This article examines the dynamical behavior of a fractional-order allelopathic plankton model subject to harvesting. The governing equations are expressed in terms of Caputo fractional derivatives with time-dependent delays. Utilizing the Banach fixed-point theorem, we derive explicit sufficient conditions ensuring the existence and uniqueness of a weighted pseudo S-asymptotically ω -periodic solution confined to an invariant set. Moreover, local Mittag-Leffler stability is established via a Lyapunov functional method combined with fractional order inequalities. The theoretical findings are supported by numerical simulations that demonstrate the applicability of the proposed framework.
Additive Bayesian Networks (ABNs) are graphical models that extend generalized linear models to multivariate settings by representing the full joint distribution of multiple dependent variables. Although ABNs have been widely applied in biological and epidemiological research, their use in road-safety studies remains largely unexplored. This paper presents one of the first applications of ABNs to human-driven crash data, demonstrating their suitability for modelling complex dependency structures in driving-risk analysis. Using 10,045 crash reports from Montgomery County (Maryland, USA), we develop a complete ABN workflow comprising prior specification, score-cache structure learning, Laplace-based marginal likelihood computation, and robustness assessment through extensive Markov Chain Monte Carlo (MCMC) exploration. The final model identifies both direct and indirect associations among behavioral (distraction, substance use), environmental (weather, surface conditions), and contextual (speed limit) factors associated with crash severity. Results illustrate how ABNs can provide an interpretable graphical representation of conditional dependencies among crash-related variables and can complement regression-based approaches by explicitly representing both direct and indirect statistical associations. This study provides a reproducible methodological template for ABN modelling and illustrates its potential for broader application in safety-critical and multivariate domains.
This article presents an alternative explicit solution to the one-dimensional Bratu problem obtained through an adaptation of the Kudryashov expansion method. The analysis revisits the classical bifurcation structure of the problem, demonstrating that the set of solutions contains zero, one, or two branches depending on the value of the critical parameter λ _c . The explicit formulation developed here provides additional analytical insight into the Bratu equation and offers a useful tool for further studies of nonlinear boundary value problems with similar exponential nonlinearities.
This paper presents a complete analysis of a nonlinear equation involving the Dirichlet-to-Neumann operator on the unit disk. For a smooth function f on the circle satisfying Af + α = e^f , where A is the Dirichlet-to-Neumann operator and α > 0 is a constant, we prove that when α is not a positive integer, the only solution is the constant function f = logα . When α is a positive integer, a rich family of non-constant solutions emerges, characterized by Fourier coefficients that vanish except at multiples of α . The analysis provides an explicit classification of solutions and illustrates the phenomenon of resonance in nonlinear boundary value problems. The methods employed—Fourier series, spectral decomposition, and inductive reasoning—offer a paradigmatic example of how symmetry and spectral theory can be combined to solve nonlinear PDEs exactly.
Modeling bivariate lifetime data with ties and complex hazard structures remains a challenging problem, particularly when classical parametric models fail to capture non-monotone failure behaviors. In this paper, we develop a flexible semi-parametric framework for bivariate distributions that incorporates a singular component to accommodate ties and allows for highly adaptable marginal hazard functions through a piecewise constant specification. The proposed approach does not assume any specific parametric form for the baseline distribution, enabling the modeling of increasing, decreasing, bathtub-shaped, and other non-standard hazard patterns. Dependence between variables is characterized through an induced copula structure, facilitating the study of dependence measures. Parameter estimation is performed using an efficient EM algorithm that involves lower-dimensional optimization and is computationally straightforward. A simulation study demonstrates the accuracy and stability of the proposed estimation procedure. Methods for selecting the number and locations of cut points are also discussed. The practical utility of the proposed framework is illustrated through an application to real data.
Motivated by the limitations of existing root-ratio methods for multiple roots, we propose a new family of iterative methods for multiple roots based on a modified root-ratio framework in which the Newton-type correction terms are regularized by replacing f'(x_n) with f'(x_n)+af(x_n) , thereby improving the flexibility of the correction step. The proposed family is shown to achieve optimal fourth-order convergence. By further incorporating the self-accelerating technique, we construct two methods with memory whose R-orders are improved from 4 to √(5)+2≈ 4.2361 and (5+√(17))/2≈ 4.5616 , respectively. Several numerical examples and dynamical tests are presented to illustrate the effectiveness of the proposed methods and to confirm the theoretical results.
