Mathematical Methods in the Applied Sciences最新文献

This article evaluates a novel class of non-autonomous differential systems involving instantaneous and non-instantaneous impulses in Hilbert spaces. Initially, we investigate the presence of a mild solution for the aforementioned system under some simple conditions on the system parameters. Next, we constructed a new set of sufficient hypotheses for investigating the approximate controllability of the nonlinear system. The evolution operator and fixed point method are frequently employed to justify the existence of a solution. At the end, an example is given to illustrate and validate the proposed theory.

In this paper, we present an innovative particle system characterized by moderate interactions, designed to accurately approximate kinetic flocking models that incorporate singular interaction forces and local alignment mechanisms. We establish the existence of weak solutions to the corresponding flocking equations and provide an error estimate for the mean-field limit. This is achieved through the regularization of singular forces and a nonlocal approximation strategy for local alignments. We show that, by selecting the regularization and localization parameters logarithmically with respect to the number of particles, the particle system effectively approximates the mean-field equation.

In this paper, we introduce a new class of stochastic fractional integrals, namely, the stochastic mean-square $��k�$-Riemann–Liouville fractional integrals, by combining elements of fractional calculus and stochastic analysis. We first establish the necessary theoretical framework by recalling fundamental concepts from both domains. A novel integral identity involving these operators is then derived, which serves as a key tool for our main results. Using this identity, we prove several $��k�$-fractional Maclaurin-type inequalities for differentiable $��s�$-convex stochastic processes. These inequalities extend classical deterministic results to the stochastic setting and generalize them via $��k�$-fractional operators, offering enhanced flexibility in modeling uncertainty and memory effects. The obtained results contribute to the growing theory of stochastic fractional analysis and provide new tools for the study of probabilistic bounds and convexity in random environments.

In this paper, we introduce an estimate of the Peano remainder for the bivariate Durrmeyer-type exponential sampling series. Thereafter, we present some approximation results about the order of convergence for the series. In doing so, the rate of approximation is improved by constructing linear combinations of these operators. Furthermore, we present an example of kernel function with the graphical illustration to verify the findings.

A novel TENO-adaptive accuracy (TENO-A) scheme is proposed to perform complex flowfield simulations containing abundant vortex field details and shock wave discontinuities. This hybrid scheme not only enhances the low dissipation property of the TENO scheme in smooth regions but also improves the discontinuity-resolving capability for shock waves while retaining the robustness of the scheme. A unified and robust discontinuous detector using the information within the global stencil width is applied to separate discontinuous and smooth regions, which is not case and parameter sensitive. In discontinuous regions, the THINC scheme with jump-like distribution is adopted to well approximate a discontinuity within a grid. Besides, in the smoothest region, the global reconstruction value is adopted as the final reconstruction value, and the accuracy is improved from the original fifth-order to sixth-order; in the general smooth region, based on the Kriging method, the candidate stencil reconstruction is employed for final reconstruction with nonlinear weights, which improves the accuracy of the candidate stencil from the original third-order to fourth-order. As a result, the dissipation is significantly reduced using both types of reconstruction, which is beneficial to resolve small-scale flow structures. A set of numerical results demonstrates that the TENO-A scheme performs better in one-dimensional and two-dimensional cases than the standard TENO scheme and is able to predict complex flowfield without the necessity of parameter tuning case by case, and the hybrid scheme can restore high-order accuracy, maintain low dissipation property, and avoid spurious oscillations.

This study investigates the dynamics of glycolytic oscillations using a time-fractional reaction-diffusion Goldbeter-Lefever model with Caputo fractional derivatives. We study both the ordinary differential system (ODE) and the spatially extended system (PDE) to understand how the fractional order $��\alpha �$ affects stability and pattern formation. The model extends the classical Goldbeter-Lefever system by including memory effects through the fractional derivative. Our results show that lowering the fractional order enlarges the stability region of the steady state and changes the onset of Turing instabilities. We provide analytical conditions using linearization and spectral methods, and confirm the results with numerical simulations using a predictor-corrector scheme and finite difference method. These results show the important role of memory and anomalous diffusion in biochemical systems. This work helps better understand how fractional dynamics affect metabolic oscillations.

This study delves into the concept of hybrid chaos, which fuses elements of chaos and order to create dynamic, adaptable systems. Chaos, often marked by unpredictable behavior, fosters creativity and innovation. Hybrid chaos, however, seeks to integrate both chaotic and ordered components to form resilient systems. Across various fields—such as creativity, biology, and technology—hybrid chaos opens up new possibilities by merging structured systems with random, chaotic elements. Moreover, the incorporation of stochastic processes further enhances the system's adaptability and complexity. This paper aims to develop novel hybrid chaotic systems by combining different chaotic systems over varying intervals, advancing our understanding of hybrid chaos and its potential applications in real-world scenarios.

This paper focuses on a discrete self-diffusion predator–prey model with a prey refuge. Firstly, through stability analysis, the parametric conditions for spatially homogeneous steady-state stability at the equilibrium points of the model are obtained. Subsequently, bifurcation theory is employed to derive the conditions under which flip bifurcation and the Neimark–Sacker bifurcation occur in the model. Furthermore, we delve into chaos control theory and unveil the conditions for the Turing instability in the discrete diffusive model under the influence of overall self-diffusion. Finally, numerical simulations are conducted to verify the theoretical analysis, and complex dynamical behaviors such as period doubling, limit cycles, periodic windows, chaotic dynamics, and pattern formation are observed.

This paper addresses a time-optimal control problem for the heat equation involving a Caputo fractional derivative. The control function is applied on a specific portion of the domain boundary. The main objective is to determine the minimal time required to achieve a prescribed average temperature within the domain. By employing spectral methods and properties of the Mittag–Leffler function, we derive an explicit estimate for the minimal time under suitable conditions.