Numerical Methods For The Eigenvalues Of Delay Differential Equations

This is an overview about numerical methods for the eigenvalues of delay differential equations (DDEs). We consider the approximation of eigenvalues for the linear DDEWe first regard the DDE as an abstract Cauchy problem, then the semi-discrete difference scheme, the partial differential equation representation of a DDE, the forward discretization of the solution operator, the Runge-Kutta (RK) method of the solution operator and the Pseudospectra approximation for the eigenvalues based on the solution operator and infinitesimal generator of the semigroup defined by the solutions are discussed.The last part of the thesis is based on the Lambert W function, an analytic approach for the linear delay differential equations with triangularizable matrices as coefficients. Here we consider some notes regarding the special case where the spectrum of the delay system (1.1.5) can be easily expressed with a formula containing a matrix version of the Lambert W function.