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Journal articles on the topic 'Delay differential equations (DDEs)'

Author: Grafiati

Published: 7 June 2025

Last updated: 2 August 2025

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1

Wahi, Pankaj, and Anindya Chatterjee. "Galerkin Projections for Delay Differential Equations." Journal of Dynamic Systems, Measurement, and Control 127, no. 1 (2004): 80–87. http://dx.doi.org/10.1115/1.1870042.

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We present a Galerkin projection technique by which finite-dimensional ordinary differential equation (ODE) approximations for delay differential equations (DDEs) can be obtained in a straightforward fashion. The technique requires neither the system to be near a bifurcation point, nor the delayed terms to have any specific restrictive form, or even the delay, nonlinearities, and/or forcing to be small. We show through several numerical examples that the systems of ODEs obtained using this procedure can accurately capture the dynamics of the DDEs under study, and that the accuracy of solutions

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2

Naveen Kashyap and Dr. B. V. Padamvar. "Differential Difference Equations in Trajectory Planning and Control for Differential Drive Robots: A Comprehensive Exploration." Journal of Advances in Science and Technology 21, no. 1 (2024): 35–42. http://dx.doi.org/10.29070/5c5t1x55.

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This paper provides a thorough investigation into the utilisation of differential difference equations (DDEs) for the purposes of trajectory planning and control in the context of differential drive four-wheeled robots. The inclusion of sensor and control delays is crucial when developing navigation strategies that are both resilient and efficient for robots functioning in dynamic environments. Differential delay equations (DDEs) provide a robust mathematical framework for representing the dynamics of such systems, facilitating precise and reliable path tracking. In order to demonstrate the ve

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3

Ahmad, M. D. Al-Eybani. "Solving Delay Differential Equations Using the Method of Steps." International Journal of Mathematics and Physical Sciences Research 13, no. 1 (2025): 46–49. https://doi.org/10.5281/zenodo.15366549.

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<strong>Abstract:</strong> Delay differential equations (DDEs) are a class of differential equations where the derivative of the unknown function at a given time depends not only on the current state but also on its values at previous times. These equations arise in numerous fields, including biology, engineering, and economics. Unlike ordinary differential equations (ODEs), DDEs incorporate time delays, making their solution more complex due to the need for a history function. A common form of a first-order linear DDE with a constant delay is: where  is the unknown function,  and &n

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4

Bachar, Mostafa. "Linearized Stability Analysis of Nonlinear Delay Differential Equations with Impulses." Axioms 13, no. 8 (2024): 524. http://dx.doi.org/10.3390/axioms13080524.

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This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear semigroups are generalized to periodic DDEs with impulses. A significant challenge arises from the need for a discontinuous initial function to obtain periodic solutions. To address this, first-kind discontinuous spaces R([a,b],Rn) are introduced for defining DDEs with impulse

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5

Jin, Gang, Houjun Qi, Zhanjie Li, Jianxin Han, and Hua Li. "A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/9490142.

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Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the pa

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Ignatius, N. Njoseh, and J. Mamadu Ebimene. "Solving Delay Differential Equations By Elzaki Transform Method." Boson Journal of Modern Physics 3, no. 1 (2017): 214–19. https://doi.org/10.5281/zenodo.3969415.

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In this paper, we implement the Elzaki transform method for the solution of delay differential equations (DDEs). The method executes the DDEs by implementing its properties on the given DDE. Also, the method treats the nonlinear terms with a well posed formula. The method is easy to implement with high level of accuracy. Also, restricted transformations, perturbation, linearization or discretization are not recognized. The resulting numerical evidences show that the method converges favourably to the analytic solution. All computational frameworks are performed with maple 18 software.

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7

Zhu, Qunxi, Yifei Shen, Dongsheng Li, and Wei Lin. "Neural Piecewise-Constant Delay Differential Equations." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 8 (2022): 9242–50. http://dx.doi.org/10.1609/aaai.v36i8.20911.

