Numerical analytical method of studying some linear functional differential equations

This paper presents the results of studies of the scalar linear functional-differential equation of a delay type $\dot x(t)=a(t)x(t-1)+b(t)x(t/q)+f(t)$, $q>1$. The main attention is being 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 the identity, is sought for. The method of polynomial quasi-solution, based on representation of an unknown function $x(t)$ as polynomial of degree $N$ is applied as the method of investigation. Substitution of this function in the original equation results in the residual $\Delta(t)=O(t^N)$, for which an accurate analytical representation is obtained. In this case, the polynomial quasi-solution is understood as exact solution in the form of polynomial of degree $N$, disturbed because of the residual of the original initial problem. The theorems of existence of polynomial quasi-solutions for the considered linear functional-differential equation and exact polynomial solutions have been proved. The results of the numerical experiment are presented.