1,269 results
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2. Periodic solution problems of neutral-type stochastic neural networks with time-varying delays.
- Author
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Famei Zheng, Xiaoliang Li, Bo Du, P, Balasubramaniam., and Peiluan Li
- Subjects
LINEAR matrix inequalities ,TIME-varying networks ,FUNCTIONAL differential equations ,DELAY differential equations ,BIDIRECTIONAL associative memories (Computer science) - Abstract
This paper is devoted to investigating a class of stochastic neutral-type neural networks with delays. By using the fixed point theorem and the properties of neutral-type operator, we obtain the existence conditions for periodic solutions of stochastic neutral-type neural networks. Furthermore, we obtain the conditions for the exponential stability of periodic solutions using Gronwall-Bellman inequality and stochastic analysis technique. Finally, a numerical example is given to show the effectiveness and merits of the present results. Our results can be used to obtain the existence and exponential stability of periodic solution to the corresponding deterministic systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. PRM volume 152 issue 2 Cover and Back matter.
- Subjects
DELAY differential equations ,SUBMANIFOLDS - Published
- 2022
- Full Text
- View/download PDF
4. Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations.
- Author
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Ademola, Adeleke Timothy, Aderogba, Adebayo Abiodun, Ogundipe, Opeoluwa Lawrence, Akinbo, Gbenga, and Onasanya, Babatunde Oluwaseun
- Subjects
LINEAR matrix inequalities ,FUNCTIONAL differential equations ,LINEAR systems ,DELAY differential equations - Abstract
In this paper, the problem of asymptotic stability of a kind of nonlinear perturbed neutral differential system with variable delay is discussed. The Lyapunov-Krasovskiˇı functional constructed, is used to obtain conditions for asymptotic stability of the nonlinear perturbed neutral differential system in terms of linear matrix inequality (LMI). The two new results (delay-independent and delay-dependent criteria) include and extend the existing results in the literature. Finally, an example of delay-dependent criteria is supplied and the simulation result is shown to justify the effectiveness and reliability of the used techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions.
- Author
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Alghamdi, Najla, Ahmad, Bashir, Alharbi, Esraa Abed, and Shammakh, Wafa
- Subjects
BOUNDARY value problems ,DELAY differential equations ,FRACTIONAL differential equations ,INTEGRO-differential equations ,FIXED point theory ,INTEGRAL operators ,FRACTIONAL integrals - Abstract
A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order βξ instead of β1; for details, see Lemma 2.2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. Finite difference method for the Riesz space distributed-order advection–diffusion equation with delay in 2D: convergence and stability
- Author
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Saedshoar Heris, Mahdi and Javidi, Mohammad
- Published
- 2024
- Full Text
- View/download PDF
7. On the qualitative behaviors of stochastic delay integro-differential equations of second order.
- Author
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Mahmoud, Ayman M. and Tunç, Cemil
- Subjects
INTEGRO-differential equations ,DELAY differential equations - Abstract
In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. Modeling and analysis of demand-supply dynamics with a collectability factor using delay differential equations in economic growth via the Caputo operator.
- Author
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Chen, Qiliang, Dipesh, Kumar, Pankaj, and Baskonus, Haci Mehmet
- Subjects
ECONOMIC expansion ,HOPF bifurcations ,DELAY differential equations ,LIMIT cycles ,OPERATING costs ,SUPPLY & demand - Abstract
In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor (τ) . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Asymptotic and Oscillatory Analysis of Fourth-Order Nonlinear Differential Equations with p -Laplacian-like Operators and Neutral Delay Arguments.
- Author
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Alatwi, Mansour, Moaaz, Osama, Albalawi, Wedad, Masood, Fahd, and El-Metwally, Hamdy
- Subjects
NONLINEAR differential equations ,DELAY differential equations ,NONLINEAR analysis ,DIFFERENTIAL equations - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. Seeds of Cross-Sector Collaboration: A Multi-Agent Evolutionary Game Theoretical Framework Illustrated by the Breeding of Salt-Tolerant Rice.