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Continuous-depth neural networks, such as the Neural Ordinary Differential Equations (ODEs), have aroused a great deal of interest from the communities of machine learning and data science in recent years, which bridge the connection between deep neural networks and dynamical systems. In this article, we introduce a new sort of continuous-depth neural network, called the Neural Piecewise-Constant Delay Differential Equations (PCDDEs). Here, unlike the recently proposed framework of the Neural Delay Differential Equations (DDEs), we transform the single delay into the piecewise-constant delay(s

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8

Naseem, Tahir. "Reduce Differential Transform Method for Analytical Approximation of Fractional Delay Differential Equation." International Journal of Emerging Multidisciplinaries: Mathematics 1, no. 2 (2022): 104–23. http://dx.doi.org/10.54938/ijemdm.2022.01.2.35.

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The study of an entirely new class of differential equations known as delay differential equations or difference differential equations has resulted from the development and application of automatic control systems (DDEs). Time delays are virtually always present in any system that uses feedback control. Because it takes a finite amount of time to sense information and then react to it, a time delay is required. This exploration was carried out for the solution of fractional delay differential equations by using the reduced differential transform method. The results are presented in a series o

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9

LUZYANINA, TATYANA, and KOEN ENGELBORGHS. "COMPUTING FLOQUET MULTIPLIERS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 12, no. 12 (2002): 2977–89. http://dx.doi.org/10.1142/s0218127402006291.

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Floquet multipliers determine the local asymptotic stability of a periodic solution and, in the context of parameter dependence, determine also its bifurcations. This paper deals with numerical aspects of the computation of the Floquet multipliers for three classes of functional differential equations: ordinary differential equations (ODEs), differential equations with constant delay (DDEs) and differential equations with state-dependent delay (sd-DDEs). Using a collocation approach for computing periodic solutions, we obtain an approximation of the (corresponding) monodromy operator, a monodr

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10

Olaniyan, Adegoke Stephen, Moshood Tolulope Kazeem, Azeez Adebayo Aweda, and Oladayo Ibukunoluwa Oladimeji. "Numerical Solution of Delay Differential Equations with Heronian Implicit Runge-Kutta Method." International Journal of Research and Innovation in Applied Science IX, no. XII (2025): 326–30. https://doi.org/10.51584/ijrias.2024.912030.

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In recent years, there has been growing interest in the numerical solution of Delay Differential Equations (DDEs). This is due to the fact that DDEs provides a good means of modelling many phenomena in diverse application fields ranging from physical sciences, economy, medicine, education to electrodynamics. Hence, the increased attention in the numerical solutions to such problems becomes a necessity. The purpose of this study is to present a numerical method that uses a polynomial interpolating function when solving DDEs. In this paper, Heronian Implicit Runge-Kutta method is considered for

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11

Ratib Anakira, N., A. K. Alomari, and I. Hashim. "Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/498902.

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We extend for the first time the applicability of the optimal homotopy asymptotic method (OHAM) to find the algorithm of approximate analytic solution of delay differential equations (DDEs). The analytical solutions for various examples of linear and nonlinear and system of initial value problems of DDEs are obtained successfully by this method. However, this approach does not depend on small or large parameters in comparison to other perturbation methods. This method provides us with a convenient way to control the convergence of approximation series. The results which are obtained revealed t

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12

Xiao, Huafeng. "The Existence of Periodic Solutions of Delay Differential Equations by E + -Conley Index Theory." Journal of Function Spaces 2022 (April 20, 2022): 1–14. http://dx.doi.org/10.1155/2022/3396716.

