Second-order Delay Differential Equations Research Articles

A class of polynomial approximation methods to second-order delay differential equations

This paper studies special high-order methods to numerically solve special second-order ordinary differential equations and second-order differential equations with constant delay, which are based on vector field expansion along the shifted and scaled Legendre polynomial orthonormal basis. In the framework of method design, we obtain the second-order perturbation results by transforming the second-order differential equations into first-order ones, and the error accuracy estimation of numerical solutions is achieved by using perturbation results for the problem under consideration. In addition, some numerical tests on specific second-order delay Hamiltonian systems are shown, the purpose of which is to verify the effectiveness of the theoretical findings.

Explicit criteria for the oscillation of Caputo type fractional-order delay differential equations

New explicit criteria for the oscillation of all solutions of first and second-order delay differential equations including the Caputo fractional derivative are presented. Examples illustrating the importance and novelty of the main results are included.

Oscillation Theorems for Nonlinear Second-Order Delay Differential Equations with Some Sublinear Neutral Terms via Canonical Transform

Abstract New sufficient conditions for oscillation of a second-order nonlinear differential equation with some sublinear neutral terms are established via canonical transform and integral averaging method. Examples are provided to illustrate the significance and novelty of the presented results.

Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equationx″(t)+p(t)x(t−τ(t))=0,t≥swhere τ:R→[0,+∞), p:R→R are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case p(t)≤0,t∈R.

Wavelets approach for the solution of nonlinear variable delay differential equations

Abstract In this study, the Laguerre wavelet-oriented numerical scheme for nonlinear first and second-order delay differential equations (DDEs) is offered. The proposed technique is dependent on the truncated series of the Laguerre wavelets approximation of an unknown function. Here, we transform the different ordered DDEs into a system of non-linear algebraic equations with the help of limit points of a sequence of collocation points. Four nonlinear illustrations are involved to prove the efficiency of the planned technique. the Obtained results are equated with the current results, indicating the proposed technique’s accuracy and efficiency.

Delay differential equation of the 2nd order and it's an oscillation yardstick

This study focuses on studying an oscillation of a second-order delay differential equation. Start work, the equation is introduced here with adequate provisions. All the previous is braced by theorems and examplesthat interpret the applicability and the firmness of the acquired provisions

A trigonometrically fitted intra-step block Falkner method for the direct integration of second-order delay differential equations with oscillatory solutions

A trigonometrically fitted intra-step block Falkner method for the direct integration of second-order delay differential equations with oscillatory solutions

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.

Oscillatory Properties of Fourth-Order Advanced Differential Equations

This paper presents a study on the oscillatory behavior of solutions to fourth-order advanced differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first and second-order delay differential equations. The results of this work contribute to a deeper understanding of fourth-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.

Index theory and multiple solutions for asymptotically linear second-order delay differential equations

This paper is concerned with the existence of periodic solutions for asymptotically linear second-order delay differential equations. We will establish an index theory for the linear system directly in the sense that we do not need to change the problem of the original linear system into the problem of an associated Hamiltonian system. By using the critical point theory and the index theory, some new existence results are obtained.

Nonoscillation and Oscillation Criteria for a Class of Second-Order Nonlinear Neutral Delay Differential Equations with Positive and Negative Coefficients

In this paper, we investigate some nonoscillatory and oscillatory solutions for a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. By means of the method of contraction mapping principle and some integral inequality techniques, we extend the recent results provided in the literature.

An a priori error analysis of the hp-version of the C 0-continuous Petrov-Galerkin method for nonlinear second-order delay differential equations

We study an hp-version of the -continuous Petrov-Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bound in the -norm that is fully explicit in the local time steps, in the local approximation degrees, and in the local regularity indexes of the exact solutions. Moreover, we prove that the -continuous Petrov-Galerkin based on special hp-version discretization can yield exponential rates of convergence for analytic solutions with initial singularities. Numerical experiments are provided to illustrate the theoretical results.

Second-order nonlinear differential equations: existence, uniqueness and global exponential stability of doubly measure pseudo-almost automorphic solutions

The main aim of this work is to study a second-order nonlinear differential equation with mixed delay and time-varying coefficients. By employing the fixed-point theorem and some properties of the doubly measure pseudo almost automorphic functions, we prove the existence, uniqueness and global exponential stability of doubly measure pseudo-almost automorphic solution to the second-order delay differential equation. Our approach generalizes the classical results on weighted pseudo almost automorphic functions. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of the obtained results.

Criteria for the Oscillation of Solutions to Linear Second-Order Delay Differential Equation with a Damping Term

The aim of this work is to present new oscillation results for a class of second-order delay differential equations with damping term. The new criterion of oscillation depends on improving the asymptotic properties of the positive solutions of the studied equation by using an iterative technique. Our results extend some of the results recently published in the literature.

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

Exact solution to a multidimensional wave equation with delay

This paper deals with a mixed problem for the wave equation with discrete delay τ>0,utt(t,x)=a12Δxu(t,x)+a22Δxu(t−τ,x)+b1u(t,x)+b2u(t−τ,x),t>τ,0≤x≤l,with Dirichlet boundary conditions. The exact infinite series solution is constructed by the method of separation of variables, where the time-dependent functions of the decomposition satisfy second-order delay differential equations. Our approach is based on and extends the work by Rodríguez, Roales and Martín (Applied Mathematics and Computation, 2012).

New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior

In this paper, we establish new sufficient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term, which are under the non-canonical condition. The results obtained complement and simplify some known results in the relevant literature. Example illustrating the results is included.

Interval oscillation criteria for damped second-order delay differential equations with nonlinearities given by Riemann-Stieltjes integral

The purpose of this paper is to investigate the oscillatory behaviour of certain types of damped second-order forced delay differential equations with nonlinearities given by Riemann-Stieltjes integral. By using the Riccati transformation, some inequalities and integral averaging technique, interval oscillation criteria of both El-Sayed type and Kong type are established. Finally, two examples are presented to illustrate the theoretical results.

An efficient method for solving second-order delay differential equation

An efficient method for solving second-order delay differential equation