In this paper, we mathematically analyzed a chemical reaction-based dynamical model to investigate codimension-two and codimension-three Bogdanov–Takens (BT) bifurcations and their impact on system dynamics. Generalized eigenvectors were first computed using normalization techniques to derive unfolding parameters for codim-two BT bifurcation. A codim-three BT bifurcation case was then obtained, with its unfolding parameters determined with the aid of the seven-step method. A control technique based on anonymous parameters and updated law was used to control oscillations in our considered dynamical system. Analytical results were validated through numerical simulations, highlighting the role of autocatalytic reactions $2 D_{1} \xrightarrow{\alpha_{4}} D_{2}$ and $2 D_{1}+D_{2} \xrightarrow{\alpha_{5}} 3 D_{1}$ in system behavior. These bifurcations generated oscillations, bi-stability, and excitability, behaviors essential for understanding and controlling complex chemical and biological systems.

In this paper, we investigate the global asymptotic stability of complex-valued neural networks (CVNNs) subject to time-varying delays and parameter uncertainties. We establish novel stability conditions that guarantee both the existence and uniqueness of equilibrium states, as well as the global convergence of the network trajectories. By constructing a suitable Lyapunov-Krasovskii functional, the approach inherently accounts for the stability of CVNNs subject to time-varying delays. Finally, numerical examples are presented to verify the theoretical findings, illustrating both the effectiveness and the practical applicability of the proposed approach.

This article introduces a novel numerical method for solving a singularly-perturbed convection-diffusion problem with mixed type boundary conditions. The approach begins by transforming the second-order boundary value problem into a system of first-order differential equations. The stability result of its corresponding differential operator vector is obtained although the system of equations is in a special form. The transformed system is approximated through a hybrid difference method, followed by a posteriori error analysis that relies on the stability result of the differential operator vector. A posteriori mesh and approximation solution are derived by designing a solution-adaptive algorithm derived from the a posteriori error bound. Finally, the second-order uniform convergence of the novel numerical method is corroborated by numerical experiments.

An adaptive moving grid method is developed to solve the time-fractional integrodifferential governing equations of viscoelastic nanofluid. The momentum equation is derived based on a dual-parameter fractional Maxwell constitutive relation, and the energy equation employs a generalized Cattaneo heat conduction relation. To improve solution accuracy, a monitor function based on the equidistribution principle is constructed, and an adaptive mesh redistribution strategy is developed in the spatial domain. The temporal fractional-order operators are approximated by the L1 algorithm and the weighted-shifted Grünwald difference method. Numerical experiments demonstrate that the adaptive grid achieves 77.6–88.4% higher accuracy compared to uniform grids at the same grid scale, along with enhanced stability in convergence. Parametric analysis indicates that increasing the fractional-order derivative in the energy equation results in a thickening of both the velocity and thermal boundary layers. Furthermore, the dual-fractional Maxwell model exhibits a thicker velocity boundary layer than its classical single-parameter counterpart. The proposed method offers an efficient and robust approach for simulating complex viscoelastic nanofluid systems with memory effects and multi-field coupling.

This paper investigates a three-species reaction-diffusion model with fractional-order derivatives and proposes an innovative numerical method for its simulation. The method integrates an optimized Gr"unwald-Letnikov discretization scheme, enhanced by a short-memory principle, with a high-order accurate nine-point compact difference scheme, enabling an efficient and stable solution of fractional operators. Rigorous convergence and stability analyses confirm the theoretical reliability of the algorithm. Through stability and Turing bifurcation analyses, the study systematically reveals the regulatory mechanism of the fractional-order exponent on the dynamic behavior of the system. The numerical results demonstrate that the present method accurately captures the effect of fractional derivatives on the formation process of spatial patterns.