- Author
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Chen, Yusheng, Sun, Zhaofa, Wang, Yanmei, Ma, Ye, and Yang, Weili
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AGRICULTURAL technology ,RICE breeding ,DELAY differential equations ,BOUNDED rationality ,INCENTIVE (Psychology) ,NUMERICAL analysis ,RICE - Abstract
In the context of global food security and the pursuit of sustainable agricultural development, fostering synergistic innovation in the seed industry is of strategic importance. However, the collaborative innovation process between seed companies, research institutions, and governments is fraught with challenges due to information asymmetry and bounded rationality within the research and development phase. This paper establishes a multi-agent evolutionary game framework, taking the breeding of salt-tolerant rice as a case study. This study, grounded in the theories of information asymmetry and bounded rationality, constructs a two-party evolutionary game model for the interaction between enterprises and research institutions under market mechanisms. It further extends this model to include government participation, forming a three-party evolutionary game model. The aim is to uncover the evolutionary trends in collaborative behavior under various policy interventions and to understand how governments can foster collaborative innovation in salt-tolerant rice breeding through policy measures. To integrate the impact of historical decisions on the evolution of collaborative innovation, this research employs a delay differential equation (DDE) algorithm that takes historical lags into account within the numerical simulation. The stability analysis and numerical simulation using the DDE algorithm reveal the risk of market failure within the collaborative innovation system for salt-tolerant rice breeding operating under market mechanisms. Government involvement can mitigate this risk by adjusting incentive and restraint mechanisms to promote the system's stability and efficiency. Simulation results further identify that the initial willingness to participate, the coefficient for the distribution of benefits, the coefficient for cost sharing, and the government's punitive and incentivizing intensities are crucial factors affecting the stability of collaborative innovation. Based on these findings, the study suggests a series of policy recommendations including enhancing the initial motivation for participation in collaborative innovation, refining mechanisms for benefit distribution and cost sharing, strengthening regulatory compliance systems, constructing incentive frameworks, and encouraging information sharing and technology exchange. These strategies aim to establish a healthy and effective ecosystem for collaborative innovation in salt-tolerant rice breeding. While this research uses salt-tolerant rice breeding as a case study, the proposed cooperative mechanisms and policy suggestions have universal applicability in various agricultural science and technology innovation scenarios, especially when research meets widespread social needs but lacks commercial profit drivers, underscoring the essential role of government incentives and support. Consequently, this research not only contributes a new perspective to the application of evolutionary game theory in agricultural science and technology innovation but also offers empirical backing for policymakers in advancing similar collaborative innovation endeavors. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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11. Analytic Solutions of Delay-Differential Equations.
- Author
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Mallet-Paret, John and Nussbaum, Roger D.
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AUTONOMOUS differential equations ,DELAY differential equations ,VOLTERRA equations ,EQUATIONS - Abstract
In 1973 Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be C ∞ smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded C ∞ solutions for which analyticity is unknown. This is the case, for example, for the equation y ′ (t) = g (y (t - 1)) + ε sin (t 2) where g is analytic and where y = 0 is a hyperbolic equilibrium when ε = 0 . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Existence for coupled Random system fractional order functional differential with infinite delay.
- Author
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Ferhat, Mohamed and Blouhi, Tayeb
- Subjects
FUNCTIONAL differential equations ,DELAY differential equations - Abstract
In this paper we prove the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay principles combined with a technique based on vector-valued metrics and convergent to zero matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2024
13. Stability Analysis in Milling Based on the Localized Differential Quadrature Method.
- Author
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Mei, Yonggang, He, Bingbing, He, Shangwen, and Ren, Xin
- Subjects
DIFFERENTIAL quadrature method ,DELAY differential equations ,FLOQUET theory - Abstract
Chatter stability analysis is an effective way to optimize the cutting parameters and achieve chatter-free machining. This paper proposes a milling chatter stability analysis method based on the localized differential quadrature method (LDQM), which has the advantages of simple principle, easy application, and high computational efficiency. The milling process, considering the regeneration effect, is modeled using linear periodic delay differential equations (DDE), then the state transition matrix during the adjacent cutting period is constructed based on the LDQM, and finally, the stability of the milling process is obtained based on the Floquet theory. The accuracy and computation efficiency of the proposed method are verified through two benchmark milling models. The simulation results demonstrate that the proposed method in this paper can accurately and quickly obtain the chatter stability lobe diagram (SLD) of the milling process, which will provide guidance for optimizing the process parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. On the qualitative analyses of nonlinear delay differential equations of third order.
