The analytical (spectral) representation of the solution of delay algebraic-differential equations

A new approach to finding analytical solutions of linear delay algebraic-differential equations is suggested. The analytical form of the solution is determined in terms of the infinite set of eigenvalues of a parametric matrix whose entries are the delay-time operators $\exp(-p\tau)$, where $p$ is the Laplace operator. In order to compute constants in the solution of the homogeneous equations, one must analytically find higher derivatives at the input of the delay operator. Issues of stopping the computation of the infinite spectrum upon determining a certain number of its components are discussed.