Delay Argument Research Articles

A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

This paper deals with the oscillation of the first-order differential equation with several delay arguments x′t+∑i=1mpitxτit=0,t≥t0, where the functions pi,τi∈Ct0,∞,R+, for every i=1,2,…,m,τit≤t for t≥t0 and limt→∞τit=∞. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.

Some New Oscillation Results for Fourth-Order Neutral Differential Equations with Delay Argument

The aim of this paper is to study the oscillatory properties of 4th-order neutral differential equations. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve well-known oscillation results in the literate. Symmetry plays an important role in determining the right way to study these equation. An example to illustrate the results is given.

Fractional optimal control problems with time-varying delay: A new delay fractional Euler–Lagrange equations

In this paper, a new Euler–Lagrange formulation is derived for fractional optimal control problems with time-varying system which is called delay fractional Euler–Lagrange equations. The delay fractional Euler–Lagrange equations are fractional differential equations with two-point boundary values which have time-varying delay argument. Despite complexity of these equations, in this paper, an attractive numerical plan is presented to solve them. In the proposed numerical scheme, by utilizing approximation operator, the new delay fractional Euler–Lagrange equations are changed to ordinary delay differential equations system with integer order, and this system is solved by using Legendre–Gauss collocation method. Finally, some numerical examples are implemented to show the efficiency and accuracy of the proposed method. The examples are focused on fractional optimal control problems of a harmonic oscillator with retarded damping as applications in engineering. The numerical result shows that the suggested method has high accuracy in comparison with the existing methods.

The best parametrization for solving the boundary value problem for the system of differential-algebraic equations with delay

In this paper, we considered the numerical approach for solving a nonlinear boundary value problem for the system of differential-algebraic equations with delay argument. The shooting method is used to solve the boundary value problem. The Newton method is used to find the parameter of shooting. To overcome the difficulties associated with the choice of the initial approximation we apply E. Lahaye’s parameter continuation method. If the curve of the solution contains limit points, the method diverges. Then to find the parameter we used the method of continuation with respect to the best parameter - the length of the curve of the solution set. The solution is constructed by advancing the sequence of values of the parameter. With a discrete continuation, the initial-value problem is transformed by a finite-difference representation of the derivatives and entering the best argument and the corresponding equation of hypersphere. The resulting system is solved using the Newton method. To find the values of the functions at the delay point Lagrange polynomial with three points is used. An example of the behavior of an elastoviscoplastic rod is considered.

Oscillation of Second-Order Emden–Fowler Neutral Differential Equations with Advanced and Delay Arguments

In this paper, we study the oscillation of the second-order Emden–Fowler neutral differential equations with advanced and delay arguments $$\begin{aligned} (r(t)(x(t)+p(t)x(\tau (t)))')'+q(t)x^\gamma (\sigma (t))=0,\ \quad t\geqslant t_0, \end{aligned}$$where $$\tau (t)\leqslant t$$, $$\sigma (t)\geqslant t$$. Sufficient conditions for oscillation of the equation are obtained by using the inequality principle, comparison principle and Riccati transform. Two examples are given to illustrate the results.

Oscillatory behavior of the second order general noncanonical differential equations

In this paper, we introduce new oscillatory criteria for the second order noncanonical differential equation with delay argument (r(t)(y′(t))α)′+p(t)yβ(τ(t))=0.Our oscillatory results essentially extend the earlier ones.

Oscillation of delay difference equations with finite non-monotone arguments

In this paper, the oscillation of difference equations with multiple non-monotone delay arguments is studied. Three criteria of these equations are obtained for oscillation. And examples are given to show the meanings of the theorems.

Oscillation of Nabla Difference Equations with Several Delay Arguments

Oscillation of Nabla Difference Equations with Several Delay Arguments

A necessary and sufficient condition for the oscillation of first order linear difference equations with several delay arguments

In this article, we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions. In particular, we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non-negative. Examples and tables are given to illustrate our results.

The Problem of Two Bodies with a Finite Velocity of Gravity

From the moment of Newton’s discoveryof the law of universal gravitation, ordinary differentialequations were used to study the motion of bodies,since it was assumed that the velocity of gravitationis infinite. However, in reality the velocity of gravityis finite, which is consistent with the theory of relativityof Einstein, which postulated that the velocity ofgravity matches the velocity of light, and the studiesconducted by S. Kopeikin and E. Fomalont on the fundamentallimit of the velocity of gravity. Due to thedelay of the gravitational field for studying the motionof bodies, the mathematical apparatus based on differentialequations with a delay argument is the mostacceptable one. These equations are used to constructand study the mathematical model of the motion of twobodies. It is shown that the motion of these bodies withidentical masses (with finite velocity of gravity!) is notcarried out in accordance with Kepler’s laws.

Stability Analysis of one Prey and Two Predators Model

We projected a 3 species Ecological model with a Prey and 2 predators. Distributed kind of delay is incorporated within the interaction of prey and second predator is taken for investigation. The system dynamics is studied at its interior equilibrium purpose with exponential kind of delay kernels. The impact of your time delay on the energising behavior of the system is studied exploitation Numerical simulation. it's shown that the delay arguments with totally different delay parameters exhibit wealthy dynamics.