In this paper, we develop some energy-preserving relaxation-type schemes to solve two-dimensional space fractional nonlinear Schrödinger equations with periodic boundary conditions. First, we change the original system into an equivalent relaxation form by introducing some new variables, which transforms the energy conservation law of the original system into quadratic invariants, and satisfies the mass conservation law of the original system. Then an implicit relaxation scheme is applied to deal with the time derivative, and the resulting semi discrete system can exactly preserve the mass and energy conservation laws. However, the obtained semi discrete system is nonlinear. Next a linear implicit relaxation scheme is directly used for the modified system to arrive at a semi discrete scheme, and the conservation of the semi discrete system is analyzed. Second, the resulting semi discrete systems are discretized by the Fourier spectral method with periodic boundary conditions, and the efficient iterative algorithms of the fully-discrete systems are given. Finally, numerical experiments of some space fractional nonlinear Schrödinger equations are given to verify the correctness of the theoretical results.

Let Ai and Bi denote the Airy functions. For a fixed k > 0, we introduce the class Airy_k($\mathcal{I}$) of k–Airy convex functions on an interval $\mathcal{I} \subset[0, \infty)$. A function f : $\mathcal{I} \rightarrow \mathbb{R}$ is called k–Airy convex on $\mathcal{I}$ if, for every subinterval $[a, b] \subset \mathcal{I}$ and all a < t < b,

We establish fundamental properties of Airy_k($\mathcal{I}$). As an application, we derive lower bounds for eigenvalues in Airy-type boundary value problems.

This study, we present a fractional-order compartmental model to describe the transmission dynamics of tick-borne encephalitis between human and vector (tick) populations. The model is first formulated in the classical integer-order sense and then extended to a fractional-order framework using the Caputo fractional derivative to incorporate memory effects. We analyzed the model with respect to positivity, boundedness, and stability, and identified the invariant region under the Caputo fractional operator. Furthermore, the model was fitted to the reported case data, and R₀ was computed using the next-generation matrix method, resulting in R₀ = 0.23. PRCC sensitivity analysis confirmed the effectiveness of vaccination in reducing disease transmission, with local and global stability established through Jacobian and Lyapunov analyses, respectively. We examined the effect of the fractional-order parameter ζ using an efficient Adams-Moulton numerical approach, and contour and 3D surface plots are used to visualize the outcomes. The simulations showed that variations in key parameters reduced disease transmission and infection levels in both populations.

We present a superconvergent and highly efficient multiscale finite element scheme for solving the Helmholtz equation in heterogeneous media. The proposed scheme is specifically designed to address the challenges posed by the multiscale nature of the Helmholtz equation, which often leads to high computational costs and numerical instabilities in traditional methods. By constructing tailored multiscale basis functions, the proposed approach effectively captures both the local fine-scale features and global macroscale behaviors of the multiscale solution. The fundamental mechanism underlying the multiscale scheme is systematically expounded. Through rigorous error analyses, we derive a second-order superconvergence result in the H¹ norm. Additionally, numerical experiments are conducted to validate the performance of the scheme. Both theoretical findings and numerical results confirm that the proposed multiscale scheme not only achieves higher solution accuracy but also exhibits superconvergence rates and significantly reduces computational costs.

This paper introduces a degenerate version of the Euler–Seidel method by incorporating a parameter λ into the classical recurrence relation. We define a degenerate Euler–Seidel matrix associated with an initial sequence and establish corresponding λ-generalized binomial identities and generating function relations. By applying this method to the degenerate Bernoulli, Euler, and Genocchi polynomials, we derive several new combinatorial identities. This work extends the classical Euler– Seidel method to the domain of degenerate special polynomials and numbers, thus providing a new framework to study their properties.

This study focused on the unsteady magnetohydrodynamic (MHD) flow and heat transfer of fractional viscoelastic nanofluids over an infinite vertical plate within a porous medium. Both ramped and isothermal wall temperature conditions were considered, along with the effects of heat injection and consumption. The momentum equation was formulated based on a dual-parameter fractional Maxwell constitutive relation, while the energy equation incorporated a fractional dual-phase-lag (DPL) model. The resulting fractional integrodifferential governing equations were solved numerically using a finite difference method combined with the L1 algorithm and the weighted-shifted Grünwald difference scheme. The accuracy of the proposed numerical scheme was verified through manufactured solutions. Numerical results show that increasing porous medium permeability enhances fluid flow, whereas a stronger magnetic field suppresses it. The effects of the phase-lag parameters on the thermal boundary layer differ under ramped and isothermal wall temperature conditions: the phase lag of the temperature gradient leads to a monotonic thickening in the former but a nonmonotonic variation in the latter, whereas the phase lag of the heat flux exhibits an opposite trend. This study provides valuable insights into the application of fractional integrodifferential models for the design and optimization of thermal systems involving nanofluids.

