On the Cauchy problem for a linear harmonic oscillator with pure delay

Abstract

In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied. In contrast to earlier works, no positivity conditions are imposed.MSC: Primary 34K06; 39A06; 39B42; secondary 34K26

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

“…where 0 as a matrix denotes the null matrix, explicit expressions for the constant matrices K m r,p defined in (8) can be obtained, as given in our next Lemma. (12), corresponding to vector form for the scalar second order delay equation (5), the constant matrices K m r,p defined in (8), with r ≥ p, can be expressed as follows, depending on being p = 0,…”

“…Different types of representations for the solution of DDE linear systems have been proposed, including solutions by definite integrals and series expansions [6], convolutions with particular solutions or fundamental systems obtained through the method of steps or by Laplace transform [3,4,[7][8][9], or infinite series involving matrix Lambert function [10]. Representations of the exact solution of problem ( 5)- (6) in the case of pure delay, that is, with a = 0, have been given in terms of delayed sine and cosine functions [11], and of delayed exponentials [12]. An expression for the exact solution of ( 5)- (6) was presented in Rodríguez et al [13], as a preliminary problem resulting from applying the method of separation of variables to an unidimensional wave equation with delay.…”

Representation of Classical Solutions to a Linear Wave Equation with Pure Delay

“…with a new unknown function ξ (cp. [11]), the initial boundary value problem (2.1), (2.2) can be written in the following simplified form with a self-adjoint operator on the right-hand side…”

“…From the practical point of view, the rapid decay condition on the Fourier coefficients of the data given in Equation (5.2) mean a sufficiently high Sobolev regularity of the data and corresponding higher order compatibility conditions at the boundary of (0, l) (cf. [11]).…”

Homodyne detection of short-range Doppler radar using a forced oscillator model

“…These are impressive and causal methods for explaining the system response, such as different circuit structures producing different system responses 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 . For example, self-excited electrical oscillator cases have been based on Adler’s equation 2 3 4 5 6 7 or the Van der Pol equation 8 9 10 11 , therein being combined with perturbation methods, which depend upon many complementary factors, including the physical characteristics of the system and the system’s elements and boundary conditions 12 13 14 15 16 . Moreover, phase-domain analysis has been used in some studies 17 18 19 .…”