Second-order Delay Differential Equations Research Articles

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.

On the qualitative behavior of the solutions to second-order neutral delay differential equations

Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented.

Numerical Approach for Solving Delay Differential Equations with Boundary Conditions

In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena with data simulation. Thus, an efficient numerical method is needed for the numerical treatment of time delay in the applications. The proposed direct block method computes the numerical solutions at two points concurrently at each computed step along the interval. The types of delays involved in this research are constant delay, pantograph delay, and time-dependent delay. The shooting technique is utilized to deal with the boundary conditions by applying a Newton-like method to guess the next initial values. The analysis of the proposed method based on the order, consistency, convergence, and stability of the method are discussed in detail. Four tested problems are presented to measure the efficiency of the developed direct multistep block method. The numerical simulation indicates that the proposed direct multistep block method performs better than existing methods in terms of accuracy, total function calls, and execution times.

On Oscillation of Second Order Delay Differential Equations with a Sublinear Neutral Term

We derive sufficient conditions for the oscillation of solutions second-order delay differential equation containing a sublinear neutral term. Our conditions differ from the earlier ones even in the special cases, linear or nonlinear, and as illustrated with an example, we not only extend but also improve several results in the literature.

Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

Abstract In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form ( r ( x ′ ) γ ) ′ ( t ) + q ( t ) x α ( τ ( t ) ) = 0 {\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0 Under the assumption ∫∞(r(n))−1/γdη=∞, we consider the two cases when γ > α and γ < α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation

We consider a singularly perturbed boundary value problem for a second-order ordinary differential equation with nonlinear right-hand side containing functions of delayed argument. We prove the existence of a solution with a transition layer that has a more sophisticated structure than the ones studied before and construct a uniform asymptotic approximation to this solution with respect to a small parameter. Vasil’eva’s method is used when constructing the asymptotic approximation, while the existence theorem is proved by combining the matching method and the asymptotic differential inequality method. Conditions for the existence of a solution with monotone internal transition and boundary layers are stated. An example illustrating the class of problems studied here is given.

Spurious roots of delay differential equations using Galerkin approximations

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design

This work revisits recent results on maximal multiplicity induced-dominancy for spectral values in reduced-order time-delay systems and extends it to the general class of second-order retarded differential equations. A parametric multiplicity-induced-dominancy property is characterized, allowing to a delayed stabilizing design with reduced complexity. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates’ delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Sensitivity of the control design with respect to the parameters uncertainties/variation is discussed. Various reduced order examples illustrate the applicative perspectives of the approach.

On oscillatory second-order nonlinear delay differential equations of neutral type

On oscillatory second-order nonlinear delay differential equations of neutral type

An [formula omitted]-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays

We present and analyze an hp-version of the discontinuous Galerkin (DG) time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bounds in the L2- and L∞-norm that are fully explicit with respect to the local time steps, the local approximation orders, and the local regularity of the exact solution. We further prove that the hp-DG scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start-up singularities. Moreover, we also propose an hp-DG scheme for the state dependent delay differential equations. We illustrate the theoretical results with a series of numerical experiments.

Qualitative Behavior of Solutions of Second Order Differential Equations

In this work, we study the oscillation of second-order delay differential equations, by employing a refinement of the generalized Riccati substitution. We establish a new oscillation criterion. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. We illustrate the results with some examples.

Uniform Convergence Method for a Delay Differential Problem with Layer Behaviour

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a linear second-order delay differential equation is examined. It is proved that it gives essentially a first-order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.

An h-p version of the continuous Petrov-Galerkin time stepping method for nonlinear second-order delay differential equations

We propose and analyze an h-p version of the continuous Petrov-Galerkin time stepping method for nonlinear second-order delay differential equations with vanishing delays. We establish a priori error estimates in the L2- and L∞-norm that are completely explicit in the local time steps, in the local polynomial degrees, and in the local regularity of the exact solutions. In particular, it is proved that, for analytic solutions with start-up singularities exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Numerical examples are given illustrating the theoretical results.

Nonlinear second-order delay differential equation

Nonlinear second-order delay differential equation

Solving Oscillatory Delay Differential Equations Using Block Hybrid Methods

A set of order condition for block explicit hybrid method up to order five is presented and, based on the order conditions, two-point block explicit hybrid method of order five for the approximation of special second order delay differential equations is derived. The method is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutions. The efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted, which clearly demonstrated the superiority of the trigonometrically fitted block hybrid method.

Necessary and Sufficient Conditions for Oscillation of Nonlinear Second-Order Delay Differential Equations

Necessary and Sufficient Conditions for Oscillation of Nonlinear Second-Order Delay Differential Equations

Periodic solutions to singular damped delay differential equations with impulses

In this paper, we discuss the existence of periodic solutions for nonautonomous second-order damped delay differential equations with singular non-linearities, in presence of impulsive effects. Simple sufficient conditions are provided that enable us to obtain positive periodic solutions. Our approach is based on a variational method. Some recent results in the literature are extended.

On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments

This paper provides a method of finding periodical solutions of the second-order neutral delay differential equations with piecewise constant arguments of the form x''(t)+px''(t-1)=qx(2[frac{t+1}{2}])+f(t), where [cdot] denotes the greatest integer function, p and q are nonzero constants, and f is a periodic function of t. This reduces the 2n-periodic solvable problem to a system of n+1 linear equations. Furthermore, by applying the well-known properties of a linear system in the algebra, all existence conditions are described for 2n-periodical solutions that render explicit formula for these solutions.

An Exponentially Fitted Spline Method for Second-Order Singularly Perturbed Delay Differential Equations

This paper deals with the singularly perturbed boundary value problem for the second-order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A fitted numerical scheme has been developed to solve the boundary value problem. The difference scheme which is shown to converge to the continuous solution uniformly with respect to the perturbation parameter is illustrated with numerical results.

Positive Non-monotone Solutions of Second-Order Delay Differential Equations

In this paper, we study the non-monotonic behavior of solutions of the second-order delay differential equations. We establish the criteria for positive solutions to be non-monotonic on a bounded interval. The results are the first to be reported on the non-monotonic behavior of solutions of functional differential equations. In the final section, two examples are given which illustrate the significance of our results.