In this paper, we investigate exponential synchronization between neural network systems with time-varying delays. The complexities introduced by such delays are handled by constructing a Lyapunov function. Using Young’s inequality and a generalized Halanay inequality, synchronization conditions are derived. Sufficient conditions are established to ensure exponential convergence using inequality-based techniques. Based on the Lyapunov stability analysis method, combined with Young’s inequality and Halanay’s inequality, a new method for the coupling scheme is proposed. Neural networks with time-varying delays are shown to synchronize exponentially under certain conditions. Moreover, the influence of delays is analyzed with the help of output coupling. Finally, theoretical results are verified through numerical simulations.
This work presents several enhancements to the Multilevel Dimension Iteration (MDI) framework for high-dimensional numerical integration. Although the original MDI algorithm reduces the computational cost from exponential to polynomial order, its applicability is largely restricted to smooth and regularly structured integrands. As a result, the original MDI algorithm struggles with complex structures, singularities, or truncated domains, as these features significantly increase computational cost and cause the algorithm to fail or degrade rapidly in higher dimensions. We identify these limitations and develop a family of extended MDI algorithms that extend the framework to a broader class of practical integration problems. Our enhanced MDI framework consists of the quadrature-adjusted MDI algorithm, which modifies the quadrature points to simplify symbolic computation; the row extraction MDI algorithm, which efficiently manages the complex matrix structures arising from the spectral decomposition of correlated assets; and the hybrid MDI algorithm, which partitions the domain effectively to handle truncation effects. The explicit solution of the multi-asset Black–Scholes equation and several comprehensive examples are used to evaluate the proposed algorithms.
In this paper, we investigate a series of harvesting problems in a predator–prey system with Beddington–DeAngelis (B-D) type functional response, focusing on bioeconomic equilibrium, maximum sustainable total yield (MSTY), and optimal economic benefits. Research shows that under the independent harvesting strategy, the MSTY exists when the Hessian matrix of the yield function Y(e_1,e_2) is negative definite; under the synchronous harvesting strategy, the MSTY can exist when appropriate parameter values are chosen. By incorporating Bang-Bang control and singular control into harvesting strategies, the system converges to the optimal equilibrium state faster than the constant effort harvesting strategy. The study employs a control parameterization method and utilizes the MISER 3 software package to solve two types of optimal control problems. These findings provide a theoretical basis for biological resource management.
We introduce a class of delayed cosine and sine matrix functions associated with the operator ℒ(X)=AX+XB and develop a systematic framework for second-order delay differential equations in matrix spaces. A generalized binomial operator is employed to construct explicit representations of solutions. We derive fundamental properties of the delayed cosine and sine functions, including differentiability, delay identities, and their role as solution kernels. A representation formula for linear delay systems is established, extending the classical Duhamel principle. For nonlinear problems, we prove existence and uniqueness results using Banach and Schauder fixed point theorems, together with global existence under growth conditions. Ulam–Hyers stability results are obtained. Finally, two numerical examples are proposed and validated through simulations.
Piecewise differential systems have received significant attention over the last decades because of their relevance in numerous applications. A key topic in the qualitative study of these systems is the existence of limit cycles, that is, periodic trajectories that are isolated from other nearby periodic trajectories. In this paper, we investigate a class of planar piecewise differential systems defined on ℝ^2 and separated by the analytic curve given by the graphic of the function y=e^x . On each side of this curve, the dynamics is governed by an arbitrary linear Hamiltonian system. The considered systems are continuous–discontinuous in the sense that the first coordinates of the corresponding vector fields agree on the switching curve, whereas the second coordinates do not necessarily coincide there. Our main result establishes that systems within this family may possess at most three limit cycles.
