Mathematical Methods in the Applied Sciences最新文献

In this paper, an efficient conformal multi-symplectic structure-preserving method is proposed for two-dimensional damped nonlinear Schrödinger equation. The method decomposes the original equation into several tractable one-dimensional subproblems by the splitting method, and then solves them separately and alternatively, and combines the resulting schemes by an appropriate method lastly. Moreover, the spatial derivatives are discretized with a high-order compact scheme which uses fewer grid nodes to achieve higher accuracy and improve the computational efficiency. Based on this framework, an efficient conformal multi-symplectic algorithm is developed for the equation. Some numerical experiments are reported to verify the effectiveness of the new scheme.

In this article, under partial information, we establish a second-order stochastic maximum principle for optimal stochastic control. The system is governed by nonlinear controlled Itô-type stochastic differential equations driven by orthogonal Teugels martingales associated with a Lévy process, and an independent Brownian motion. We establish pointwise second-order necessary conditions for the optimal control under incomplete information. The control domain is assumed to be bounded and convex. It is required that the control process is adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. The proof of our main result is based on a variational approach using the stochastic calculus of orthogonal Teugels martingales. Finally, an example is studied to illustrate our theoretical results.

This article investigates the dynamics of an SIS epidemic reaction-diffusion model with logistic source and mass action infection mechanism. The uniform boundedness of the solution is obtained. The stability of the trivial and disease-free equilibria is discussed. For homogeneous environments, global attractivity of equilibria is rigorously demonstrated through Lyapunov function methods. In heterogeneous environments, the existence of endemic equilibrium is established by the uniform persistence theory. Moreover, the asymptotic behaviors of the endemic equilibrium for large or small diffusion rate are characterized. Our results reveal that restricting the movement of infected individuals while reducing the growth rate of susceptible populations can effectively suppress disease prevalence. These findings provide some theoretical foundations for formulating effective disease control strategies.

Extended Chebyshev spaces (ECT-spaces) are of significant importance and interest from a design perspective, as they provide valuable shape parameters that enhance control. In this investigation, we introduce a recursively defined generalized Bernstein basis (GB-basis) based on the blossom approach, together with a closed-form expression. This basis serves as a natural extension of the classical Bernstein basis functions of degree $�\; ��\; �n��$. The new GB-basis extends the Bernstein basis to a broader context involving non-polynomial bases, including fractional, exponential, and trigonometric functions. We prove that the new GB-basis possesses shape-preserving properties, which play a fundamental role in computer-aided geometric design (CAGD) and computer graphics. A new corner-cutting algorithm is developed to efficiently compute the corresponding generalized Bézier curves (GB-curves), rational curves (RGB-curves), and surfaces (GB-surfaces). Moreover, the shape parameter can predictably adjust the shape of the new curves and surfaces without altering the control points. As the shape parameter increases, the resulting curves and surfaces approach the control polygon or control net in a predictable manner. To address challenges in creating complex curves, we derive necessary and sufficient conditions for $�\; ��\; �{�\; �C�}^{�\; �1�}��$ continuity. Additionally, the GB-basis enables the construction of sharp-pointed curves. Experimental results demonstrate the superiority and efficiency of the new GB-basis over the classical Bernstein basis.

This article investigates the existence and uniqueness of solutions to mixed variational-hemivariational inequalities. Unlike existing approaches in published literature on mixed variational-hemivariational inequalities, our method for analyzing existence and uniqueness is easy to understand and apply. Moreover, a projection iterative algorithm is provided for solving such inequalities. First, the existence of a solution for an auxiliary hemivariational problem is established. Subsequently, two sequences corresponding to the problem variables are constructed via projection operators. By leveraging the projection theorem and operator properties, the convergence of both sequences is rigorously demonstrated, thereby proving that the limit of the sequences constitutes the unique solution to the mixed variational-hemivariational inequality. Finally, we provide examples such that $�\; ��\; �X�\; �,�\; �Y�\; �,�\; �𝒳�\; �,�\; �\Lambda �\; �,�\; �\gamma �\; �,�\; �T�\; �,�\; �A�\; �,�\; �b�\; �,�\; �\phi �\; �,�\; �\psi �\; �,�\; �J��$ satisfy the working hypotheses to illustrate the theory.

This paper investigates the uniform stabilization of semilinear acoustic systems with memory damping, where the kernel function is derived from a generalized positive definite kernel (GPDK). By exploiting the properties of the GPDK and the structure of the acoustic model, we relax the classical requirements of kernel nonnegativity and monotonicity, and establish sufficient conditions for asymptotic, exponential, and polynomial stability.

This paper establishes the well-posedness and stability properties of a coupled thermo-viscoelastic Timoshenko–Boltzmann system featuring two nonlocal mechanisms: (i) shear deformation with Boltzmann-type viscoelastic memory, and (ii) heat conduction governed by the Gurtin–Pipkin nonlocal thermal law. Through semigroup operator theory, we first prove the existence of a unique solution flow in an appropriate history-weighted energy space. Subsequently, using energy methods with memory-dependent functional constructions, we characterize the energy decay rates in terms of the relaxation kernel's dissipation strength. When the viscoelastic kernel decays exponentially, the system exhibits exponential stabilization. For weakly decaying kernels, we establish the polynomial decay rate. These results quantify how the interplay between viscoelastic memory dissipation and nonlocal thermal damping governs the system's stabilization dynamics.