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In this paper, the E + -Conley index theory has been used to study the existence of periodic solutions of nonautonomous delay differential equations (in short, DDEs). The variational structure for DDEs is built, and the existence of periodic solutions of DDEs is transferred to that of critical points of the associated function. When DDEs are 2 π -nonresonant, some sufficient conditions are obtained to guarantee the existence of periodic solutions. When the system is 2 π -resonant at infinity, by making use a second disturbing of the original functional, some sufficient conditions are obtained

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13

Yann Seong, Hoo, Zanariah Abdul Majid, and Fudziah Ismail. "Solving Second-Order Delay Differential Equations by Direct Adams-Moulton Method." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/261240.

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This paper will consider the implementation of fifth-order direct method in the form of Adams-Moulton method for solving directly second-order delay differential equations (DDEs). The proposed direct method approximates the solutions using constant step size. The delay differential equations will be treated in their original forms without being reduced to systems of first-order ordinary differential equations (ODEs). Numerical results are presented to show that the proposed direct method is suitable for solving second-order delay differential equations.

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14

Barde, Aminu. "A Natural Homotopy Analysis Method for Nonlinear Delay Differential Equations." Science Proceedings Series 1, no. 2 (2019): 86–90. http://dx.doi.org/10.31580/sps.v1i2.680.

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Delay differential equation (DDEs) is a type of functional differential equation arising in numerous applications from different areas of studies, for example biology, engineering population dynamics, medicine, physics, control theory, and many others. However, determining the solution of delay differential equations has become a difficult task more especially the nonlinear type. Therefore, this work proposes a new analytical method for solving non-linear delay differential equations. The new method is combination of Natural transform and Homotopy analysis method. The approach gives solutions

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15

Yi, S., P. W. Nelson, and A. G. Ulsoy. "Eigenvalue Assignment via the Lambert W Function for Control of Time-delay Systems." Journal of Vibration and Control 16, no. 7-8 (2010): 961–82. http://dx.doi.org/10.1177/1077546309341102.

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In this paper, we consider the problem of feedback controller design via eigenvalue assignment for linear time-invariant systems of linear delay differential equations (DDEs) with a single delay. Unlike ordinary differential equations (ODEs), DDEs have an infinite eigenspectrum, and it is not feasible to assign all closed-loop eigenvalues. However, we can assign a critical subset of them using a solution to linear systems of DDEs in terms of the matrix Lambert W function. The solution has an analytical form expressed in terms of the parameters of the DDE, and is similar to the state transition

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16

Calio, F., E. Marchetti, and R. Pavani. "Peculiar spline collocation method for solving rough and stiff delay differential problems." Journal of Numerical Analysis and Approximation Theory 33, no. 1 (2004): 25–37. http://dx.doi.org/10.33993/jnaat331-756.

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As well known, solutions of delay differential equations (DDEs) are characterized by low regularity. In particular solutions of neutral delay differential equations (NDDEs) frequently exhibit discontinuities in the first derivative so that the differential problems become rough. The aim of this paper is to approximate the solutions of such rough delay differential problems by means of a peculiar deficient spline collocation method. Significant numerical examples are provided to enlighten the features of the proposed method.

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17

Shifeng, Wu. "A CLASS OF BOUNDARY VALUE METHODS FOR THE COMPLEX DELAY DIFFERENTIAL EQUATION." International Journal of Soft Computing, Mathematics and Control (IJSCMC) 6, no. 2/3 (2017): 15 to 25. https://doi.org/10.5281/zenodo.3362008.

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In this paper, a class of boundary value methods (BVMs) for delay differential equations (DDEs) is considered. The delay dependent stable regions of the extended trapezoidal rules of second kind (ETR2s), which are a class of BVMs, are displayed for the test equation of DDEs. Furthermore, it is showed ETR2s cannot preserve the delay-dependent stability of the complex coefficient test equation considered. Some numerical experiments are given to confirm the theoretical results.   AMS 2000 Mathematics Subject Classification: 65L20, 65M12

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18

Jin, Gang, Xinyu Zhang, Kaifei Zhang, et al. "Stability Analysis Method for Periodic Delay Differential Equations with Multiple Distributed and Time-Varying Delays." Mathematical Problems in Engineering 2020 (July 25, 2020): 1–9. http://dx.doi.org/10.1155/2020/1982363.

