First-order Delay Differential Equations Research Articles

THEORETICAL ANALYSIS AND NUMERICAL SOLUTION OF LINEAR AND NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper, a collocation method based on the Haar wavelet is presented for the solution of both linear and nonlinear first-order neutral delay differential equations. The Haar functions are used to approximate the first-order derivative, and the approximate solution is obtained by using initial condition and integration. Some examples from the literature are used to test the suggested method efficiency and applicability. A comparison of exact and approximate solutions is given in figures for different numbers of collocation points. The root mean square and maximum absolute errors are calculated for different numbers of collocation points. The rate of convergence is calculated which is approximately equal to 2. The comparison of the present method with the other numerical methods is also given. The results demonstrate that the Haar wavelet collocation method is simple and effective for solving first-order linear and nonlinear neutral delay differential equations.

Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.

Oscillatory Properties Test for Even-Order Di§erential Equations of Neutral type

This paper presents a study on the oscillatory behavior of solutions to even-order neutral differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first-order delay differential equations, Riccati technique and integral averages technique. The results of this work contribute to a deeper understanding of even-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area.

Second-Order Neutral Differential Equations with Distributed Deviating Arguments: Oscillatory Behavior

In this paper, new criteria for a class oscillation of second-order delay differential equations with distributed deviating arguments were established. Our method mainly depends on making sharper estimates for the non-oscillatory solutions of the studied equation. By using the Ricati technique and comparison theorems that compare the studied equations with first-order delay differential equations, we obtained new and less restrictive conditions that ensure the oscillation of all solutions of the studied equation. Further, we give an illustrative example.

On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations

ABSTRACT The authors examine the oscillatory behavior of solutions to a class of third order half-linear delay differential equations. The results are obtained by a comparison with first-order linear delay differential equations whose oscillatory characters are known. An example is provided to illustrate the results.

Fractional-Step Method with Interpolation for Solving a System of First-Order 2D Hyperbolic Delay Differential Equations

In this article, we consider a delayed system of first-order hyperbolic differential equations. The presence of the delay term in first-order hyperbolic delay differential equations poses significant challenges in both analysis and numerical solutions. The delay term also makes it more difficult to use standard numerical methods for solving differential equations, as these methods often require that the differential equation be evaluated at the current time step. To overcome these challenges, specialized numerical methods and analytical techniques have been developed for solving first-order hyperbolic delay differential equations. We investigated and presented analytical results, such as the maximum principle and stability results. The propagation of discontinuities in the solution was also discussed, providing a framework for understanding its behavior. We presented a fractional-step method using a backward finite difference scheme and showed that the scheme is almost first-order convergent in space and time through the derivation of the error estimate. Additionally, we demonstrated an application of the proposed method to the problem of variable delay differential equations. We demonstrated the practical application of the proposed method to solving variable delay differential equations. The proposed algorithm is based on a numerical approximation method that utilizes a finite difference scheme to discretize the differential equation. We validated our theoretical results through numerical experiments.

A Computational Block Method with Five Hybrid-Points for Differential Equations Containing a Piece-wise Constant Delay

AbstractThis paper presents an efficient numerical approach for first-order delay differential equations containing a piece-wise constant delay. The strategy is based on a five-point hybrid block method that has been developed for ordinary differential equations. We will use the interpolation technique for the evaluation of delay terms that are not defined at the grid points. The main characteristics are discussed, including zero stability, local truncation errors, convergence and stability region. The method is easy to implement, and the numerical experiments show the efficiency and accuracy of the proposed method, compared to other methods appeared in the literature.

On periodic solutions of a recruitment model with iterative terms and a nonlinear harvesting

We consider a first-order delay differential equation involving iterative terms. We prove the existence of positive periodic and bounded solutions by utilizing the Schauder's fixed point theorem combined with the Green's functions method. Furthermore, by virtue of the Banach contraction principle, the uniqueness and stability of the solution are also analyzed. Our new results are illustrated with two examples that show the feasibility of our main findings.

