutions are sought in the form u=∑n=1NΦn(x)Ψn(t), with the functions Φn(x)Φn(x) and Ψn(t)Ψn(t) to be determined in the analysis. Most of the equations considered contain one or two arbitrary functions of a single argument or one arbitrary function of two arguments of special form. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for non-linear delay PDEs. Some results are extended to non-linear delay partial differential equations of any order.