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Dynamic stability problems leading to delay differential equations (DDEs) are found in many different fields of science and engineering. In this paper, a method for stability analysis of periodic DDEs with multiple distributed and time-varying delays is proposed, based on the well-known semidiscretization method. In order to verify the correctness of the proposed method, two typical application examples, i.e., milling process with a variable helix cutter and milling process with variable spindle speed, which can be, respectively, described by DDEs with the multidistributed and time-varying del

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19

Liping Liu and Tamás Kalmár-Nagy. "High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations." Journal of Vibration and Control 16, no. 7-8 (2010): 1189–208. http://dx.doi.org/10.1177/1077546309341134.

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This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical

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20

Nishiguchi, Junya. "Mild solutions, variation of constants formula, and linearized stability for delay differential equations." Electronic Journal of Qualitative Theory of Differential Equations, no. 32 (2023): 1–77. http://dx.doi.org/10.14232/ejqtde.2023.1.32.

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The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is def

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21

C. Kayelvizhi and A. Emimal Kanaga Pushpam. "Subdomain collocation method based on successive integration technique for solving delay differential equations." International Journal of Science and Research Archive 11, no. 2 (2024): 382–90. http://dx.doi.org/10.30574/ijsra.2024.11.2.0429.

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The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). 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 DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solut

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22

Kumar Subramanium. "Analyzing the Stability of Neural Networks with Delay Differential Equations." Modern Dynamics: Mathematical Progressions 1, no. 1 (2024): 14–17. http://dx.doi.org/10.36676/mdmp.v1.i1.04.

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Neural networks are powerful computational models widely used in various domains, including machine learning, neuroscience, and control systems. However, the stability analysis of neural networks with time delays remains a challenging problem due to the complex interactions between neurons and the presence of delayed feedback loops. In this paper, we propose a novel approach to analyze the stability of neural networks using delay differential equations (DDEs). We begin by formulating a mathematical model of the neural network dynamics, incorporating time delays to account for the finite propag

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23

Veerabathiran, G., G. Jagan Kumar, and Siriluk Donganont. "Stability Analysis of Parkinson's Disease Model with Multiple Delay Differential Equations using Laplace Transform Method." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5562. https://doi.org/10.29020/nybg.ejpam.v18i2.5562.

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In this paper, we investigate the stability analysis of Parkinson's disease model multiple delay differential equations (DDEs) utilizing the Laplace transform method. Delay differential equations are often encountered in a wide range of scientific and engineering applications, such as signal processing, control systems, and population dynamics. These equations are formulated using delayed arguments. The analysis and solution of these equations are frequently made more difficult by the existence of delays. Here, we simplify the process of locating explicit solutions by converting the DDEs into

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Tunç, Osman. "Stability tests and solution estimates for non-linear differential equations." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 13, no. 1 (2023): 92–103. http://dx.doi.org/10.11121/ijocta.2023.1251.

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This article deals with certain systems of delay differential equations (DDEs) and a system of ordinary differential equations (ODEs). Here, five new theorems are proved on the fundamental properties of solutions of these systems. The results on the properties of solutions consist of sufficient conditions and they dealt with uniformly asymptotically stability (UAS), instability and integrability of solutions of unperturbed systems of DDEs, boundedness of solutions of a perturbed system of DDEs at infinity and exponentially stability (ES) of solutions of a system of nonlinear ODEs. Here, the te

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Keane, Andrew, Bernd Krauskopf, and Henk A. Dijkstra. "The effect of state dependence in a delay differential equation model for the El Niño Southern Oscillation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2153 (2019): 20180121. http://dx.doi.org/10.1098/rsta.2018.0121.

