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Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using an integral inequality due to Komornik, the general decay outcome is obtained in the case b=0$$ b=0 $$. [ABSTRACT FROM AUTHOR]

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]

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1-type macrophages with pro-inflammatory properties), and pro-tumor cells (M2-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters τ1 and τ2. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters. [ABSTRACT FROM AUTHOR]

For the family of nonautonomous delay equations, we show that the generator of the evolution semigroup obtained from any such equation gives rise to an autonomous delay equation on a higher-dimensional space. We call it an induced delay equation. More significantly, we show that this equation can be used to study some important properties of the original dynamics in four main directions: the characterization of the hyperbolicity of the original delay equation via the hyperbolicity of the induced delay equation; the description of the spectral properties and of the corresponding Lyapunov exponents; the equivalence between the hyperbolicity of the induced delay equation and an admissibility property for its nonlinear perturbations; and, finally, the robustness of hyperbolicity under sufficiently small linear perturbations. The proofs of some of these results depend on a new variation of constants formula for the nonlinear perturbations of the induced delay equation that is also established in our work. [ABSTRACT FROM AUTHOR]

In this paper, we present a new approach on how the multiple time-scales perturbation method can be applied to differential-delay equations such that approximations of the solutions can be obtained which are accurate on long time-scales. It will be shown how approximations can be constructed which branch off from solutions of differential-delay equations at the unperturbed level (and not from solutions of ordinary differential equations at the unperturbed level as in the classical approach in the literature). This implies that infinitely many roots of the characteristic equation for the unperturbed differential-delay equation are taken into account and that the approximations satisfy initial conditions which are given on a time-interval (determined by the delay). Simple and more advanced examples are treated in detail to show how the method based on differential and difference operators can be applied. [ABSTRACT FROM AUTHOR]

Motivated by the recent work by Ma and Magal [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] on the global stability property of the Gurtin–MacCamy's population model, we consider a family of scalar nonlinear convolution equations with unimodal nonlinearities. In particular, we relate the Ivanov and Sharkovsky analysis of singularly perturbed delay differential equations in [https://doi.org/10.1007/978-3-642-61243-5%5f5] to the asymptotic behavior of solutions of the Gurtin–MacCamy's system. According to the classification proposed in [https://doi.org/10.1007/978-3-642-61243-5%5f5], we can distinguish three fundamental kinds of continuous solutions of our equations, namely, solutions of the asymptotically constant type, relaxation type, and turbulent type. We present various conditions assuring that all solutions belong to the first of these three classes. In the setting of unimodal convolution equations, these conditions suggest a generalized version of the famous Wright's conjecture. [ABSTRACT FROM AUTHOR]

Differential equations with state-dependent delays specify a semiflow of continuously differentiable solution operators, in general, only on an associated submanifold of the Banach space C1([−h, 0],ℝn). We extend a recent result on the simplicity of these solution manifolds to systems in which the delay is given by the state only implicitly in an extra equation. These algebraic delay systems appear in various applications. [ABSTRACT FROM AUTHOR]

Summary: In this article, an event‐triggered recursive filtering problem is studied for a class of nonlinear multi‐rate systems (MRSs) with random transmission delays (RTDs). The RTDs are described by utilizing random variables with a known probability distribution and the Kronecker δ$$ \delta $$ function. To facilitate further study, the MRS is converted into a single‐rate one by adopting an iteration equation approach. To address the challenge of filter design caused by different measurement sampling periods, a modified prediction method of measurements is given. Moreover, an event‐triggered mechanism (ETM) is introduced to regulate the innovation transmission frequency. The objective of the addressed filtering problem is to design a recursive distributed filtering method for MRSs subject to ETM and RTDs, where a minimum upper bound on the filter error covariance is obtained. Moreover, the filter gain matrix is formulated by resorting to the solutions to matrix difference equations. Besides, the boundedness in the mean‐square sense of the filtering error is analyzed and a sufficient condition is provided. Finally, simulations with comparison experiments are presented to demonstrate the effectiveness of the newly proposed theoretical results. [ABSTRACT FROM AUTHOR]

