Solving first-order delay differential equation by symmetry lie group

Abstract

The use of delay differential equations has become more popular among researches in the modeling of natural phenomena. A general solution for such models is still largely lacking. This paper develops a symmetry Lie groups method to find a general solution of first order delay differential equation of the form x'(t) = F(t; x(ґ)) + G(t; x(t-ґ)), where ґ is constant and ґ > 0. In this method a symmetry condition is applied on the given equation and some assumptions on infinitesimals are made to find the general solution. After the new coordinate system is found in which the solution depends on only one of the variables, the solution is then expressed in the original coordinates. The paper further generalizes this method to non-linear delay differential equations. At the end we applied these concept on Houseflies model, on which analysis of this model has not been carried out. It is thus shown that the symmetry Lie groups method is valid and feasible to the study of linear and non-linear delay differential equations.