Some New Oscillation Results for Higher-Order Nonlinear Differential Equations with a Nonlinear Neutral Term

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results.

New oscillation results for higher order nonlinear differential equations with a nonlinear neutral terms

The paper deals with the oscillation of higher order nonlinear differential equations with a nonlinear neutral term. The main results are proved via utilizing an integral criterion as well as a comparison theorem with first-order delay differential equation whose oscillatory properties are known. The proposed theorems improve, extend, and simplify existing ones in the literature. The results are associated with four numerical examples.

Nonlinear effects of instantaneous and delayed state dependence in a delayed feedback loop

<p style='text-indent:20px;'>We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We present the bifurcation diagram in the parameter plane of the two feedback strengths showing how periodic orbits bifurcate from a curve of Hopf bifurcations and disappear along a curve where both period and amplitude grow beyond bound as the orbits become saw-tooth shaped. We then 'switch on' the delay within the state-dependent feedback term, reflected by a parameter <inline-formula><tex-math id="M1">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula>. Our main conclusion is that the new parameter <inline-formula><tex-math id="M2">\begin{document}$ b $\end{document}</tex-math></inline-formula> has an immediate effect: as soon as <inline-formula><tex-math id="M3">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula> the bifurcation diagram for <inline-formula><tex-math id="M4">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> changes qualitatively and, specifically, the nature of the limiting saw-tooth shaped periodic orbits changes. Moreover, we show — numerically and through center manifold analysis — that a degeneracy at <inline-formula><tex-math id="M5">\begin{document}$ b = 1/3 $\end{document}</tex-math></inline-formula> of an equilibrium with a double real eigenvalue zero leads to a further qualitative change and acts as an organizing center for the bifurcation diagram. Our results demonstrate that state dependence in delayed feedback terms may give rise to new dynamics and, moreover, that the observed dynamics may change significantly when the state-dependent feedback depends on past states of the system. This is expected to have implications for models arising in different application contexts, such as models of human balancing and conceptual climate models of delayed action oscillator type.</p>

Finite difference methods with linear interpolation for solving a coupled system of hyperbolic delay differential equations

In this article a system of first-order hyperbolic delay differential equations is considered. The maximum principle is proved for the problem considered. Further, the stability of the solution is established as an application of the maximum principle. The propagation of the discontinuity of the solution is also established. A piecewise uniform mesh is designed for solving the problem. On the piecewise uniform mesh, conditionally stable and unconditionally stable finite difference methods with piecewise linear interpolation are suggested to solve the problem. It is proved that the methods are consistent, stable, and convergent. Numerical illustrative examples are given to validate the theoretical results.

Explicit values of the oscillation bounds for linear delay differential equations with monotone argument

The problem of finding the oscillation bounds for first-order linear delay differential equations has been in the focus of the oscillation theory for a long time. Although numerous estimates for the oscillation bounds are available in the literature, their explicit values were not known. In this paper, we give the oscillation bounds explicitly in terms of the real branches of the Lambert [Formula: see text] function.

A partly sharp oscillation criterion for first-order delay differential equations

In this paper, a partly sharp oscillation criterion is established for the first-order delay differential equationx′(t)+p(t)x(τ(t))=0,t≥T0,where p(t) is continuous and p(t)≥0 for all t≥T0;τ(t) is continuous, non-decreasing, τ(t)<t for all t≥T0, and limt→∞τ(t)=+∞. By improving techniques from previous researches, we show that when α=lim inft→∞∫τ(t)tp(s)ds∈(0,1e], all solutions of the equation are oscillatory iflim supt→∞∫τ(t)tp(s)ds>2α+2λ1−1under no additional constraints, where λ1 is the smaller root of the equation λ=eαλ. This result is proved to be weaker than previous ones, and sharp when α∈(0,ln22) by constructing a specific example.