Summary: This study investigates the issue of containment control (CC) of multi‐agent systems (MASs) that are heterogeneous, nonlinear, and fractional‐order while taking practical scenarios into account, such as uncertainties, unknown nonlinearities, time‐varying state delays, and distributed time‐varying delays. A fully distributed adaptive observer is created for each follower in accordance with the communication topology information among agents. This observer is designed to estimate the convex combination information of the leaders. The approach utilizes interval type‐2 fuzzy set theory and adaptive methods to design a distributed containment control strategy that only uses the self‐information of the followers and neighbor nodes' information. Sufficient conditions for implementing containment control are given. A new Lyapunov‐Krasovskii functional (LKF) is constructed, and the stability of the system with feedback loop is proven using fractional‐order theory and inequality methods. Lastly, a simulation example is presented to illustrate the efficacy of the proposed approach. [ABSTRACT FROM AUTHOR]

Climate warming is predicted to increase mean temperatures and thermal extremes on a global scale. Because their body temperature depends on the environmental temperature, ectotherms bear the full brunt of climate warming. Predicting the impact of climate warming on ectotherm diversity and distributions requires a framework that can translate temperature effects on ectotherm life‐history traits into population‐ and community‐level outcomes. Here we present a mechanistic theoretical framework that can predict the fundamental thermal niche and climate envelope of ectotherm species based on how temperature affects the underlying life‐history traits. The advantage of this framework is twofold. First, it can translate temperature effects on the phenotypic traits of individual organisms to population‐level patterns observed in nature. Second, it can predict thermal niches and climate envelopes based solely on trait response data and, hence, completely independently of any population‐level information. We find that the temperature at which the intrinsic growth rate is maximized exceeds the temperature at which abundance is maximized under density‐dependent growth. As a result, the temperature at which a species will increase the fastest when rare is lower than the temperature at which it will recover from a perturbation the fastest when abundant. We test model predictions using data from a naturalized–invasive interaction to identify the temperatures at which the invasive can most easily invade the naturalized's habitat and the naturalized is most likely to resist the invasive. The framework is sufficiently mechanistic to yield reliable predictions for individual species and sufficiently broad to apply across a range of ectothermic taxa. This ability to predict the thermal niche before a species encounters a new thermal environment is essential to mitigating some of the major effects of climate change on ectotherm populations around the globe. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In this paper, we consider one-dimensional nonlinear piezoelectric beams with thermal and magnetic effects in the presence of a distributed delay term acting on the heat equation. First, we show that the system is well-posed in the sense of a semigroup. Through the construction of an appropriate Lyapunov functional, we establish a general decay result for the solutions of the system, for which the exponential and polynomial decays are only special cases, under a suitable assumption on the weight of the delay that the damping effect through heat conduction is strong enough to stabilize the system even in the presence of a time delay. Furthermore, our result does not depend on any relationship between system parameters. [ABSTRACT FROM AUTHOR]

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]

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]

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term. [ABSTRACT FROM AUTHOR]

In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order γ ∈ (1/2, 1) in Lp spaces with p ≥ 2, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent γ. The second section was devoted to examining the regularity of time. As a result, we found that, for each π ∈ (0, γ-1/2), the solution to the considered problem has a F-Hölder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm. [ABSTRACT FROM AUTHOR]

This work explores a new abstract model using multi-term fractional differential operator and delay effect. The model is defined by the existence of a delay parameter and insertion of multi term fractional differential operators in the input function. For the solution of the newly formulated problem, we utilized fixed-point theorems, which provide a powerful analytical tool for establishing the existence and uniqueness of solutions in functional spaces. Additionally, we investigate the stability properties of the implicit multi-term fractional delay differential equation (IMTFDDE) and employ the concept of Ulam-Hyers stability to assess their behaviour. The Ulam-Hyers stability framework is a contemporary approach that characterizes the sensitivity of solutions in functional spaces. Furthermore, we use an example to show how to apply our results. Lastly, for the real implementation of this research endeavour, we provided a real model devoted to modeling the dispersion of pollutants in a river while accounting for delayed effects and strip boundary circumstances. [ABSTRACT FROM AUTHOR]

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically. [ABSTRACT FROM AUTHOR]

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]

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay, and variable exponents. Under a suitable hypothesis the blow-up and growth of solutions are proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case of absence of the source term f 1 = f 2 = 0 . [ABSTRACT FROM AUTHOR]

