Solution of a system of linear delay differential equations using the matrix Lambert function

An approach for the analytical solution to systems of delay differential equations (DDEs) has been developed using the matrix Lambert function. To generalize the Lambert function method for scalar DDEs, we introduce a new matrix, Q when the coefficient matrices in a system of DDEs do not commute. The solution has the form of an infinite series of modes written in terms of the matrix Lambert functions. The essential advantage of this approach is the similarity with the concept of the state transition matrix in linear ordinary differential equations (ODEs), enabling its use for general classes of linear delay differential equations. Examples are presented to illustrate by comparison to numerical methods.