Solutions Of Delay Differential Equations Research Articles

Transient Reviving Dynamics with an Exact Solution for Delay Differential Equations

Transient Reviving Dynamics with an Exact Solution for Delay Differential Equations

Bridging machine learning and weighted residual methods for delay differential equations of fractional order

In this research, an attempt has been made to build a bridge between a machine learning model called Least-Squares Support Vector Regression (LS-SVR) and Weighted Residual Methods (WRMs) to solve delay differential equations in fractional derivative order. Our machine learning algorithm, which is influenced by the collocation method, uses Legendre polynomials as a kernel function to approximate the solution of linear and nonlinear delay differential equations, including the Lane–Emden pantograph equation and some fractional cases. In the proposed algorithm, the estimated solution of delay differential equations is approximated as a linear combination of M degrees of Legendre polynomials and a set of weights that are learned during the fitting process. The roots of the Legendre functions are utilized as training data to develop the algorithm. The effectiveness of our method is demonstrated through the use of numerical graphs and tables. A comparative analysis of the numerical results has shown that the proposed method yields better accuracy and convergence compared to other numerical techniques.

Stability of Solutions of Delay Differential Equations

Stability of Solutions of Delay Differential Equations

Oscillatory Behaviour of Solutions of Delay Differential Equations of Fractional Order

Oscillatory Behaviour of Solutions of Delay Differential Equations of Fractional Order

Existence, uniqueness and approximation of solutions to Carathéodory delay differential equations

In this paper we address the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with Carathéodory type right-hand side functions. We provide construction of the randomized Euler scheme for DDEs and investigate its error. We also report results of numerical experiments.

High accuracy breaking point detection for stiff DDEs

We have recently introduced a very high order Discontinuous Galerkin (DG) method for the solution of stiff ordinary differential equations (ODEs) and delay differential equations (DDEs). In this communication we focus on the detection of breaking points frequently arising in the solution of stiff DDEs. Breaking points are discontinuities appearing in the solution of DDEs and/or in some of its derivatives. Breaking points affect the efficiency of the numerical method by forcing very small time steps in their vicinity. They also locally affect the solution accuracy. We therefore introduce a very simple breaking point detection algorithm in order to further improve the efficiency our DG method.

Determine the Solution of Delay Differential Equations using Runge-Kutta Methods with Cubic-Spline Interpolation

This paper describes some iterations for term delay in Delay Differential Equation (DDE), which is causing a huge number of iteration calculations. Time-delay was approximated using Cubic-Spline Interpolation, so DDE can rewrite as Differential Equations. Then, Runge-Kutta methods have been used to determine the solution of Differential equations from DDE.

Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback.

We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model.

Chebyshev technique in the numerical solution of Delay Differential Equations

نتحرى عن استقرار الحل العددي للتأخير المعادلة التفاضلية باستخدام طريقة Runge-kutta الصريحة عندما يكون التأخير تم تقريبه باستخدام Chebyshev متعدد الحدود بدلاً من استخدام Lagrange و Hirmate interpellation كثيرات حدود.

Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

We deal with approximation of solutions of delay differential equations (DDEs) via the classical Euler algorithm. We investigate the pointwise error of the Euler scheme under nonstandard assumptions imposed on the right-hand side function f. Namely, we assume that f is globally of at most linear growth, satisfies globally one-side Lipschitz condition but it is only locally Hölder continuous. We provide a detailed error analysis of the Euler algorithm under such nonstandard regularity conditions. Moreover, we report results of numerical experiments.

Global Attractivity of Solutions of Nonlinear Delay Difference Equations

Abstract The asymptotic behavior of solutions of delay difference equations is an important research topic in the field of qualitative analysis of difference equations and the key to grasp the evolution trend of various dynamic systems As we all know, most difference equations cannot find exact solutions Therefore, it is necessary to study the asymptotic behavior of the solution of the delay difference equation by indirect method. In this paper, improved the form of the equation in the existing reference, a single delay is changed into a multi delay equation, and the conditions required for the theorem to be established are weakened, so that the theorem can be applied to a wider range and still holds under the original conditions sufficient conditions for the asymptotic behavior of solutions independent of delay are obtained. The conclusion is proved by constructor method and counter proof method The relevant results in the existing literature are extended and improved to make the results more general. The results are applied to a kind of hematopoietic stem cell model. The asymptotic behavior of the solution is intuitively displayed in the form of chart to verify the conclusion.

Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications

In the article, we have proposed a new type of hybrid iterative scheme which is a hybrid of Picard and Thakur et al. repetitive schemes. This new hybrid iterative scheme converges faster than all leading schemes like Picard-S∗ hybrid, Picard-S, Picard-Ishikawa hybrid, Picard-Mann hybrid, Thakur et al. and Abbas and Nazir, S-iterative, Ishikawa and Mann iterative schemes for contraction mapping. By using the Picard-Thakur hybrid iterative scheme, we can find the solution of delay differential equations and also prove some convergence results for nonexpansive mapping in a uniformly convex Banach space.