The economic impact of Human Immunodeficiency Virus (HIV) goes beyond individual levels and it has a significant influence on communities and nations worldwide. Studying the transmission patterns in HIV dynamics is crucial for understanding the tracking behavior and informing policymakers about the possible control of this viral infection. Various approaches have been adopted to explore how the virus interacts with the immune system. Models involving differential equations with delays have become prevalent across various scientific and technical domains over the past few decades. In this study, we present a novel mathematical model comprising a system of delay differential equations to describe the dynamics of intramural HIV infection. The model characterizes three distinct cell sub-populations and the HIV virus. By incorporating time delay between the viral entry into target cells and the subsequent production of new virions, our model provides a comprehensive understanding of the infection process. Our study focuses on investigating the stability of two crucial equilibrium states the infection-free and endemic equilibriums. To analyze the infection-free equilibrium, we utilize the LaSalle invariance principle. Further, we prove that if reproduction is less than unity, the disease free equilibrium is locally and globally asymptotically stable. To ensure numerical accuracy and preservation of essential properties from the continuous mathematical model, we use a spectral scheme having a higher-order accuracy. This scheme effectively captures the underlying dynamics and enables efficient numerical simulations. [ABSTRACT FROM AUTHOR]

We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α ∈ (1 , 2) , which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented. [ABSTRACT FROM AUTHOR]

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]

In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions' sense. We also provide a sufficient condition for the optimality of the control. [ABSTRACT FROM AUTHOR]

Optimal disturbances of a number of typical stationary solutions of the hepatitis B virus infection dynamics model have been found. Specifically optimal disturbances have been found for stationary solutions corresponding to various forms of the chronic course of the disease, including those corresponding to the regime of low-level virus persistence. The influence of small optimal disturbances of individual groups of variables on the stationary solution is studied. The possibility of transition from stable stationary solutions corresponding to chronic forms of hepatitis B to stable stationary solutions corresponding to the state of functional recovery or a healthy organism using optimal disturbances is studied. Optimal disturbances in this study were constructed on the basis of generalized therapeutic drugs characterized by one-compartment and two-compartment pharmacokinetics. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the degenerate fractional differential equations with finite delay: Dα(Mu)(t)+cDβ(Mu)(t)$D^\alpha (Mu)(t) + cD^\beta (Mu)(t)$=Au(t)+Fut+f(t),(0≤t≤2π)$= Au(t) + Fu_t + f(t),(0\le t\le 2\pi)$ on Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, where A$A$ and M$M$ are closed linear operators in a complex Banach space X$X$ satisfying D(A)⊂D(M)$D(A)\subset D(M)$, c∈C$c\in \mathbb {C}$ and 0<β<α$0 < \beta < \alpha$ are fixed, ut(s)=u(t+s)$u_t(s) = u(t+s)$ when 0≤t≤2π,−2π≤s≤0$0\le t\le 2\pi , -2\pi \le s\le 0$, and the delay operator F$F$ is a bounded linear operator from Lp([−2π,0];X)$L^p([-2\pi ,0]; X)$ (resp. Bp,qs([−2π,0];X)$B_{p,q}^s([-2\pi ,0]; X)$) into X$X$. Using known operator‐valued Fourier multiplier theorems on Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, we completely characterize the Lp$L^p$‐well‐posedness and the Bp,qs$B_{p,q}^s$‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied. [ABSTRACT FROM AUTHOR]

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]

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]

This article deals with the numerical analysis of a class of singularly perturbed delay Volterra integro-differential equations exhibiting multiple boundary layers. The discretization of the considered problem is done using an implicit difference scheme for the differential term and a composite numerical integration rule for the integral term. The analysis of the discrete scheme consists of two parts. First, we establish an a priori error estimate that is used to prove robust convergence of the discrete scheme on Shishkin and Bakhvalov type meshes. Next, we establish the maximum norm a posteriori error estimate that involves difference derivatives of the approximate solution. The derived a posteriori error estimate gives the computable and guaranteed upper bound on the error. Numerical experiments confirm the theory. [ABSTRACT FROM AUTHOR]

In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order p$$ p $$ even if the order conditions of order p$$ p $$ hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods. [ABSTRACT FROM AUTHOR]

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]

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]

A rapidly converging domain decomposition algorithm is introduced for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers. Firstly, a space-time decomposition of the original problem is considered. Subsequently, an iterative process is proposed, wherein the exchange of information to neighboring subdomains is accomplished through the utilization of piecewise-linear interpolation. It is shown that the algorithm provides uniformly convergent numerical approximations to the solution. Our analysis utilizes some novel auxiliary problems, barrier functions, and a new maximum principle result. More importantly, we showed that only one iteration is needed for small values of the perturbation parameter. Some numerical results supporting the theory and demonstrating the effectiveness of the algorithm are presented. [ABSTRACT FROM AUTHOR]