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Delay differential equations (DDEs) have been used successfully in the past to model climate systems at a conceptual level. An important aspect of these models is the existence of feedback loops that feature a delay time, usually associated with the time required to transport energy through the atmosphere and/or oceans across the globe. So far, such delays are generally assumed to be constant. Recent studies have demonstrated that even simple DDEs with non-constant delay times, which change depending on the state of the system, can produce surprisingly rich dynamical behaviour. Here, we presen

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Mechee, M., F. Ismail, N. Senu, and Z. Siri. "Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/830317.

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Runge-Kutta-Nyström (RKN) method is adapted for solving the special second order delay differential equations (DDEs). The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK) method. Numerical results show that the RKN method is more efficient in terms of accurac

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CHUNG, K. W., C. L. CHAN, and J. XU. "A PERTURBATION-INCREMENTAL METHOD FOR DELAY DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 16, no. 09 (2006): 2529–44. http://dx.doi.org/10.1142/s0218127406016239.

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A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of periodic solutions of nonlinear systems of delay differential equations (DDEs). Periodic solutions can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at a period-doubling bifurcation is made simple by the present scheme as a parameter is simply increased from zero to a small positive value so that a solution on the new branch is obtained. Subsequent continuation of an emanating branch is also discussed. T

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Mosaad, Peter Nazier, Martin Fränzle, and Bai Xue. "Model Checking Delay Differential Equations Against Metric Interval Temporal Logic." Scientific Annals of Computer Science XXVII, no. 1 (2017): 77–109. https://doi.org/10.7561/SACS.2017.1.77.

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Delay differential equations (DDEs) play an important role in the modeling of dynamic processes. Delays arise in contemporary control schemes like networked distributed control and can cause deterioration of control performance, invalidating both stability and safety properties. This induces an interest in DDE especially in the area of modeling and verification of embedded control. In this article, we present an approach aiming at automatic safety verification of a simple class of DDEs against requirements expressed in a linear-time temporal logic. As requirements specification language, we ex

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Dhivyadharshini, B., and R. Senthamarai. "Modeling Rugose Spiraling Whitefly Infestation on Coconut Trees Using Delay Differential Equations: Analysis via HPM." European Journal of Pure and Applied Mathematics 17, no. 3 (2024): 1908–36. http://dx.doi.org/10.29020/nybg.ejpam.v17i3.5249.

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In this paper, a delay - induced pest control model is proposed. We have introduced a time delay in healthy trees and whitefly population in the infected tree density of the proposed system of equations to reduce the probability of healthy trees becoming infected, as well as the level of infection. We have analyzed the impact of time delay on the stability of the equilibrium and establish requirements to verify its asymptotic stability over all delays. The solutions of this system of non-linear ordinary differential equations(ODEs) and delay differential equations(DDEs) are presented by using

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30

Brunner, Hermann. "The numerical analysis of functional integral and integro-differential equations of Volterra type." Acta Numerica 13 (May 2004): 55–145. https://doi.org/10.1017/s0962492904000170.

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The qualitative and quantitative analysis of numerical methods for delay differential equations is now quite well understood, as reflected in the recent monograph by Bellen and Zennaro (2003). This is in remarkable contrast to the situation in the numerical analysis of functional equations, in which delays occur in connection with memory terms described by Volterra integral operators. The complexity of the convergence and asymptotic stability analyses has its roots in new ‘dimensions’ not present in DDEs: the problems have distributed delays; kernels in the Volterra operators may be weakly sin

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Ali, Ishtiaq. "Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method." AIMS Mathematics 7, no. 4 (2022): 4946–59. http://dx.doi.org/10.3934/math.2022275.

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<abstract> <p>Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of

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Luzyanina, Tatyana, Koen Engelborghs, Kurt Lust, and Dirk Roose. "Computation, Continuation and Bifurcation Analysis of Periodic Solutions of Delay Differential Equations." International Journal of Bifurcation and Chaos 07, no. 11 (1997): 2547–60. http://dx.doi.org/10.1142/s0218127497001709.