Semianalytical Approach for the Approximate Solution of Delay Differential Equations

In this analysis, we develop a new approach to investigate the semianalytical solution of the delay differential equations. Mohand transform coupled with the homotopy perturbation method is called Mohand homotopy perturbation transform method (MHPTM) and performs the solution results in the form of series. The beauty of this approach is that it does not need to compute the values of the Lagrange multiplier as in the variational iteration method, and also, there is no need to implement the convolution theorem as in the Laplace transform. The main purpose of this scheme is to reduce the less computational work and the error analysis of the problems than others studied in the literature. Some illustrated examples are interpreted to confirm the accuracy of the newly developed scheme.

On Mathematical Aspects of Evolution of Dislocation Density in Metallic Materials

This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into delay differential equation (DDE) with strong (in general, H\"older) nonlinearity. We prove upper error bounds for the explicit Euler method, under the assumption that the right-hand side function is H\"older continuous and monotone which allows us to compare accuracy of other numerical methods in our model (e.g. Runge-Kutta), in particular when explicit formulas for solutions are not known. Finally, we test the above results in simulations of real industrial process.

Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation

Numerical approach of two-derivative Runge-Kutta type method with three-stage fifth-order (TDRKT3(5)) is developed and proposed for solving a special type of third-order delay differential equations (DDEs) with constant delay. An algorithm based on Newton interpolation and hybrid with the TDRKT method is built to approximate the solution of third-order DDEs. In this paper, three-stage fifth-order called TDRKT3(5) method with single third derivative and multiple evaluations of the fourth derivative is highlighted to solve third-order pantograph type delay differential equations directly with the aid of the Newton interpolation method. Stability analysis of TDRKT3(5) method is investigated. The numerical experiments illustrate high efficiency and validity of the new method for solving a special class of third-order DDEs and some future works are recommended by extending proposed method to solve fractional and singularly perturbed delay differential equations.

Delay Differential Equations of Fourth-Order: Oscillation and Asymptotic Properties of Solutions

In this work, by using the comparison method and Riccati transformation, we obtain some oscillation criteria of solutions of delay differential equations of fourth-order in canonical form. These criteria complement those results in the literature. We give two examples to illustrate the main results. Symmetry plays an essential role in determining the correct methods for solutions to differential equations.

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

In this article, we presented an asymptotic SDFEM for singularly perturbed convection diffusion type differential difference equations with point source term. First, the solution is decomposed into two functions, among them one is the solution of delay differential equation and other one is the solution of differential equation with point source. Furthermore, using the asymptotic expansion approximation, the delay differential equation is modified as a nondelay differential equations. Streamline diffusion finite element methods are applied to approximate the solutions of the two problems. We prove that the present method gives an almost second-order convergence in maximum norm and square integrable norm, whereas first-order convergence in H 1 norm. Numerical results are presented to validate the theoretical results.

Special fitted finite difference scheme for delay differential equation with dual boundary layers

In this paper, we have proposed a fitted special finite difference method for the solution of delay differential equations with dual boundary layers. The delay differential equation is replaced by an asymptotically equivalent singular perturbation problem using the Taylor’s series expansion. Then, a fitted special finite difference scheme is described to get accurate solution to the problem. The method is demonstrated by implementing on several model problems by taking various values for the delay parameter and perturbation parameter . To show the effect of delay on the boundary layer or oscillatory behaviour of the solution, several numerical problems are carried out in this chapter. To demonstrate the effect on the layer behaviour, the solution of the problems are shown graphically. We observed that when the order of the coefficient of the delay parameter is of o(1), the delay affects the boundary layer solution but maintains the layer behaviour and as the delay increases, the thickness of the left boundary layer decreases while that of the right boundary layer increases.

Approximation of mild solutions of delay differential equations on Banach spaces

In this work we study an approximation of a mild solution y of a semilinear first order abstract differential problem with delay, which depends of an initial history condition and an unbounded closed linear operator A generating a C0-semigroup on a Banach space X. The approximation considers the mild solutions (zδ)δ>0 of the corresponding family of differential equations with piecewise constant argument, varying the semilinear term with a parameter δ. Our main results is about the obtaining of the solution zδ in terms of a difference equation on X and conditions to ensure uniform convergence of zδ to y as δ→0, on compact and unbounded intervals. We obtain explicit exponential decay estimates for the error function using the stability of the semigroup and the Halanay's inequality. Also with a new idea and method we prove that the approximation is stable and there exists a preservation of asymptotic stability between the solution of delayed differential equation and its corresponding difference equation, obtained by piecewise constant argument.

A New Multi-Step Method for Solving Delay Differential Equations using Lagrange Interpolation

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).