论 DDE 和 PDDE θ 方法的稳定性

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2024-06-25 DOI:10.1016/j.apnum.2024.06.018

Alejandro Rodríguez-Fernández , Jesús Martín-Vaquero

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引用次数: 0

摘要

本文基于检验方程 y′(t)=-Ay(t)+By(t-τ)（其中 τ 为常数延迟，A 为正定矩阵）研究了延迟微分方程的 θ 方法的稳定性。主要考虑矩阵 A 和 B 不能同时对角化的情况，并利用值域概念证明了这些方法无条件稳定性的充分条件和另一个也能保证其稳定性的条件，但取决于步长。对于矩阵 A 和 B 同时可对角化的情况，所获得的结果也进行了简化，并与其他针对一般情况的类似著作进行了比较。此外，还介绍了几个数值示例，这些示例将本文讨论的理论应用于带有扩散项和延迟项的偏延迟微分方程给出的抛物问题。

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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On the stability of θ-methods for DDEs and PDDEs

In this paper, the stability of θ-methods for delay differential equations is studied based on the test equation $y^{'} (t) = - A y (t) + B y (t - τ)$ , where τ is a constant delay and A is a positive definite matrix. It is mainly considered the case where the matrices A and B are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices A and B are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.