Uniform polynomial decay of semi-discrete scheme for 1-D wave equation with nonlinear internal dissipation

IF 3.8 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2025-09-25 DOI:10.1016/j.cnsns.2025.109364

Bao-Zhu Guo , Ya-Ting Wang

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引用次数: 0

Abstract

This paper investigates the preservation of uniform stability for a spatial semi-discrete finite difference scheme applied to a one-dimensional wave equation with nonlinear internal damping. The discretization process employs the order reduction method. Unlike previous studies, this system focuses on a nonlinear system, with absolute stability being a special case. Due to the nonlinearity, the system may not always exhibit exponential stability, but it can be polynomially stable in some cases. After semi-discretization, the system transforms into an infinitely large number of lumped parameter nonlinear systems. Achieving uniform exponential stability, particularly polynomial stability, is a challenging endeavor. We demonstrate that when the nonlinear function satisfies certain conditions at the origin and infinity, the system possesses polynomial stability. These properties are preserved during the semi-discretization process. The mathematical proofs are similar to those for the continuous counterpart in many ways.

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非线性内耗散一维波动方程半离散格式的一致多项式衰减

研究了具有非线性内阻尼的一维波动方程的空间半离散有限差分格式保持一致稳定性的问题。离散化过程采用降阶法。与以往的研究不同的是，该系统关注的是一个非线性系统，绝对稳定是一个特例。由于非线性，系统可能并不总是表现出指数稳定性，但在某些情况下它可能是多项式稳定的。系统经过半离散化后，转化为无限大数量的集总参数非线性系统。实现一致的指数稳定性，特别是多项式稳定性，是一项具有挑战性的工作。证明了当非线性函数在原点和无穷远处满足一定条件时，系统具有多项式稳定性。这些性质在半离散化过程中保持不变。数学证明在许多方面与连续对应物的数学证明相似。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.