Linear Neutral Delay Differential Equations Research Articles

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.

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.

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.

Convergence of the Euler Method for Impulsive Neutral Delay Differential Equations

In this paper, we are concerned with a fixed stepsize Euler method for a class of linear impulsive neutral delay differential equations. By taking the partition nodes for the Euler scheme and employing the linear interpolation, we strictly prove the method is convergent of order one. Two examples illustrating the efficiency results are also presented.

A new approaching method for linear neutral delay differential equations by using Clique polynomials

A new approaching method for linear neutral delay differential equations by using Clique polynomials

Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates.

Analytic solutions of linear neutral and non-neutral delay differential equations using the Laplace transform method: featuring higher order poles and resonance

In this article, we extend the Laplace transform method to obtain analytic solutions for linear RDDEs and NDDEs which have real and complex poles of higher order. Furthermore, we present first-order linear DDEs that feature resonance phenomena. The procedure is similar to the one where all of the poles are order one, but requires one to use the appropriate modifications when using Cauchy’s residue theorem for the poles of higher order. The process for obtaining the solution relies on computing the relevant infinite sequence of poles and then determining the Laplace inverse, via the Cauchy residue theorem. For RDDEs, the poles can be obtained in terms of the Lambert W function, but for NDDEs,the complex poles, in most cases, must be computed numerically. We found that an important feature of first-order linear RDDES and NDDES with poles of higher order is that it is possible to incite the resonance phenomena, which in the counterpart ordinary differential equation cannot occur. We show that despite the presence of higher order poles or resonance phenomena, the solutions generated by the Laplace transform method for linear RDDEs and NDDEs that have higher order poles are still accurate.

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Asymptotic-numerical solvers for linear neutral delay differential equations with high-frequency extrinsic oscillations

We present a method to compute efficiently and easily solutions of systems of linear neutral delay differential equations with highly oscillatory forcing terms. This method is based on asymptotic expansions in inverse powers of a perturbed oscillatory parameter. Each term of the asymptotic expansion is derived by recursion. The cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided and show that with few terms of the asymptotic expansion, the solutions are approximated with high accuracy.

Accuracy of the Laplace transform method for linear neutral delay differential equations

In this paper, we study and solve linear neutral delay differential equations. We investigate the reliability and accuracy of applying the Laplace transform to obtain the solutions of linear neutral delay differential equations. These types of equations are more difficult to solve because the time delay appears in the derivative of the state variable. We rely on computer algebra and numerical methods to determine the poles which are required for computing the inverse Laplace transform. The form of the resulting solution is a non-harmonic Fourier series. A sufficient degree of accuracy can often be achieved by using a relatively small number of terms in the associated truncated series. We compare these solutions with the solutions obtained by the classical method of steps and numerical solutions obtained by the discretization of the linear delay differential equations. It is shown that the Laplace transform method provides very reliable and accurate solutions.

A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations

In this paper, we present a new method for solving linear neutral delay differential equations. We derive and illustrate the main features of this novel approach, that combines the Laplace transform method with (harmonic) Fourier series theory. Linear neutral delay differential equations are generally more difficult to solve because the time delay appears in the derivative of the state variable. We rely on computer algebra and numerical methods to implement the method. In addition, we derive an approximate formula for the location of the complex poles, which are required for computing the inverse Laplace transform. The form of the resulting solution, when only the Laplace method is used is a non-harmonic Fourier series. The accuracy of this solution can be improved by including more terms in the associated truncated series, but the convergence to the correct solution is slow. The main goal of this paper is to present a modified method which enables us to account for the terms which are excluded from these truncated Laplace series. That is, the terms in the tail of the infinite series. We include several examples where we compare the solutions generated by the standard Laplace method and the proposed Laplace-Fourier approach. Both solutions require using Cauchy’s residue theorem and finding the real and complex poles. It is shown that the Laplace-Fourier solution provides more accurate solutions than the conventional Laplace transform solution. Finally, since the Laplace-Fourier method generates a solution which is valid for all times, it allows us to accurately approximate the solution at any point with a single calculation.

