Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay

•Nonlinear delay PDEs are considered.•Reaction-diffusion and wave-type PDEs are studied.•New indirect methods for constructing exact solutions are described.•New functional separable solutions are presented.•New generalized traveling-wave solutions are obtained. We describe new indirect methods for constructing exact solutions of nonlinear PDEs with delay. Features of the proposed methods are illustrated by nonlinear delay reaction-diffusion and wave-type equations with variable coefficients. All the presented equations contain from three to seven arbitrary functions, which depend on the spatial variable and/or unknown quantity. We obtain new generalized traveling-wave solutions and functional separable solutions in explicit or implicit form for equations with constant delay as well as exact solutions with time-varying delay (a special case of which are pantograph-type PDEs with proportional delay). We show that the proposed methods can also be applied to construct exact solutions for nonlinear systems of coupled delay PDEs and higher-order delay PDEs. The considered equations and their exact solutions can be used to formulate test problems to check the adequacy and estimate the accuracy of numerical and approximate analytical methods of solving nonlinear initial-boundary value problems for delay PDEs.