On the complete group classification of the reaction–diffusion equation with a delay

The reaction–diffusion delay differential equationut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ))ut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ)) arises in many applications in the sciences. Group analysis is applied in the study of this equation. A new definition of an equivalence Lie group for delay differential equations is given. As for the Lie group theory of differential equations, the determining equations for the equivalence and admitted Lie groups are constructed. The general solutions of the determining equations are obtained. The complete group classification of the reaction–diffusion equation with delay is presented in the manuscript and the invariant solutions of this equation are constructed.