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We present a new numerical method for the efficient computation of periodic solutions of nonlinear systems of Delay Differential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows effective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.

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LI, JUN YU, and ZAI HUA WANG. "LOCAL HOPF BIFURCATION OF COMPLEX NONLINEAR SYSTEMS WITH TIME-DELAY." International Journal of Bifurcation and Chaos 19, no. 03 (2009): 1069–79. http://dx.doi.org/10.1142/s0218127409023494.

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A Hopf bifurcation can occur commonly in the time-delay systems described by delay differential equations (DDEs), even in first order autonomous DDEs. Compared with the intensive study of Hopf bifurcation for DDEs with real coefficients, which was mainly done using the center manifold reduction, perturbation methods and so on, little effort has been made directly for the Hopf bifurcation of DDEs with complex coefficients. In this paper, a generalization of the newly developed "pseudo-oscillator analysis" for the Hopf bifurcation of scalar real nonlinear DDEs is presented for scalar complex non

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Verdugo, Anael. "Linear Analysis of an Integro-Differential Delay Equation Model." International Journal of Differential Equations 2018 (2018): 1–6. http://dx.doi.org/10.1155/2018/5035402.

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This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a

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Jaaffar, Nur Tasnem, Zanariah Abdul Majid, and Norazak Senu. "Numerical Approach for Solving Delay Differential Equations with Boundary Conditions." Mathematics 8, no. 7 (2020): 1073. http://dx.doi.org/10.3390/math8071073.

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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

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Kim, Pilkee, Wan Sun, and Jong Won Seok. "High-Dimensional Harmonic Balance Analysis for a Turning Process with State-Dependent Delay." Advanced Materials Research 655-657 (January 2013): 515–20. http://dx.doi.org/10.4028/www.scientific.net/amr.655-657.515.

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The present study considers the state-dependent delay differential equations (SD-DDEs) for the turning process. In general, series expansion of the SD-DDEs turning system is essential in the nonlinear analysis such as the conventional methods of multiple scales and harmonic balance. Unfortunately, the mathematical theory of SD-DDEs, especially for those with an implicit function of delay, was just recently developed and any rigorous mathematical theory has not yet been proven. As one approach for the nonlinear analysis of the SD-DDEs, physically reasonable results could be obtained by extendin

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Vilu, Subashini, Rokiah Rozita Ahmad, Ummul Khair Salma Din, and Mohd Almie Alias. "RESOLVING DELAY DIFFERENTIAL EQUATIONS WITH HOMOTOPY PERTURBATION AND SUMUDU TRANSFORM." Jurnal Teknologi 85, no. 3 (2023): 145–51. http://dx.doi.org/10.11113/jurnalteknologi.v85.18937.

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A novel proposition has been introduced in this study for resolving delay differential equations (DDEs) of nature that is a composite in reference to Homotopy perturbation method (HPM) along with Sumudu transform. A rare transform called the Sumudu transform is used alongside the perturbation theory. Demonstration of this new methodology is shown by solving a few numerical cases. Reducing the complication of computational tasks associated to the conservative means is the objective of this research. Results display the amount of valuation being reduced and is as good as in the previous studies

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Ahmad, N. A., N. Senu, Z. B. Ibrahim, and M. Othman. "Stability Analysis of Diagonally Implicit Two Derivative Runge-Kutta methods for Solving Delay Differential Equations." Malaysian Journal of Mathematical Sciences 16, no. 2 (2022): 215–35. http://dx.doi.org/10.47836/mjms.16.2.04.

