Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay

Many mathematical models in science are described by delay differential equations. Recent developments of the theory of delay differential equations allow one to derive a method for studying this class of equations by the group analysis method. So far there have been few investigations of delay differential equations by group analysis method. The present article studies the delay partial differential equation∂u∂t(x,t)+u(x,t)∂u∂x(x,t)=G(u(x,t-τ),u(x,t)).The complete group classification of this delay equation with the functional G=G(u(x,t)-u(x,t-τ))+H(u)G=G(u(x,t)-u(x,t-τ))+H(u) is given in this article. The classification is considered with respect to the functions GG and HH.

► An application of group analysis to delay partial differential equation is provided.
► The nonhomogeneous inviscid Burgers equation with delay is considered.
► The complete group classification of the equation is developed.
► All representations of invariant solutions of the equation are presented.