We propose a modeling framework to develop a continuum description of opinion dynamics on networks as an alternative to other models, like the ones based on graphons. In a nutshell, the continuum model that we propose aimed to approximate the distribution of opinions and the probability that two given opinions are connected. To illustrate our framework, we focused on a simple model of consensus dynamics on a network and derived a continuum description using techniques inspired by mean-field limits. We also discussed the limitations of this approach and suggested extensions to account for dynamic networks with evolving connections, stochastic effects, and directional interactions.

Green cryptocurrencies and artificial intelligence (AI)-coins have emerged as prominent digital asset classes amid growing interest in sustainable finance and technological innovation. Despite their rapid development, the empirical relationship between these two segments remains largely unexplored. To address this gap, assess dynamic risk connectedness, and identify net transmitter and net receiver spillovers between green and AI cryptocurrencies, we first use a time-varying parameter vector autoregressive framework to characterize the dynamic interdependencies across time and frequency. To further account for nonlinear dependence and improve interpretability for portfolio construction, we integrate eXtreme Gradient Boosting (XGBoost) with SHapley Additive exPlanations (SHAP) and implement an AI-driven portfolio-optimization strategy spanning green cryptocurrencies and AI tokens. Empirically, the central findings indicate that Ethereum (ETH), the dominant green cryptocurrency, persistently serves as a net transmitter of volatility to other green assets and to the five major AI-crypto markets. SHAP-based attribution further shows that the portfolio-contribution metric (Z_p), together with Ripple (XRP) and ETH, ranks among the most influential features. Overall, the proposed XGBoost-based approach captures nonlinear interactions among momentum, liquidity, and volatility, enhancing forecasting and portfolio allocation. When evaluated out-of-sample over 438 daily-rebalanced trading days (August 13, 2024–October 24, 2025), the machine learning methodology attains an annualized Sharpe ratio of approximately 2.08 at mean daily volatility near 1.04%. It also achieves the highest terminal wealth among the benchmark strategies (USD 17,812 from a USD 10,000 baseline), yielding a 78.1% cumulative return and superior risk-adjusted performance relative to competing strategies. These findings provide economically intuitive evidence of cross‑segment spillovers and yield policy-relevant implications for machine-learning-based portfolio allocation across sustainable and AI cryptocurrency markets. They also offer useful insights for regulators with respect to systemic risk surveillance, contagion monitoring, and the preservation of market integrity.

In this article, we focused on the bilinear neural network method to obtain the exact solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation. This method can also be applied to solve other nonlinear partial differential equations (NPDEs). Relying on Hirota bilinear transformation and Bell polynomial theories, we successfully constructed one-hidden- layer and multiple-hidden-layers models with assigned activation functions. Thereby, we derived many exact solutions for this equation including three-wave, rogue wave, and interaction solutions. To intuitively reflect features of these solutions, we depicted their three-dimensional, density, and curve graphs.

Recently, by combining type 2 Bernoulli and Euler polynomials with the central Bell polynomials, the central Bell-based type 2 Bernoulli and Euler polynomials of order α were considered, and many of their properties, formulas, and applications were investigated. The main aim of this work is to consider higher-order central Bell-based type 2 Bernoulli and Euler polynomials of complex variable, by which, both sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order λ are introduced by treating the imaginary and real components separately. Then, diverse summation formulas, differential formulas, addition formulas, and correlation formulas with new and existing old polynomials and numbers are derived in a systematic way. Also, several intriguing connections of sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order λ with the bivariate and one-variable central Bell polynomials, and the classical Stirling and central factorial numbers of the second kinds are investigated in detail. Moreover, the first few members of the new polynomials are provided by the lists, and the distributions of zeros of the new polynomials are illustrated by graphical representations, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.