- Author
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Erduri, Sultan and Tunç, Cemil
- Subjects
NONLINEAR differential equations ,DELAY differential equations ,NONLINEAR analysis ,DIFFERENTIAL equations - Abstract
The authors of this paper examine uniformly asymptotically stability (UAS). uniformly boundedness (UB). ultimately uniformly boundedness (UUB) and existence of periodic solutions (EPSs) of certain nonlinear delay differential equations (DDEs) of third order with multiple constant delays. They prove two results and give a corollary on these properties of solutions. The main results of this paper have sufficient conditions and the technique of the proofs for these results is based on construction of a proper Lyapunov- KrasovskiT function (LKF). This paper has novel results to the theory of DDEs of third order and fills some gaps related to the mentioned concepts. [ABSTRACT FROM AUTHOR]
- Published
- 2022
15. Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials.
- Author
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Abd-Elhameedi, Waleed Mohamed, Youssri, Youssri Hassan, and Atta, Ahmed Gamal
- Subjects
FRACTIONAL differential equations ,DELAY differential equations ,CHEBYSHEV polynomials ,LINEAR systems ,POLYNOMIALS - Abstract
Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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16. Hopf bifurcation and stability analysis of a delay differential equation model for biodegradation of a class of microcystins.
- Author
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Luyao Zhao, Mou Li, and Wanbiao Ma
- Subjects
HOPF bifurcations ,DELAY differential equations ,MICROCYSTINS ,LOTKA-Volterra equations ,BIODEGRADATION ,SPHINGOMONAS - Abstract
In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. Maximum likelihood inference for a class of discrete-time Markov switching time series models with multiple delays.
- Author
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Martínez-Ordoñez, José. A., López-Santiago, Javier, and Miguez, Joaquín
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TIME series analysis ,MAXIMUM likelihood detection ,EL Nino ,DELAY differential equations ,STOCHASTIC differential equations ,MAXIMUM likelihood statistics ,NONLINEAR dynamical systems ,STOCHASTIC systems ,PARAMETER estimation - Abstract
Autoregressive Markov switching (ARMS) time series models are used to represent real-world signals whose dynamics may change over time. They have found application in many areas of the natural and social sciences, as well as in engineering. In general, inference in this kind of systems involves two problems: (a) detecting the number of distinct dynamical models that the signal may adopt and (b) estimating any unknown parameters in these models. In this paper, we introduce a new class of nonlinear ARMS time series models with delays that includes, among others, many systems resulting from the discretisation of stochastic delay differential equations (DDEs). Remarkably, this class includes cases in which the discretisation time grid is not necessarily aligned with the delays of the DDE, resulting in discrete-time ARMS models with real (non-integer) delays. The incorporation of real, possibly long, delays is a key departure compared to typical ARMS models in the literature. We describe methods for the maximum likelihood detection of the number of dynamical modes and the estimation of unknown parameters (including the possibly non-integer delays) and illustrate their application with a nonlinear ARMS model of El Niño–southern oscillation (ENSO) phenomenon. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. Stability and Numerical Analysis of a Coupled System of Piecewise Atangana–Baleanu Fractional Differential Equations with Delays.
- Author
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Almalahi, Mohammed A., Aldwoah, K. A., Shah, Kamal, and Abdeljawad, Thabet
- Abstract
This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. Optimal energy decay for a viscoelastic Kirchhoff equation with distributed delay acting on nonlinear frictional damping.
- Author
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Mohammed, Aili and Khemmoudj, Ammar
- Subjects
WAVE equation ,CONVEX functions ,EQUATIONS ,DELAY differential equations - Abstract
In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. An Investigation of a Nonlinear Delay Functional Equation with a Quadratic Functional Integral Constraint.
- Author
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El-Sayed, Ahmed M. A., Ba-Ali, Malak M. S., and Hamdallah, Eman M. A.
