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

The problem area of the present paper is an initial-value problem with the initial function for the linear differential difference equation of neutral type. The problem is stated, which is bound up with finding an initial function such that the solution of the initial-value problem, 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 the case of its substitution into the initial problem, there appears some incorrectness in the sense of dimension of 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 as initial function for the initial-value problem in question, then the solution generated will have smoothness at the abutment points not smaller than degree $N$.