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The stability properties of fourth and fifth-order Diagonally Implicit Two Derivative Runge-Kutta method (DITDRK) combined with Lagrange interpolation when applied to the linear Delay Differential Equations (DDEs) are investigated. This type of stability is known as P-stability and Q-stability. Their stability regions for (λ,μ∈R) and (μ∈C,λ=0) are determined. The superiority of the DITDRK methods over other same order existing Diagonally Implicit Runge-Kutta (DIRK) methods when solving DDEs problems are clearly demonstrated by plotting the efficiency curves of the log of both maximum errors ve

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Wahi, Pankaj, and Anindya Chatterjee. "Asymptotics for the Characteristic Roots of Delayed Dynamic Systems." Journal of Applied Mechanics 72, no. 4 (2004): 475–83. http://dx.doi.org/10.1115/1.1875492.

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Delayed dynamical systems appear in many areas of science and engineering. Analysis of general nonlinear delayed systems often begins with the linearized delay differential equation (DDE). The study of these linearized constant coefficient DDEs involves transcendental characteristic equations, which have infinitely many complex roots not obtainable in closed form. Here, after motivating our study with a well-known delayed dynamical system model for tool vibrations in metal cutting, we obtain asymptotic expressions for the large characteristic roots of several delayed systems. These include fir

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Olvera, D., A. Elías-Zúñiga, L. N. López de Lacalle, and C. A. Rodríguez. "Approximate Solutions of Delay Differential Equations with Constant and Variable Coefficients by the Enhanced Multistage Homotopy Perturbation Method." Abstract and Applied Analysis 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/382475.

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We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differential equations (DDEs) with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration sol

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Asl, Farshid Maghami, and A. Galip Ulsoy. "Analysis of a System of Linear Delay Differential Equations." Journal of Dynamic Systems, Measurement, and Control 125, no. 2 (2003): 215–23. http://dx.doi.org/10.1115/1.1568121.

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A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different

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Tariq, Muhammad, Muhammad Suleman, Sher khan Awan, Prem Kuma, and Asif Ali Shaikh. "Delay Differential-Algebraic Equations (DDAEs)." Babylonian Journal of Mathematics 2023 (August 25, 2023): 40–44. http://dx.doi.org/10.58496/bjm/2023/008.

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Delay differential-algebraic equations (DDAEs) are an important class of mathematical models that broaden standard differential-algebraic equations (DAEs) to incorporate discrete time delays. The time lag terms pose significant analytical and computational challenges. This paper provides a comprehensive overview of current and emerging methods for solving DDAEs and systems of DDAEs. Generalized Taylor series techniques, linear multistep methods, and reduction to ordinary differential equations are examined for numerically integrating DDAEs. Stability, convergence, and accuracy considerations a

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Bibi, Seema, Rashid Nawaz, Laiq Zada, Negar Alam, and Nicholas Fewster Young. "A Semi-Analytical Framework for higher-Order Delay Differential Equations: Utilizing Optimal Auxiliary Functions." Global Journal of Sciences 1, no. 1 (2024): 1–13. https://doi.org/10.48165/gjs.2024.1101.

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Delay differential equations (DDEs) are extensively utilized in fields such as control systems, biology, and engineering to model processes where current states depend on past states, effectively accounting for time lags. Key applications include population dynamics, epidemic modeling, and economic systems, where delayed responses significantly influence system behavior. This paper presents the first extension of the Optimal Auxiliary Functions Method (OAFM) to second-order and third-order DDEs. The strength of this method lies in its convergence control parameters and auxiliary functions. Not

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Bellour, A., M. Bousselsal, and H. Laib. "Numerical Solution of Second-Order Linear Delay Differential and Integro-Differential Equations by Using Taylor Collocation Method." International Journal of Computational Methods 17, no. 09 (2019): 1950070. http://dx.doi.org/10.1142/s0219876219500701.

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The main purpose of this work is to provide a numerical approach for linear second-order differential and integro-differential equations with constant delay. An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution [Formula: see text] for approximating the solution of second-order linear DDEs and DIDEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the validity of this algorithm.

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Sherman, Michelle, Gilbert Kerr, and Gilberto González-Parra. "Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method." Mathematical and Computational Applications 27, no. 5 (2022): 81. http://dx.doi.org/10.3390/mca27050081.