In this paper, a new approaching technique is offered to unravel multi-pantograph-type delay differential equations. The suggested new method is a collocation method based on integration and Boubaker polynomials. As the main idea of the method, the process starts by approaching the first derivative function in the equation in the form of truncated Boubaker series. Then this approximating form is composed in the matrix form. The unknown function is then obtained by integrating the approximate derivative function and expressing it as a matrix. Using the approximation, the matrix forms for the proportionally delayed terms in the equation are derived. In addition, operational matrix forms are constructed for convenience in the method. By using these matrix forms and matrix operations, the problem is reduced to a system of algebraic linear equations. The method is illustrated through numerical implementations and compared with existing techniques in the literature. The results demonstrate the effectiveness and reliability of the proposed approach, highlighting its superiority over other methods. [ABSTRACT FROM AUTHOR]

We construct and analyze a second-order parameter uniform numerical method for parabolic singularly perturbed space-delay problems with interior turning point. The considered problem's solution possesses an interior layer in addition to twin boundary layers due to the presence of delay. Some theoretical estimates on derivatives of the analytical solution, which are useful for conducting the error analysis, are given. The proposed technique employs an upwind scheme on a fitted Bakhvalov–Shishkin mesh in the spatial variable and implicit-Euler scheme on a uniform mesh in the time variable. This discretization of the problem is shown to be uniformly convergent of O (Δ τ + − 1) , where Δ τ is the step size in the temporal direction and K denotes the number of mesh-intervals in the spatial direction. Further, to improve the accuracy, we make use of Richardson extrapolation and establish parameter-uniform convergence of O ((Δ τ) 2 + − 2) for the resulting scheme. Numerical experiments are performed over two test problems for validation of the theoretical predictions. [ABSTRACT FROM AUTHOR]

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]

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]

Developing a model of fractional differential systems and studying the existence and stability of a solution is considered one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to presenting a hybrid system of delay fractional differential equations with boundary integral conditions, namely Atangana-Baleanu-Caputo (ABC) differential equations. Also, the fixed point (FP) technique has been applied to investigate the existence of solutions for the proposed hybrid system. Moreover, stability results have been studied for the solution of the desired system in the sense of Hyers-Ulam (HU). Finally, to support our findings, we provide two examples with different parameter values. [ABSTRACT FROM AUTHOR]

Protective antigen (PA) is a protein produced by Bacillus anthracis. It forms part of the anthrax toxin and is a key immunogen in US and UK anthrax vaccines. In this study, we have conducted experiments to quantify PA in the supernatants of cultures of B. anthracis Sterne strain, which is the strain used in the manufacture of the UK anthrax vaccine. Then, for the first time, we quantify PA production and degradation via mathematical modelling and Bayesian statistical techniques, making use of this new experimental data as well as two other independent published data sets. We propose a single mathematical model, in terms of delay differential equations (DDEs), which can explain the in vitro dynamics of all three data sets. Since we did not heat activate the B. anthracis spores prior to inoculation, germination occurred much slower in our experiments, allowing us to calibrate two additional parameters with respect to the other data sets. Our model is able to distinguish between natural PA decay and that triggered by bacteria via proteases. There is promising consistency between the different independent data sets for most of the parameter estimates. The quantitative characterisation of B. anthracis PA production and degradation obtained here will contribute towards the ambition to include a realistic description of toxin dynamics, the host immune response, and anti-toxin treatments in future mechanistic models of anthrax infection. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example. [ABSTRACT FROM AUTHOR]

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]

In this paper, a new effective technique for the investigation of the higher-order trinomial differential equations y (n) (t) + p (t) y (τ (t)) + q (t) y (σ (t)) = 0 is established. We offer new criteria for so-called property (A) and oscillation of the considered equation. Examples are provided to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

In this work, we consider the one-dimensional thermoelastic Bresse system by addressing the aspects of nonlinear damping and distributed delay term acting on the first and the second equations. We prove a stability result without the common assumption regarding wave speeds under Neumann boundary conditions. We discover a new relationship between the decay rate of the solution and the growth of ϖ at infinity. Our results were achieved using the multiplier method and the perturbed modified energy, named Lyapunov functions together with some properties of convex functions. [ABSTRACT FROM AUTHOR]