Smooth solutions of an initial-value problem for some differential difference equations

Abstract

The subject matter of this paper is an initial-value problem with an initial function for a linear differential difference equation of neutral type. The problem is to find an initial function such that the solution generated by this function has some given smoothness at the points multiple of the delay. The problem is solved using a method of polynomial quasisolutions, which is based on a representation of the unknown function in the form of a polynomial of some degree. Substituting this into the initial problem yields some incorrectness in the sense of degree of polynomials, which is compensated for by introducing some residual into the equation. For this residual, an exact analytical formula as a measure of disturbance of the initial-value problem is obtained. It is shown that if a polynomial quasisolution of degree N is chosen as an initial function for the initial-value problem in question, the solution generated will have smoothness not lower than N at the abutment points.