Numerical analytical method of studying some linear functional differential equations

This paper presents the results of studying a scalar linear functional differential equation of a delay type ?(t) = a(t)x(t ? 1) + b(t)x(t/q) + f(t), q > 1. Primary attention is given to the original problem with the initial point, when the initial condition is specified at the initial point, and the classical solution, whose substitution into the original equation transforms it into an identity, is sought. The method of polynomial quasi-solutions, based on representation of an unknown function x(t) as a polynomial of degree N, is applied as the method of investigation. Substitution of this function into the original equation yields a residual Δ(t) = O(t ^N ), for which an accurate analytical representation is obtained. In this case, the polynomial quasi-solution is understood as an exact solution in the form of a polynomial of degree N, disturbed because of the residual of the original initial problem. Theorems of existence of polynomial quasi-solutions for the considered linear functional differential equation and exact polynomial solutions have been proved. Results of a numerical experiment are presented.