A high order method for oscillatory delay differential equations

This paper addresses the oscillatory properties of a class of second-order neutral delay differential equations (NDDEs) and introduces a numerical scheme based on a fourth-order predictor-corrector multistep method for solving NDDEs of this type. An example in the family of oscillatory equations has been integrated by using the proposed numerical scheme. Results confirm the theoretical assumptions.


Introduction
Numerous phenomena in physics as well as in biosciences are mathematically described by means of delay differential equations (DDEs), also in case of oscillatory behavior [1]; for instance, the cardiovascular system (CVS) has been depicted by several models based on DDEs with an increasing complex grade.In [2], a cardiovascular model with singularity in derivative function is presented; in [3] the role played, in the nonlinear dynamics, by the delay due to the autonomiccardiac regulation, that exhibits the relationship between the autonomic nervous system (ANS) and the CVS, has been discussed and analyzed in terms of stability.
Recent letterature is reach in physics-based models describing biological and medical systems through DDE, where, also, a qualitative analysis of oscillation and nonoscillation properties has been carried out, as, for instance, in [4], where delay logistic models and their modifications are investigated.In [4], furthermore, a survey on the influence of initial conditions, i.e. surroundings, on the dynamic evolution of biological and ecological systems is illustrated.
In the contest of a qualitative analysis, it is noteworthy the research on mathematical models holding nonlinear delay in a single species or in species with interactions, but also the delay models, such as autonomous ones, where coefficients are constants.For sake of completeness, [5] deals with the asymptotic properties for functional differential equations (FDEs).
In order to represent the dynamic models and, then, reaching their integration, novel and advanced numerical schemes are usually required.Nevertheless, nowadays, identifying the proper numerical approaches still represents an open problem.In this paper, a numerical integration scheme is discussed in order to numerically approach a second-order neutral delay differential equation with oscillatory properties.In details, a predictor-corrector multistep method has been considered, where Adams-Bashfort and Adams-Moulton scheme have been combined in order to gain a fourth-order method.

The Model
This paper is focused on a family of second-order Delay Differential Equations (DDEs) whose structure is given as when w(y) = x(y) In 1978, Brands, in [6], proved that a necessary and sufficient condition for being the equation oscillatory for each ϑ(y), bounded delay, is that the equation x (y) + q(y)x(y) = 0 is oscillatory.In [7,8] the more general equation has been investigated and in case of lim y→∞ A(y) = ∞ and lim y→∞ A(y) < ∞, new oscillation criteria are proposed by Chatzarakis et al.

Numerical Approach
The aim of this section is to present a numerical scheme for approximation of second-order Delay Differential Equations (DDEs) whose structure is given by for y ∈ [y 0 , +∞).In Equation ( 5), x(y) involves also the delay function, and thus, in general, it could be defined the so called history function, as ∀y ≤ y 0 x(y) = x 0 (y) (6) Let us introduce the vectorial function where Then, the equation ( 4) can be reduced to the system The numerical integration procedure is based on a predictor corrector multistep method [11], In details: • Corrector: 3 Steps Adams-Moulton method (O(h 4 )) where F represents the vectorial function associated with the differential equation defining the investigated case, η j the numerical approximation of the vector X at the node t j , for all j = 1, . . ., n, denoting the discretization of the independent variable y through equidistant points.
A numerical scheme of the forth order, for solving ordinary differential equations has been adopted and in particular, in this work, it has been re-written for delay differential equation including the delay as known term with initial conditions defined as constant functions.

Results
The equation ( 4) has been numerically integrated in the range [4,1000], with a constant discretization step h = 0.001.The Figure 1 depicts the sign of the solution, in order to put into evidence its zeros.Indeed, they could not be appreciated in the solution plot, due to the sudden growth of the function modulus, as it can be noticed in Table 1, where the amplitude of solution peaks between consecutive zeros is displayed.
For sake of completeness, Table 2 displays the zeros of the solution The oscillatory behavior of the solution is confirmed, and in particular, it can be observed that the distance between consecutive zeros increases with the independent variable, maintaining a ratio that is around 3.

Conclusions
This paper treats the numerical integration of second-order nonlinear neutral delay differential equations belonging to the family (1).First of all, some oscillation criteria have been cited from literature.Additionally, the Adams-Bashforth-Moulton predictor-corrector numerical scheme to estimate solutions of this class of NDDE has been introduced.This latter has been applied for the numerical integration of a case study in the family of DDEs investigated; the proposed example confirms that, under assumptions (A1)-(A4), equation (1) exhibits oscillatory behavior [8].

Table 1 .
Sign of the numerical solution of 4 versus t, t ∈ [4, 1000] Peaks of the solution of equation 4