Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time

•Heat/diffusion differential-difference equations with delay are considered.•Solutions to a Stokes problem with a periodic boundary condition are given.•Exact solutions to non-linear differential-difference heat equations are obtained.•Non-linear systems of coupled reaction–diffusion equations are considered.•Some equations and systems contain arbitrary functions.

We consider heat and diffusion equations with finite relaxation time which ensure a finite speed of propagation of disturbances. We use the Cattaneo–Vernotte model for the heat flux and obtain a number of exact solutions to the corresponding linear differential-difference heat equation. We also give exact solutions to two one-dimensional Stokes problem for a differential-difference mass/heat transfer equation (without a source and with a linear source) with a periodic boundary condition.We describe a number of exact solutions to non-linear differential-difference heat equations of the formT¯t=div[f(T)∇T]+g(T¯),T¯=T(x,t+τ),where ττ is the relaxation time. In addition, we obtain some exact solutions to non-linear systems of two coupled reaction–diffusion equations with finite relaxation time and present several exact solutions of non-linear reaction–diffusion equations with time-varying delay of the formut=kuxx+F(u,w),w=u(x,t−τ),where τ=τ(t)τ=τ(t).All equations in question contain arbitrary functions or free parameters. Their solutions can be used to solve certain problems and test numerical methods for non-linear partial differential-difference equations (delay partial differential equations).