The numerical solution of time-dependent systems is often carried out using a step-wise updating procedure known as marching in time or simply time-marching methods. They require completing previous steps to yield an update for the next time step. This imposes a barrier in implementing the algorithm in a computer program, which prevents parallel speed-up by increasing the serial fraction of the program. This paper analyzes a method based on the space-time discretization of a time-dependent system using Discrete Picard Iterations (DPI). Unlike time marching schemes, the updating barrier is removed in the current formulation by converting the solution procedure into a set of larger matrix–vector products that correspond to the simultaneous computation of spatial residuals at various time levels. The stability and accuracy of the proposed method are investigated. The accuracy and computation time are compared to various popular time-marching methods involving structured and higher order/spectral unstructured grids in one and two-dimensional space. It is shown that when the inverse of the Chebyshev differentiation matrix is used in the DPI framework, the resulting scheme is superior in both accuracy and CPU time. Various forms of iterative methods and preconditionings are implemented and evaluated. This includes GMRES, IDR(s), and Conjugate Gradient methods. It is shown that the stabilized Bi-conjugate gradient method produces the best iterative solver when a band-LU preconditioner is used. This method, which relies heavily on matrix multiplication, is strategically consistent with the recent development in the hardware implementation of matrix multiplication accelerators on ASICs and FPGAs.
In this paper, we investigate a discrete-time Kolmogorov-type predator–prey model with a ratio-dependent functional response that incorporates both predator interference and handling time. In contrast to its continuous-time counterpart, the discrete model exhibits a wider variety of dynamical behaviors. We first establish the positivity and boundedness of solutions, determine all equilibrium points, and study their local stability properties. Our analysis shows that the coexistence equilibrium can undergo transcritical, period-doubling, and Neimark–Sacker bifurcations, whereas the boundary equilibria may experience transcritical and period-doubling bifurcations. In particular, the intersection of the period-doubling and Neimark–Sacker bifurcation curves gives rise to a codimension-two 1:2 resonance point, which plays a central role in the emergence of quasi-periodic oscillations, higher-period orbits, and chaotic dynamics. The onset of chaos is further characterized by means of the maximum Lyapunov exponent. To support the theoretical results, numerical simulations are presented in the form of bifurcation diagrams and phase portraits. In addition, a state-feedback control strategy is proposed to stabilize the coexistence equilibrium and suppress chaotic oscillations, showing that the chaotic behavior of the system can be effectively controlled.
In this paper, we apply the Homotopy Analysis Method (HAM) to fractal ordinary and partial differential equations defined on fractal curves. After briefly reviewing the essential tools of fractal calculus, we adapt the HAM framework to the fractal setting and show how the convergence–control parameter improves the accuracy of the homotopy series. Several illustrative examples of nonlinear fractal ODEs and PDEs are solved, demonstrating the efficiency and flexibility of the method for analytic approximation on fractal geometries.
In the present study, we used a deterministic model to simulate the propagation of an epidemic by dividing the population into five sub classes: susceptible, vaccinated, exposed, infectious, and recovered. The fuzzy SVEIR model has been used to examine two equilibria mathematically: endemic equilibrium and disease-free equilibrium. Fuzzy analysis has been done by taking into account the infection rate, disease-related death rate, and recovery rate as membership functions of fuzzy numbers. In order to determine the stability of the disease, stability analyses for endemic equilibrium and equilibrium in the absence of disease were done with respect to the reproduction number. The system is locally asymptotically stable at the disease-free equilibrium point if the fundamental reproduction number is smaller than 1, else it is unstable. LaSalle’s invariance principle enables us to determine that the model is globally asymptotically stable when the reproduction number R_0<1 . Additionally, we examine the sensitivity of the parameters, offer suggestions for illness prevention, and clarify the viability of the model, using the Euler approach, numerical stimulation is discovered for various h values.