- Subjects
FUNCTIONAL equations ,INTEGRAL equations ,QUADRATIC equations ,NONLINEAR equations ,INTEGRALS ,DELAY differential equations ,NONLINEAR functional analysis - Abstract
This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L 1 [ 0 , T ] and, secondly, the existence of nonincreasing solutions in unbounded interval L 1 (R +) . Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ ∈ L 1 [ 0 , T ] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ ∈ L 1 (R +) on some parameters. Finally, some examples are provided to verify our investigation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
21. STOCHASTIC STABILIZATION OF NEUTRAL STOCHASTIC DELAY SYSTEMS BASED ON DISCRETE OBSERVATIONS.
- Author
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FANGZHE WAN, XUEYAN ZHAO, FEIQI DENG, PEILIN YU, and XIONGDING LIU
- Subjects
STOCHASTIC systems ,DISCRETE systems ,LINEAR systems ,GRONWALL inequalities ,MATHEMATICAL series ,DELAY differential equations - Abstract
This paper addresses the problem of stochastic stabilization for the neutral stochastic delay systems (NSDSs) based on discrete observations. Specifically, we design sampled-data based controllers (SDBCs) to stabilize NSDSs. To conquer the difficulties caused by the neutral term and stochastic stabilization, we introduce a new nominal system. By using the Lyapunov function method, we discuss the stability of the nominal system and provide a stability criterion. For the part of SDBC design, there are two methods: the lifting technique method (LTM) and the input delay method (IDM). The LTM is more effective for linear delay-free systems, but there is little research on the LTM for nonlinear stochastic delay systems. In this research, we combine the LTM with the equivalence technique for NSDSs, resulting in improved results. Additionally, to overcome the difficulty caused by LTM, we propose a series of mathematical tools, such as Gronwall's inequality of the discrete version. We compared our method with the IDM presented in X. Mao, IEEE Trans. Automat. Control, 61 (2016), pp. 1619--1624., and our method performs better. Finally, to showcase the efficiency and correctness of our proposed approach, we provide an application. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
22. Kneser-Type Oscillation Criteria for Half-Linear Delay Differential Equations of Third Order.
- Author
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Masood, Fahd, Cesarano, Clemente, Moaaz, Osama, Askar, Sameh S., Alshamrani, Ahmad M., and El-Metwally, Hamdy
- Subjects
DELAY differential equations ,OSCILLATIONS - Abstract
This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
23. Optimum Chebyshev filter with an equalised group delay response.
- Author
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Živaljević, Dragana, Stamenković, Negovan, and Stojanović, Nikola
- Subjects
CASCADE connections ,NONLINEAR equations ,NEWTON-Raphson method ,CHEBYSHEV polynomials ,IMPULSE response ,DELAY differential equations - Abstract
The first part of the paper discusses designing of the lowpass filter that gives both a constant group delay over the passband and a sharp cut-off slope, which is one of the most difficult challenges in continuous-time signal processing. Furthermore, equalisation of the group delay distortion of the Chebyshev filter has been performed through a cascade connection of the allpass filters. The Newton–Raphson method is applied for solving the system of nonlinear equations that calculate the allpass filter parameters for equalising the constant group delay in an equiripple sense. This procedure gives nearly symmetric impulse response of the resulting filtering system, as well as a longer duration. In the second part of the paper, two solutions for synthesising polynomial filter transfer functions are discussed: filter with monotonic amplitude characteristic (LSM) and filter using Chebyshev polynomial. For the purpose of comparison, the passband ripple factor of the Chebyshev filter is necessary to be adjusted so that both filters have the same stopband insertion loss, group delay and transient response. It is shown that Chebyshev filter is preferable since the passband magnitude distortions and sensitivity to the element deviation tolerance are significantly superior. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