Improved oscillation criteria for first-order delay differential equations with variable delay

This article concerns the oscillation of solutions to the delay differential equation \(x'(t)+p(t)x(\tau(t))=0\). Conditions for oscillation have been stated as lower bounds for the limit superior and limit inferior of \(\int_\tau^t p\). In this article we match the bound for the best case in [7], without using one of their hypotheses. Then assuming that hypothesis, we obtain a bound lower than the one in [12]. Then we apply our results to an equation with several delays. We employ iterated estimates of the solution. For more information see https://ejde.math.txstate.edu/Volumes/2021/32/abstr.html

Oscillation tests for nonlinear differential equations with nonmonotone delays

In this paper, our aim is to investigate a class of first-order nonlinear delay differential equations with several deviating arguments. In addition, we present some sufficient conditions for the oscillatory solutions of these equations. Differing from other studies in the literature, delay terms are not necessarily monotone. Finally, we give examples to demonstrate the results.

Further results on Ulam stability for a system of first-order nonsingular delay differential equations

AbstractThis paper is concerned with a system governed by nonsingular delay differential equations. We study theβ-Ulam-type stability of the mentioned system. The investigations are carried out over compact and unbounded intervals. Before proceeding to the main results, we convert the system into an equivalent integral equation and then establish an existence theorem for the addressed system. To justify the application of the reported results, an example along with graphical representation is illustrated at the end of the paper.

A new oscillation criterion for first-order delay differential equations by iteration

A new oscillation criterion is established for the first-order delay differential equationx′(t)+p(t)x(τ(t))=0,t≥t0,where p(t),τ(t)∈C([t0,+∞),[0,+∞)),τ(t) is non-decreasing, τ(t) < t and limt→∞τ(t)=∞. By discretizing not only x(t) but also ∫τ(t)tp(s)ds, we show that all solutions of the equation are oscillatory iflim supt→∞∫τ(t)tp(s)ds>1+lnλ1λ1−α221−α−α22+α3λ13!1−α−α22+α3λ13!+α4λ124!1−α−α22+α3λ13!+α4λ124!+α5λ135!1−α−⋱,(*)where α=lim inft→∞∫τ(t)tp(s)ds∈(0,1e] and λ1 is the smaller root of the equation λ=eαλ. The right-hand side of (*) is proved to be strictly smaller than previous ones with an illustrative example. In particular, when α=1e, our result leads to lim supt→∞∫τ(t)tp(s)ds>0.5621.

Oscillatory Properties of Odd-Order Delay Differential Equations with Distribution Deviating Arguments

Throughout this work, new criteria for the asymptotic behavior and oscillation of a class of odd-order delay differential equations with distributed deviating arguments are established. Our method is essentially based on establishing sharper estimates for positive solutions of the studied equation, using an iterative technique. Moreover, the iterative technique allows us to test the oscillation, even when the related results fail to apply. By establishing new comparison theorems that compare the nth-order equations with one or a couple of first-order delay differential equations, we obtain new conditions for oscillation of all solutions of the studied equation. To show the importance of our results, we provide two examples.

Absolutely stable difference scheme for a general class of singular perturbation problems

This paper presents an absolutely stable noniterative difference scheme for solving a general class of singular perturbation problems having left, right, internal, or twin boundary layers. The original two-point second-order singular perturbation problem is approximated by a first-order delay differential equation with a variable deviating argument. This delay differential equation is transformed into a three-term difference equation that can be solved using the Thomas algorithm. The uniqueness and stability analysis are discussed, showing that the method is absolutely stable. An optimal estimate for the deviating argument is obtained to take advantage of the second-order accuracy of the central finite difference method in addition to the absolute stability property. Several problems having left, right, interior, or twin boundary layers are considered to validate and illustrate the method. The numerical results confirm that the deviating argument can stabilize the unstable discretized differential equation and that the new approach is effective in solving the considered class of singular perturbation problems.