Smooth Solutions of Linear Functional Differential Equations of Neutral Type

Abstract

The paper considers an initial-value problem with the initial function for the linear functional differential equation of neutral type with constant coefficients. The problem is stated, which is bound up with finding an initial function such that the solution of the initial-value problem, which is generated by this function, possesses some desired smoothness at the points multiple to the delay. For the purpose of solving this problem we use the method of polynomial quasi-solutions, whose basis is formed by the concept of an unknown function of the form of a polynomial of some degree. In case of its substitution into the initial problem, there appears some incorrectness in the sense of dimension of the polynomials, which is compensated by introducing into the equation some residual, for which a precise analytical formula, which characterizes the measure of disturbance of the considered initial-value problem. It is shown that if a polynomial quasi-solution of degree N has been chosen in the capacity of the initial function for the initial-value problem under scrutiny, then the solution generated will have the smoothness at the abutment points not smaller than the degree N.