24. Spectral collocation methods for fractional multipantograph delay differential equations*.
- Author
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Shi, Xiulian, Wang, Keyan, and Sun, Hui
- Subjects
COLLOCATION methods ,VOLTERRA equations ,INTEGRAL equations ,FRACTIONAL differential equations ,DELAY differential equations - Abstract
In this paper, we propose and analyze a spectral collocation method for the numerical solutions of fractional multipantograph delay differential equations. The fractional derivatives are described in the Caputo sense. We present that some suitable variable transformations can convert the equations to a Volterra integral equation defined on the standard interval [−1, 1]. Then the Jacobi–Gauss points are used as collocation nodes, and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence analysis of the proposed method is investigated in the infinity norm and weighted L
2 norm. To perform the numerical simulations, some test examples are investigated, and numerical results are presented. Further, we provide the comparative study of the proposed method with some existing numerical methods. [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
25. New Monotonic Properties for Solutions of a Class of Functional Differential Equations and Their Applications.
- Author
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Masood, Fahd, Moaaz, Osama, AlNemer, Ghada, and El-Metwally, Hamdy
- Subjects
FUNCTIONAL differential equations ,MONOTONE operators ,DELAY differential equations ,OSCILLATIONS - Abstract
This paper delves into the enhancement of asymptotic and oscillatory behaviors in solutions to even-order neutral differential equations with multiple delays. The main objective is to establish improved inequalities to advance the understanding of oscillation theory for these equations. The paper's approach is centered on improving the understanding of the intricate relationship between solutions and their corresponding functions. This is achieved by harnessing the modified monotonic properties of positive solutions, which provide valuable insights into oscillation behavior. Furthermore, leveraging the symmetry between positive and negative solutions, we derived criteria that ensure oscillation for all solutions, with a specific emphasis on excluding only positive solutions. To illustrate the significance of our findings, we provide an illustrative example. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. On the Asymptotic Stability a System with Nonlinear Perturbations and Constant Delay.
- Author
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Gözen, Melek
- Subjects
STABILITY of nonlinear systems ,DELAY differential equations ,INTEGRO-differential equations ,NONLINEAR systems ,LINEAR matrix inequalities - Abstract
In this paper, we consider a nonlinear perturbed system of neutral delay integro-differential equations (NDIDEs). We prove two new theorems, Theorems 1 and 2, such that these theorems include sufficient conditions and are related to asymptotical stability of zero solution of the perturbed system of NDIDEs. The technique of the proofs depends upon the definitions of two new and more suitable Lyapunov-Krasovskiĭ functionals (LKFs). When we compared the results of this paper with those found in the literature, our results improve and extend some classical results, and contribute new contents to the topic of NDIDEs and to the profession. [ABSTRACT FROM AUTHOR]
- Published
- 2022
27. Strong delayed negative feedback.
- Author
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Erneux, Thomas
- Subjects
HOPF bifurcations ,NONLINEAR theories ,DELAY differential equations ,NUMERICAL analysis ,MATHEMATICAL models - Abstract
In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Periodicity, stability, and synchronization of solutions of hybrid coupled dynamic equations with multiple delays.
- Author
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Agrawal, Divya, Dhama, Soniya, Kostić, Marko, and Abbas, Syed
- Subjects
- *
TOPOLOGICAL degree , *SYNCHRONIZATION , *EQUATIONS , *MATHEMATICAL models , *DYNAMICAL systems , *DIFFERENCE equations , *DELAY differential equations - Abstract
This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations.
- Author
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Zaky, M. A., Babatin, M., Hammad, M., Akgül, A., and Hendy, A. S.
- Subjects
COLLOCATION methods ,PANTOGRAPH ,FRACTIONAL differential equations ,NONLINEAR systems ,FRACTIONAL calculus ,DELAY differential equations ,LOGARITHMIC functions ,HAMILTON-Jacobi equations - Abstract
Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. A Survey on the Oscillation of First-Order Retarded Differential Equations.
- Author
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Stavroulakis, Ioannis P.
- Subjects
DELAY differential equations ,DIFFERENTIAL equations ,OSCILLATIONS - Abstract
In this paper, a survey of the most interesting conditions for the oscillation of all solutions to first-order linear differential equations with a retarded argument is presented in chronological order, especially in the case when well-known oscillation conditions are not satisfied. The essential improvement and the importance of these oscillation conditions is also indicated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Frequently Hypercyclic Semigroup Generated by Some Partial Differential Equations with Delay Operator.