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In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear delay differential equations (DDEs). We computed the analytical solutions that are obtained by using the Laplace transform method and the method of steps. The accuracy of the Laplace method solutions was determined (or assessed) by comparing them with those obtained by the method of steps. The Laplace transform met

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Bandopadhya, Dibakar. "Application of Lambert W-function for solving time-delayed response of smart material actuator under alternating electric potential." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230, no. 13 (2015): 2135–44. http://dx.doi.org/10.1177/0954406215590640.

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An active actuator of electro-active polymer (EAP), i.e. ionic polymer metal composite (IPMC), is subjected to alternating electric potential to investigate and study the time-delayed vibration characteristics. A generalized mathematical model of the actuator is obtained assuming multi-mode excitation and applying the Hamilton’s principle. Lambert W-function is then applied and a closed form solution of the transcendental characteristic equation of delay differential equation (DDE) is obtained. Delay differential equations (DDEs) are then solved taking into account the experimental data and ph

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Moaaz, Osama, Ioannis Dassios, Haifa Bin Jebreen, and Ali Muhib. "Criteria for the Nonexistence of Kneser Solutions of DDEs and Their Applications in Oscillation Theory." Applied Sciences 11, no. 1 (2021): 425. http://dx.doi.org/10.3390/app11010425.

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The objective of this study was to improve existing oscillation criteria for delay differential equations (DDEs) of the fourth order by establishing new criteria for the nonexistence of so-called Kneser solutions. The new criteria are characterized by taking into account the effect of delay argument. All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example.

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Tahir, A., and H. Habib. "Sufficient Conditions for Nonoscillation of Delay Differential Equations with Positive and Negative Coefficients using Schauder’s Fixed Point Theorem." Nigerian Journal of Basic and Applied Sciences 30, no. 2 (2023): 65–73. http://dx.doi.org/10.4314/njbas.v30i2.9.

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Schauder’s fixed point theorem and its applications to delay differential equations (DDEs) cannot be over emphasized. In this work, interesting results regarding the sufficient conditions for nonoscillation of more generalized forms of DDEs are established and provides improvements on the results obtained in the past. However, applying the theorem and the characteristic equation of DDE with constant coefficients helps to determine those conditions. Lastly, to ascertain a claim from an existing literature for a solution that is on oscillatory and has such larger solutions, comparison theorem is

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Bahgat, Mohamed S. M., and A. M. Sebaq. "An Analytical Computational Algorithm for Solving a System of Multipantograph DDEs Using Laplace Variational Iteration Algorithm." Advances in Astronomy 2021 (June 12, 2021): 1–16. http://dx.doi.org/10.1155/2021/7741166.

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In this research, an approximation symbolic algorithm is suggested to obtain an approximate solution of multipantograph system of type delay differential equations (DDEs) using a combination of Laplace transform and variational iteration algorithm (VIA). The corresponding convergence results are acquired, and an efficient algorithm for choosing a feasible Lagrange multiplier is designed in the solving process. The application of the Laplace variational iteration algorithm (LVIA) for the problems is clarified. With graphics and tables, LVIA approximates to a high degree of accuracy with a few n

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Green, Kirk, and Bernd Krauskopf. "Bifurcation Analysis of Frequency Locking in a Semiconductor Laser with Phase-Conjugate Feedback." International Journal of Bifurcation and Chaos 13, no. 09 (2003): 2589–601. http://dx.doi.org/10.1142/s0218127403008107.

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We present a detailed study of the external-cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback. Mathematically, lasers with feedback are modeled by delay differential equations (DDEs) with an infinite-dimensional phase space. We employ new numerical bifurcation tools for DDEs to continue steady states and periodic orbits, irrespective of their stability. In this way, we show that the periodic orbits corresponding to the ECMs are connected to the steady state solution associated with the locking range of the laser. We also identify symmetric and nonsymmetric homoclinic o

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