Results on the Behavior of the Solutions for Linear Impulsive Neutral Delay Differential Equations with Constant Coefficients

We have given some results regarding the behavior of solutions for first order linear impulsive neutral delay differential equations with constant coefficients. These results were obtained using two different real roots of the corresponding characteristic equation. Finally, two examples are given for solutions of impulsive neutral delay differential equations.

Collocation methods based on Gegenbauer and Bernoulli wavelets for solving neutral delay differential equations

In this paper, we introduce two different methods based on Gegenbauer wavelet and Bernoulli wavelet for the solution of neutral delay differential equations. These methods convert linear and nonlinear neutral delay differential equations into system of linear and nonlinear algebraic equations, respectively. After solving these equations, we get the approximate solutions. Here, we have used the Gegenbauer wavelet (for different values of μ) and Bernoulli wavelet and seen that both methods converge fast. We present six test problems consisting of five linear and one nonlinear, to illustrate the accuracy of present methods. Further, we compared our results with the results of existing methods present in the literature and have seen that our methods give more accurate results.

Hybrid Multistep Block Method for Solving Neutral Delay Differential Equations

The initial-value problem for first order single linear neutral delay differential equations (NDDEs) of constant and pantograph delay types have been solved by using hybrid multistep block method. The method has been derived by applying Taylor series interpolation polynomial and implementing the predictor-corrector formulas in PE(CE)m mode where m is the number of iterations for the proposed method. Both types of NDDEs will be solved at two-point simultaneously including the off-step point with constant step-size. In order to find the solution for NDDEs, the delay solutions of the unknown function will be interpolated using Lagrange interpolation polynomial and the derivative of the delay solutions will be obtained by applying divided difference formula. The order, consistency and convergence of the proposed method have been discussed in detail in the methods section. The properties of stability region for NDDEs have also been analysed. Numerical results presented have concluded that the proposed method is comparable with the existing method and is assumed to be reliable for solving first order NDDEs with constant and pantograph delay.

二阶线性脉冲中立型时滞微分方程的稳定性

二阶线性脉冲中立型时滞微分方程的稳定性

Oscillatory Properties of Third-Order Neutral Delay Differential Equations with Noncanonical Operators

In the paper, we study the oscillatory and asymptotic properties of solutions to a class of third-order linear neutral delay differential equations with noncanonical operators. Via the application of comparison principles with associated first and second-order delay differential inequalities, we offer new criteria for the oscillation of all solutions to a given differential equation. Our technique essentially simplifies the process of investigation and reduces the number of conditions required in previously known results. The strength of the newly obtained results is illustrated on the Euler type equations.

English

English

Stability of linear impulsive neutral delay differential equations with constant coefficients

In this article, we have developed a new method regarding asymptotic behavior and stability for the solutions of first order linear impulsive neutral delay differential equations with constant coefficients and constant delays. The asymptotic behavior of the solutions and the stability of the zero solution were investigated using a real root suitable for the characteristic equation. Furthermore, we obtained similar results for impulsive delay differential equations. Finally, three examples are given for the stability of the zero solution of the impulsive neutral delay differential equation.

The continuous Galerkin finite element methods for linear neutral delay differential equations

In the paper, the superconvergence of continuous Galerkin finite element methods (CGFEMs) for linear delay differential equations of neutral type is presented. By the orthogonal analysis method and under the suitable condition, it is proven that the finite element solution is superconvergent at the nodal points and Lobatto points. Numerical experiments further confirm the effectiveness and the superconvergence of the CGFEMs.

Haar wavelet series solution for solving neutral delay differential equations

We solved linear and nonlinear neutral delay differential equations using Haar wavelet series. We apply Haar wavelet and obtain its integration for neutral delay differential equations with respect to the collocation points to obtain the numerical solution of the problems. Six problems are solved and results are compared with the exact solution and existing methods from literature to show the accuracy and applicability of the Haar wavelet series. On increasing the level of resolutions maximum absolute error decreases uniformly.