- Author
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Hung, Cheng-Hung
- Subjects
DELAY differential equations ,OPERATOR equations ,CELL cycle - Abstract
In this paper, under appropriate hypotheses, we have the existence of a solution semigroup of partial differential equations with delay operator. These equations are used to describe time–age-structured cell cycle model. We also prove that the solution semigroup is a frequently hypercyclic semigroup. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Some Oscillatory Criteria for Second-Order Emden–Fowler Neutral Delay Differential Equations.
- Author
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Tian, Haifeng and Guo, Rongrong
- Subjects
DELAY differential equations ,INTEGRAL inequalities ,OSCILLATIONS - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Bifurcation Analysis of a Class of Two-Delay Lotka–Volterra Predation Models with Coefficient-Dependent Delay.
- Author
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Li, Xiuling and Fan, Haotian
- Subjects
HOPF bifurcations ,DELAY differential equations ,PREDATION - Abstract
In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Cross-Convolution Approach for Delay Estimation in Fractional-Order Time-Delay Systems.
- Author
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Asiri, Sharefa and Liu, Da-Yan
- Subjects
- *
TIME delay estimation , *DELAY differential equations , *COMPUTER simulation , *FRACTIONAL calculus - Abstract
Several real-life problems that involve a time delay are modeled using fractional time-delay systems. However, most studies related to these systems assume that the delay is already known, which is not the case in practical scenarios where the delay is often uncertain or unknown. To address this issue, this paper proposes an algebraic and robust method to estimate the input delay for a class of fractional time-delay systems in a noisy environment, by applying a cross-convolution approach. Besides, a filtering methodology is incorporated with the proposed approach to enhance its efficacy. In addition, this paper presents novel theories on convolution in the field of fractional calculus. Finally, the performance of the proposed approach is demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds.
- Author
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Kürkçü, Ömür Kıvanç and Sezer, Mehmet
- Subjects
VOLTERRA equations ,INTEGRO-differential equations ,DELAY differential equations ,BENCHMARK problems (Computer science) ,POLYNOMIALS ,PROBLEM solving - Abstract
Purpose: This study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points. Design/methodology/approach: Thanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled. Findings: All computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question. Originality/value: This paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments.
- Author
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Zhang, Yuhang, Song, Minghui, Liu, Mingzhu, and Zhao, Bowen
- Subjects
EULER method ,EXPONENTIAL stability ,DELAY differential equations - Abstract
This work develops the Milstein scheme for commutative stochastic differential equations with piecewise continuous arguments (SDEPCAs), which can be viewed as stochastic differential equations with time-dependent and piecewise continuous delay. As far as we know, although there have been several papers investigating the convergence and stability for different numerical methods on SDEPCAs, all of these methods are Euler-type methods and the convergence orders do not exceed 1/2. Accordingly, we first construct the Milstein scheme for SDEPCAs in this work and then show its convergence order can reach 1. Moreover, we prove that the Milstein method can preserve the stability of SDEPCAs. In the last section, we provide several numerical examples to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. A new result on averaging principle for Caputo-type fractional delay stochastic differential equations with Brownian motion.
- Author
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Zou, Jing and Luo, Danfeng
- Subjects
STOCHASTIC differential equations ,DELAY differential equations ,EQUATIONS of motion ,BROWNIAN motion ,JENSEN'S inequality ,LAPLACE transformation - Abstract
In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. On solving some stochastic delay differential equations by Daubechies wavelet.
- Author
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Shariati, Nasim Madah, Yaghouti, Mohammadreza, and Alipanah, Amjad
- Subjects
STOCHASTIC differential equations ,NUMERICAL solutions to stochastic differential equations ,DIFFERENTIAL equations ,DELAY differential equations ,COLLOCATION methods ,VIBRATION (Mechanics) - Abstract
There are numerous phenomena in real world that their practical modeling in mathematics language deals with differential equations. Stochastic terms have significant roles in estimating behavior of these events such as machine tool vibrations, biology, traffic dynamics, neural networks and so on, it cannot be reasonable to consider an event without its past information, and it ends up with equations including time delays. The numerical solution of stochastic delay differential equations has always been of interest to researchers. In this paper, by using scaling function of the Daubechies wavelet, a collocation method is presented to solve a class of stochastic delay differential equations and it is tried to demonstrate the advantages of using this type of wavelet. Convergence of the method is provided and numerical experiments are reported to show application of the method in practice. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. AN ITERATIVE METHOD FOR THE QUALITATIVE ANALYSIS OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS.
