奇异摄动 Volterra 积分微分方程参数一致的数值方法

奇异摄动 Volterra 积分微分方程参数一致的数值方法

A Parameter-Uniform Numerical Method for a Singularly Perturbed Volterra Integro-Differential Equation

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数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 576-583.

刘利斌^*(),廖仪戈(),隆广庆()

南宁师范大学应用数学中心南宁 530100

收稿日期:2022-10-10 修回日期:2024-10-06 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 刘利斌 E-mail:liulibin969@163.com;lyg199600@163.com;longgq@amss.ac.cn
作者简介:廖仪戈，E-mail: lyg199600@163.com;|隆广庆，E-mail: longgq@amss.ac.cn
基金资助:
国家自然科学基金(12361087);国家自然科学基金(12261062)

Libin Liu^*(),Yige Liao(),Guangqing Long()

Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100

Received:2022-10-10 Revised:2024-10-06 Online:2025-04-26 Published:2025-04-09
Contact: Libin Liu E-mail:liulibin969@163.com;lyg199600@163.com;longgq@amss.ac.cn
Supported by:
NSFC(12361087);NSFC(12261062)

摘要：

针对一类奇异摄动 Volterra 积分微分方程, 在 Vulanović-Bakhvalov 网格上构造了一个一阶参数一致收敛的有限差分格式. 进一步, 基于 Richardson 外推技术, 将数值格式的收敛阶从 $O(N^{-1})$ 提高到 $O(N^{-2})$ , 其中 $N$ 是网格剖分数. 最后, 数值实验证明了数值方法的有效性.

关键词: 奇异摄动, Volterra 积分微分方程, Richardson 外推, Vulanovi?-Bakhvalov 网格

Abstract:

A singularly perturbed Volterra integro-differential equation is considered. The problem is discretized by using a simple first-order finite difference scheme on a Vulanović-Bakhvalov mesh, the accuracy of which is first-order uniformly convergent with respect to the perturbation parameter $\varepsilon$ . Furthermore, based on the Richardson extrapolation technique, the $\varepsilon$ -uniform accuracy of the presented approximation scheme can be improved from $O(N^{-1})$ to $O(N^{-2})$ , where $N$ is the number of mesh intervals. Finally, the theoretical finds are illustrated by two numerical experiments.

Key words: singularly perturbed, Volterra integro-differential equation, Richardson extrapolation, Vulanovi?-Bakhvalov mesh

中图分类号:

O241.8

刘利斌,廖仪戈,隆广庆. 奇异摄动 Volterra 积分微分方程参数一致的数值方法[J]. 数学物理学报, 2025, 45(2): 576-583.

Libin Liu,Yige Liao,Guangqing Long. A Parameter-Uniform Numerical Method for a Singularly Perturbed Volterra Integro-Differential Equation[J]. Acta mathematica scientia,Series A, 2025, 45(2): 576-583.

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