Lie Group Analysis of Second-Order Non-Linear Neutral Delay Differential Equations

Lie group analysis is applied to second order neutral delay differential equations (NDDEs) to study the properties of the solution by the classification scheme. NDDE is a delay differential equation which contains the derivatives of the unknown function both with and without delays. It turns out that in many cases where retarded delay differential equation (RDDE) fail to model a problem, NDDE provides a solution. This paper extends the classification of second order non-linear RDDE to solvable Lie algebra to that for second order non-linear NDDE. In this classification the second order extension of the general infinitesimal generator acting on second order neutral delay is used to determine the determining equations. Then the resulting equations are solved, and the solvable Lie algebra is obtained, satisfying the inclusion property. Finally, one-parameter Lie groups which are corresponding to NDDEs are determined. This approach provides a theoretical background for constructing invariant solutions.