Spectral Analysis And Solution Expansion Of A Class Of Delay Differential Equations

The study of delay differential equation (DDE) has attracted many experts and scholars because of the wide practical background of it, and many good results have been gained. However, those results mainly focus on the qualitative analysis of the equations, namely the existence, uniqueness, boundedness and stability of the system solutions, but there is a lack of quantitative analysis of the equations. However, if we expand the solution of a system, then we can get a more explicit understanding of the inner structure, convergence property and dynamic behavior of the solution and give a more satisfying numerical simulation as well. That is to say, the qualitative analysis of the system is developed into quantitative analysis.The aim of the present paper is to provide the asymptotic expansion of the solution to DDE.The model we study describes the vibration of machine tools with time-delay in high precision cutting process in practice. The linearization of this model yields to the equation below:where u(t) denotes the direction of the corresponding oscillation mode, and k,α, k_s,m,N are all physical parameters, the meanings of which refer to [8]. Reference [8] studied the stability of it. Whereas, we mainly discuss the spectrum analysis of the system operator and the expansion of the solution. First, by applying functional analysis, we rewrite the second order delay differential equation into abstract evolutionary equation in a Hilbert state space and prove the two equations are equivalent and well-posed. Then with the method provided in [9], we give the asymptotic expression of the eigenvalues of the system by a detailed spectrum analysis. After that, we discuss the properties of the eigenvectors and prove that they cannot form a basis for the state space. However, we still obtain the asymptotic expansion of the solution of the equation according to the eigenvectors. In this way, we solve the problem of solution expansion using semigroup theory of linear operators. Therefore, we assert that the qualitative analysis of the system is indeed developed into quantitative analysis.At the end of this paper, we provide a numerical simulation of the solution to illustrate our results intuitionistically and expressly. By comparing the traditional iterations with our method, we indicate the advantages of our solution expansion. We find that the solution of the system drives to stability as t becomes lager, so our numerical simulation is more precise than those iterations. But when t is small, the result has no theoretic verification by now, thus we suggest the traditional iterations for simulation in this situation.Although we study the model of vibration of machine tools with time-delay, the method we use in this paper can also be applied to the analysis of other models with delay because of its generality.