The generating equations method: Constructing exact solutions to delay reaction–diffusion systems and other non-linear coupled delay PDEs

Abstract

We present a new method for constructing exact solutions to non-linear systems of coupled delay reaction–diffusion equations of the formut=k1uxx+F(u,u¯,w,w¯),wt=k2wxx+G(u,u¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), and τ is the delay time. The method relies on using generalized and functional separable solutions, obtained for single delay reaction–diffusion equations (called generating equations), to construct exact solutions of more complex systems of coupled delay equations. All of the systems considered contain two or more arbitrary functions of two or three arguments. We show that the generating equations method can also be used for (i) non-linear reaction–diffusion systems with time-varying delay τ=τ(t), (ii) delay reaction–diffusion systems with varying transfer coefficients, (iii) multicomponent delay reaction–diffusion systems, (iv) multidimensional delay reaction–diffusion systems, and (v) non-linear delay systems of higher-order PDEs. The results may be suitable for solving certain model problems and testing approximate analytical and numerical methods for some classes of similar and more complex non-linear coupled delay systems.