- Author
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Basu, R. and Lather, J.
- Subjects
- *
DELAY differential equations , *NONLINEAR analysis , *DIFFERENTIAL equations - Abstract
In this paper, the authors studied sufficient conditions to understand the oscillatory behavior of solutions of nonlinear neutral delay (and advanced) differential equation (E) with deviating arguments. Employing Banach fixed point theorem, the authors established the existence of bounded positive solutions of (E). The authors solved various examples using MATLAB software to understand the applications of the main theorems. Moreover, the authors analyzed the effect of delay terms on the behavior of solutions of (E). This paper has improved the results obtained in Basu [2] and Chatzarakis [11]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Complex global dynamics of conditionally stable slopes: effect of initial conditions.
- Author
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Prekrat, D., Todorović-Vasović, N. K., Vasović, N., and Kostić, S.
- Subjects
DELAY differential equations ,STOCHASTIC differential equations ,STOCHASTIC systems ,NUMERICAL analysis ,LANDSLIDES - Abstract
In the present paper, we investigate the effect of the initial conditions on the dynamics of the spring-block landslide model. The time evolution of the studied model, which is governed by a system of stochastic delay differential equations, is analyzed in the mean-field approximation, which qualitatively exhibits the same dynamics as the initial model. The results of the numerical analysis show that changing the initial conditions has different effects in different parts of the parameter space of the model. Namely, moving away from the fixed-point initial conditions has a stabilizing effect on the dynamics when the noise, the friction parameters a (higher values) and c as well as the spring stiffness k are taken into account. The stabilization manifests itself in a complete suppression of the unstable dynamics or a partial limitation of the effect of some friction parameters. On the other hand, the destabilizing effect of changing the initial conditions occurs for the lower values of the friction parameters a and for b. The main feature of destabilization is the complete suppression of the sliding regime or a larger parameter range with a transient oscillatory regime. Our approach underlines the importance of analyzing the influence of initial conditions on landslide dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Asymptotic behaviour and boundedness of solutions for third-order stochastic differential equation with multi-delay.
- Author
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Mahmoud, A. M., Eisa, D. A., Taie, R. O. A., and Bakhit, D. A. M.
- Subjects
STOCHASTIC differential equations ,DELAY differential equations - Abstract
In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. A SECOND-ORDER NUMERICAL METHOD FOR PSEUDO-PARABOLIC EQUATIONS HAVING BOTH LAYER BEHAVIOR AND DELAY PARAMETER.
- Author
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GUNES, Baransel and DURU, Hakkı
- Subjects
APPROXIMATION error ,ANALYTICAL solutions ,DELAY differential equations ,EQUATIONS - Abstract
In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Periodic solutions in reversible systems in second order systems with distributed delays.
- Author
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Yameng Duan, Krawcewicz, Wieslaw, and Huafeng Xiao
- Subjects
NONLINEAR differential equations ,DELAY differential equations - Abstract
In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Two Different Analytical Approaches for Solving the Pantograph Delay Equation with Variable Coefficient of Exponential Order.
- Author
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Alrebdi, Reem and Al-Jeaid, Hind K.
- Subjects
PANTOGRAPH ,LINEAR differential equations ,DELAY differential equations ,DECOMPOSITION method ,REGULAR graphs ,EQUATIONS - Abstract
The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Exact solution of system of multi-photograph type delay differential equations via new algorithm based on homotopy perturbation method.