In this paper, we have extended a classical mathematical model of contact spread infection based on the one-host two parasitoid competition model with fractional order values to account for the spread of infection. We have investigated contact spread infection of pathogen transmission on the host-parasitoid competition model using Caputo and fractal–fractional C–F fractional derivatives. Fractal–fractional operators improve the predictive capacity of infection by utilizing the memory effect. We have calculated the existence and uniqueness of the proposed fractional competition model. Moreover, this study exhibits the chaotic behavior under different values of fractional order and parameters based on the possibility of obtaining new chaotic behaviors. We have illustrated some maximal bifurcation diagrams of the fractional competition model with different parameter values. Numerical outcomes are carried out by two numerical schemes: Adams-Bashforth and Newton polynomial based technique, with Caputo and Caputo–Fabrizio (C–F) fractional operators. We have gained some useful insights about the fractional host-parasitoid competition model at various parameters and fractional orders based on the graphical results. In contact spread infections, transient and long-term dynamics are highly complex, as shown by computational simulations and steady-state analyses. Using this model for integer and non-integer orders, we can gain a better understanding of the complexity of the host-parasitoid competition model.
In this work, we present an accurate and efficient algorithm (based on neural networks) for computing the lower and upper critical values of the Durbin–Watson test for first-order autocorrelation, tailored for large sample sizes T and a large number of regressors k - a setting increasingly common in big-data applications. The critical values are obtained by numerically inverting the cumulative distribution function (CDF) using an adaptive Newton scheme. Evaluation of the CDF relies on numerical integration of the characteristic function of the underlying random variables. Because computing this characteristic function becomes expensive for large T, we develop a computationally cheap yet accurate approximation of the integrand. Since the convergence of the root-finding procedure depends strongly on the choice of initial values, we train neural networks on a precomputed dataset of critical values. For any given pair (T, k), the trained model provides accurate initial guesses, which significantly accelerate the adaptive Newton method. We validate our approach by recomputing previously published critical values and show that several of them are imprecise or internally inconsistent. Finally, we demonstrate that the proposed algorithm computes critical values within a fraction of a second on a standard laptop, even for sample sizes as large as 2 · 10^6 .
We propose a Stochastic Computation System (SCS) as a mathematical modelling framework intended to place deterministic, stochastic and coherence-enhanced two-level computational regimes within a common parametrized language. The central object is a probabilistic bit (prob-bit), represented by a triple consisting of a logical probability, a stochastic dispersion parameter and a normalized coherence coordinate. This parametrization separates logical bias, finite-resource statistical dispersion and phase-sensitive coherence effects, while allowing binary, classical stochastic and quantum-compatible regimes to be described as limiting or admissible subsets of a common state description. The paper formalizes SCS states, measurement maps and admissible SCS channels at an abstract level. Particular attention is given to Bernoulli parameter estimation as a minimal task for comparing resource–accuracy behaviour across regimes. The classical independent-sampling error law and its effective-sample-size modification for correlated stochastic bitstreams are distinguished from the coherence-enhanced ( 1/R ) benchmark motivated by quantum amplitude estimation. Thus, the proposed framework enables model-level quantitative comparisons under explicitly stated assumptions on dispersion, coherence and resource counting. The SCS formalism is not intended to replace Boolean algebra, stochastic-process theory, generalized probabilistic theories or density-matrix models; rather, it provides a reduced computational parametrization useful for analysing finite-resource trade-offs in stochastic and partially coherent two-level systems.
This paper presents a mathematical and computational method for approximating the solution of nonlinear gas transmission systems, with application to the Irish gas network. Starting from the governing partial differential equations, we derive a steady-state formulation under equilibrium assumptions, leading to a system of nonlinear algebraic equations. To approximate the solution efficiently, we propose a tailored method in which we use a physically informed estimate and construct a linear system whose solution closely approximates the nonlinear model. This approach enables fast computation of steady-state pressures and flows and provides a practical tool for gas network analysis. A numerical example illustrates the effectiveness of the method under realistic conditions.
We construct approximate solutions of the 2D Cauchy problem for domains bounded by nonsmooth curves. We reduce these solutions to the set of boundary value problems for analytic functions and integral equations. Our construction of an approximate solution is based on introduction of analytic functions and application of approximate conformal mappings. We reduce the initial problem to the set of boundary value problems and systems of linear algebraic equations. We also give estimates of the approximate solution error and some examples of the solution.
