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

Abstract

The reaction-diffusion delay differential equation 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.

On the complete group classification of the one‐dimensional nonlinear Klein–Gordon equation with a delay

“…For finding an equivalence Lie group of partial differential equations, one can apply the infinitesimal approach [32] . ‡ For delay differential equations, the notion of an equivalence Lie group was introduced in [38,39]: it is a Lie group corresponding to an infinitesimal generator, which satisfies determining equations. This Lie group of transformations provides a set of potential equivalence transformations.…”

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

“…Such solutions are dealt with in many studies (e.g., see the papers [2][3][4][5][6][7] and references in them). A complete group analysis of the non-linear delay equation (1) was carried out in [29]; four equations of the form (1) were found to admit invariant solutions (two of these equations have degenerate solutions that are linear in x). The studies [30][31][32][33] describe a number of simple separable, generalized separable, and functional separable exact solutions to Eq.…”

Symmetry analysis of the nonlinear two-dimensional Klein-Gordon equation with a time-varying delay

“…A number of studies have dealt with the nonlinear delay reaction–diffusion equations of the form and the nonlinear delay Klein–Gordon equation where . From the physical viewpoint, the delay is responsible for inertia in mass/heat transfer processes, that is, the system does not respond to an action immediately at the same time t when the action is applied, but at a relaxation time τ later.…”

“…A number of studies [7,[12][13][14][15][16][17] have dealt with the nonlinear delay reaction-diffusion equations of the form…”

“…In [28], complete group classification of second-order delay ordinary differential equations is presented. In [14,17], group classifications of the delay reaction-diffusion equation (1) and the one-dimensional nonlinear delay Klein-Gordon equation (2) are given, and all invariant solutions are found.…”