- Author
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Anakira, Nidal Ratib, Mohammed, Mohammed Jasim, Irianto, Irianto Irianto, Amourah, Ala Ali, and Oqilat, Osama Nasser
- Subjects
DELAY differential equations ,PADE approximant ,DIFFERENTIAL equations ,ALGORITHMS ,NONLINEAR systems - Abstract
A new algorithm is proposed in this paper to explain how the modified homotopy perturbation approach can be successfully implemented based on the Pade approximants and the Laplace transform in order to acquire the accurate solutions of a nonlinear system of multi-photograph delay differential equations. The method that has been suggested has the benefit of giving exact solutions, and it is simple to implement analytically on the issues that have been presented. Examples have been provided to demonstrate that this strategy may be utilized and is successful in its application. The results show that the method that was described could be used to solve many different types of differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. The Green Development in Saline–Alkali Lands: The Evolutionary Game Framework of Small Farmers, Family Farms, and Seed Industry Enterprises.
- Author
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Chen, Yusheng, Sun, Zhaofa, Wang, Yanmei, Ma, Ye, and Zhou, Yongwei
- Subjects
SUSTAINABLE development ,SEED industry ,FAMILY farms ,SALT-tolerant crops ,DELAY differential equations ,MONETARY incentives ,BOUNDED rationality ,INFORMATION asymmetry - Abstract
Amid global climate change and population growth, the prevalence of saline–alkali lands significantly hampers sustainable agricultural development. This study employs theories of asymmetric information and bounded rationality to construct an evolutionary game model, analyzing the interactions among small farmers, family farms, and seed industry enterprises in the context of saline–alkali land management. It investigates the strategic choices and dynamics of these stakeholders under the influence of economic incentives and risk perceptions, with a focus on how government policies can foster green development. Utilizing Delay Differential Equations (DDEs) for simulations, this study highlights the risk of "market failure" without government intervention and underscores the need for government participation to stabilize and improve the efficiency of the green development process. The findings reveal that factors such as initial willingness to participate, the economic viability of salt-tolerant crops, seed pricing, research and development costs, and the design of incentive policies are crucial for sustainable land use. Accordingly, the paper proposes specific policy measures to enhance green development, including strengthening information dissemination and technical training, increasing the economic attractiveness of salt-tolerant crops, alleviating research and development pressures on seed companies, and optimizing economic incentives. This study provides a theoretical and policy framework for the sustainable management of saline–alkali lands, offering insights into the behavioral choices of agricultural stakeholders and supporting government strategies for agricultural and environmental protection. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. A higher order compact numerical approach for singularly perturbed parabolic problem with retarded term.
- Author
-
Babu, Gajendra, Sharma, Kapil K., and Bansal, Komal
- Subjects
FINITE difference method ,FINITE differences ,DELAY differential equations - Abstract
In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Solving a System of One-Dimensional Hyperbolic Delay Differential Equations Using the Method of Lines and Runge-Kutta Methods.
- Author
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Karthick, S., Subburayan, V., and Agarwal, Ravi P.
- Subjects
HYPERBOLIC differential equations ,ORDINARY differential equations ,PARTIAL differential equations ,RUNGE-Kutta formulas ,DIFFERENTIAL equations ,DELAY differential equations ,MAXIMUM principles (Mathematics) - Abstract
In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O (Δ t + h ¯ 4) , where Δ t is the time step and h ¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback.
- Author
-
Mei, Zisong and Wang, Zaihua
- Subjects
DELAY differential equations ,PENDULUMS ,PSYCHOLOGICAL feedback ,CLOSED loop systems - Abstract
This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. Characterization of exponential polynomial as solution of certain type of nonlinear delay-differential equation.
- Author
-
BANERJEE, Abhijit and BISWAS, Tania
- Subjects
NONLINEAR equations ,POLYNOMIALS ,DELAY differential equations ,DIFFERENTIAL-difference equations ,NEVANLINNA theory ,MATHEMATICS - Abstract
In this paper, we have characterized the nature and form of solutions of the following nonlinear delaydifferential equation where bi C, L(z, f) are a linear delay-differential polynomial of f; n is positive integers; q, Q and P respectively are nonzero, nonconstant and any polynomials. Different special cases of our result will accommodate all the results of [J. Math. Anal. Appl., 452(2017), 1128-1144; Mediterr. J. Math., 13(2016), 3015-3027; Open Math., 18(2020), 1292-1301]. Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occur and consequently the same are automatically applicable to the previous results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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