Prey Predator Dynamics in Three Species Ecological Model

In this paper we proposed a three species Ecological model with a Prey and Two predators. Distributed type of delay is incorporated in the interaction of prey and second predator is taken for investigation. The system dynamics is studied at its interior equilibrium point with exponential type of delay kernel. The effect of Time delay on the dynamical behavior of the system is studied using Numerical simulation. It is shown that the delay arguments with different weight parameters exhibit rich dynamics and the system is asymptotically stable.

Oscillation of fourth-order strongly noncanonical differential equations with delay argument

The aim of this paper is to study oscillatory properties of the fourth-order strongly noncanonical equation of the form \t\t\t(r3(t)(r2(t)(r1(t)y′(t))′)′)′+p(t)y(τ(t))=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(r_{3}(t) \\bigl(r_{2}(t) \\bigl(r_{1}(t)y'(t) \\bigr)' \\bigr)' \\bigr)'+p(t)y \\bigl( \\tau (t) \\bigr)=0, $$\\end{document} where int ^{infty }frac{1}{r_{i}(s)},mathrm {d}{s}<infty , i=1,2,3. Reducing possible classes of the nonoscillatory solutions, new oscillatory criteria are established.

A Robust Numerical Scheme for Singularly Perturbed Delay Differential Equations with Turning Point

In this article, we propose a parameter uniform numerical method for singularly perturbed delay differential equations with turning point. Using interpolation, a new algorithm is developed to tackle the retarded term. A fitted operator finite difference scheme using Il’in Allen Southwell fitting factor is used for numerical discretization. Bounds on the analytical solution and its derivatives are obtained. The efficiency of the proposed numerical method is illustrated via applying the proposed method on some test problems. The solution is proved to be stable and ε-uniform error estimates are derived. It is shown that the delay argument has significant effect on the solution behavior.

Dynamics of Regulatory Mechanisms of the Human Organism on the Basic Hierarchical Levels of the Organization

Based on the methodology of the regulatory, the main methods of mathematical modeling of the regulatory mechanisms of living systems at the molecular-genetic, cellular and organ-tissue levels are presented. The constructed mathematical models of the regulatory of the cardiovascular system, the liver and the thyroid gland are presented in the form of systems of functional-differential equations with delay arguments, and also results of qualitative analysis for these models are obtained. Computer models have been developed for quantitative analysis and assessment of the state of biological systems. The most common regularities in the functioning of systems and various behavioral regimes, such as stationary, self-oscillating, dynamic chaos and a “black hole” are revealed.

Nonstationary Vibrations of Electroelastic Cylindrical Shell in Acoustic Layer

The propagation of a pressure wave in a fluid bounded by two parallel plane boundaries generated by an infinitely long cylindrical electroelastic shell submerged into the fluid is considered. To describe the motion of the shell and the processes in the fluid, the equations of the linear theory of shells generalized to the case of electromechanics and the acoustic approximation are used. The problem-solving method is based on application of the image source method, the method of separation of variables, and Laplace integral transform. The method is used to reduce the problem to an infinite system of Volterra equations with delay arguments, which is numerically solved using the reduction method and regularizing procedures. The hydrodynamic pressure is calculated for the case where either step-wise or sinusoidal electric pulse load is applied to the continuous electrodes of the shell.

Oscillation criterion for first-order linear differential equations with several delay arguments

By using iterated estimates involving all delay arguments, we establish an oscillation criterion for first-order linear differential equations with several delay arguments. This criterion is focused on the interaction among the delay arguments, instead of converting the original equation into a single delay equation and using existing results. Several examples illustrate the results obtained.

Investigating Dynamics of One Weakly Nonlinear System with Delay Argument

Investigating Dynamics of One Weakly Nonlinear System with Delay Argument

Oscillations of Solutions of Neutral Nonlinear Differential Equations

This paper aims to establish a new class of differential equations and study the oscillatory behavior of a kind of first-order neutral nonlinear differential equation with time delay arguments. The oscillatory properties of the solutions of the type of first order neutral functional differential equations applied in chemomedical problems are studied. Sufficient conditions for the oscillations of solutions of the above equations are obtained. Also, some results which demonstrate in literature [1-4] will be extended, and the paper focuses on expanding the main finding of literature [2, 3]. Moreover, a new kind of method to be used to discuss the properties of oscillation of the first-order neutral nonlinear differential equations and some theorems are obtained in the paper.

An Efficient Nonstandard Finite Difference Scheme for a Class of Fractional Chaotic Systems

In this paper, we formulate a new nonstandard finite difference (NSFD) scheme to study the dynamic treatments of a class of fractional chaotic systems. To design the new proposed scheme, an appropriate nonlocal framework is applied for the discretization of nonlinear terms. This method is easy to implement and preserves some important physical properties of the considered model, e.g., fixed points and their stability. Additionally, this scheme is explicit and inexpensive to solve fractional differential equations (FDEs). From a practical point of view, the stability analysis and chaotic behavior of three novel fractional systems are provided by the proposed approach. Numerical simulations and comparative results confirm that this scheme is also successful for the fractional chaotic systems with delay arguments.