Neutral Delay Differential Equations Research Articles

Over-order multiplicities and their application in controlling delay dynamics. On zeros’ distribution of linear combinations of Kummer hypergeometric functions

A series of recent works have shown that, for a system of linear functional differential equations, a spectral value having a multiplicity exceeding the order of the system tends to correspond to the spectral abscissa of the system, a property called MID for multiplicity-induced-dominancy. In particular, when this multiplicity coincides with the degree of the characteristic quasipolynomial, this property is called generic MID (GMID), in opposition to the intermediate MID (IMID), which corresponds to a multiplicity strictly smaller than the degree. The GMID has been fully characterized for single-delay retarded as well as neutral delay-differential equations thanks to the representation of the corresponding quasipolynomial in terms of a Kummer hypergeometric function. However, apart from partial results, in full generality, no result of the literature enables the characterization of the dominance of a spectral value having an intermediate multiplicity, which is essentially due to the lack of existing results among the open literature pertaining to linear combinations of Kummer functions’ zeros distribution. In this work, we overcome this difficulty and we further investigate the MID to cover the so-called over-order MID, that is, the cases where the multiplicity is larger than the order of the corresponding differential equation.

Comparison Theorems for Oscillation of Higher-Order Neutral Delay Differential Equations

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons.

Finite time stability analysis for fractional stochastic neutral delay differential equations

Finite time stability analysis for fractional stochastic neutral delay differential equations

Ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations with semi-Markovian switching signals

The ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations (NSDDEs) are investigated in this paper. Different from many previous works, the highly nonlinear NSDDEs with semi-Markov switching signals are considered for the first time in this paper. Meanwhile, the time delay function in this paper is only required to meet much more relaxed restrictions, which can invalidate plentiful methods with requirements on its derivatives for the stability of NSDDEs. To overcome this difficulty, several novel techniques to tackle NSDDEs with such a delay are developed. Furthermore, a crucial property on the ergodic semi-Markov process is established, which plays a key role in the proof on the existence and boundedness of global solution. The generalized Khasminskii-type theorems are established for the existence and uniqueness of global solution by applying new methods and this property, and the criteria for boundedness and stability are also provided. In particular, feasible measures are provided to deal with the neutral term in our stability criteria. Finally, the effectiveness is verified by a numerical example.

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

AbstractThis article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Stability analysis of linear fractional neutral delay differential equations

Stability analysis of linear fractional neutral delay differential equations

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Some Oscillatory Criteria for Second-Order Emden–Fowler Neutral Delay Differential Equations

In this paper, by using the Riccati transformation and integral inequality technique, we establish several oscillation criteria for second-order Emden–Fowler neutral delay differential equations under the canonical case and non-canonical case, respectively. Compared with some recent results reported in the literature, we extend the range of the neutral coefficient. Therefore, our results generalize to some of the results presented in the literature. Furthermore, several examples are provided to illustrate our conclusions.

Analyze the Eigenvalues of a Neutral Delay Differential Equation by Employing the Generalized Lambert W Function

Analyze the Eigenvalues of a Neutral Delay Differential Equation by Employing the Generalized Lambert W Function

Boundedness and stability of nonlinear hybrid neutral stochastic delay differential equation with Lévy jumps under different structures

This paper investigates the boundedness and stability of a class of nonlinear hybrid neutral stochastic differential delay systems with Lévy jumps and different structures. The coefficients in this system satisfy the local Lipschitz condition and a suitable Khasminskii-type condition, and the state space of the system is separated into two subsets, the existence uniqueness, asymptotic boundedness, and exponential stability of the system are obtained by designing a new Lyapunov function and applying the M-matrix technique as well as dealing with the non-differentiable delay function. Different with the existing work, we not only consider the neutral term, but also the case of the delay function being bounded and non-differentiable. At last, numerical examples are performed to manifest the obtained results.

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.

Solving neutral delay differential equations using least square method based on successive integration technique

The main objective of this work is to propose the Least square method (LSM) using successive integration technique for solving Neutral delay differential equations (NDDEs). Continuous LSM and Discrete LSM have been presented by adopting different orthogonal polynomials as weighted basis functions. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear NDDEs have been provided to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems.

Solving Neutral Delay Differential Equations Using Galerkin Weighted Residual Method Based on Successive Integration Technique and its Residual Error Correction

Objectives: The main objectives of this work are to solve Neutral Delay Differential Equations (NDDEs) using Galerkin weighted residual method based on successive integration technique and to obtain the Estimation of Error using Residual function. Methods: The Galerkin weighted residual method based on successive integration technique is proposed to obtain approximate solutions of the NDDEs. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered. Findings: Numerical examples of linear and nonlinear NDDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. Novelty: From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems. Keywords: Galerkin Weighted Residual method, Polynomials, Hermite, Bernoulli, Chebyshev, Fibonacci, Successive integration technique, Neutral Delay Differential Equations

Neutral delay differential equationmodel of an optically injected Kerr cavity.

A neutral delay differential equation(NDDE) model of a Kerr cavity with external coherent injection is developed that can be considered as a generalization of the Ikeda map with second- and higher-order dispersion being taken into account. It is shown that this model has solutions in the form of dissipative solitons both in the low dissipation limit, where the model can be reduced to the Lugiato-Lefever equation(LLE), and beyond this limit, where the soliton is eventually destroyed by the Cherenkov radiation. Unlike the standard LLE, the NDDE model is able to describe the overlap of multiple resonances associated with different cavity modes.

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.

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.

Asymptotic and Oscillatory Analysis of Fourth-Order Nonlinear Differential Equations with p-Laplacian-like Operators and Neutral Delay Arguments

This paper delves into the asymptotic and oscillatory behavior of all classes of solutions of fourth-order nonlinear neutral delay differential equations in the noncanonical form with damping terms. This research aims to improve the relationships between the solutions of these equations and their corresponding functions and derivatives. By refining these relationships, we unveil new insights into the asymptotic properties governing these solutions. These insights lead to the establishment of improved conditions that ensure the nonexistence of any positive solutions to the studied equation, thus obtaining improved oscillation criteria. In light of the broader context, our findings extend and build upon the existing literature in the field of neutral differential equations. To emphasize the importance of the results and their applicability, this paper concludes with some examples.

A Legendre wavelet collocation method for solving neutral delay differential equations

The Legendre wavelet based method has been employed in this paper to investigate neutral delay differential equations. The highest order derivative is approximated by Legendre wavelet using the integral operator technique. Then integrations of Legendre wavelet are used to approximate the lower order derivatives and unknown function. To get an algebraic system of linear or nonlinear equations, approximated values of unknown function and its derivatives are substituted in neutral delay differential equations. On solving the developed system, we get unknown wavelet coefficients and subsequently the approximate solution. To analyze the theoretical usability of the approach, the upper bound of error norm is established. Moreover, the theoretical results are confirmed through few numerical experiments. A comparison of the results of presented method with method available in literature is given to conclude the superiority of the proposed method.

Note on oscillation of neutral differential equations with multiple delays

This note is a reaction on a recently published sufficient condition for oscillation of all solutions of a neutral delay differential equation. It is shown by a counterexample that the result is not correct and the problem is explained in details. Several, more general, classes of neutral differential equations with time-dependent discrete, distributed as well as mixed delays are considered. New sufficient conditions for oscillation of all their solutions are proved. Applications are given for illustration.