基于再生核方法求解奇异摄动延迟微分方程

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1232-1245.

基于再生核方法求解奇异摄动延迟微分方程

单雨晴¹, 于文欣¹, 冉翠平², 牛晶¹^,^*()

¹ 哈尔滨师范大学数学科学学院哈尔滨 150000
² 东营市胜利第一中学物理组山东东营 257100

收稿日期:2025-02-20 修回日期:2025-06-27 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 牛晶 E-mail:njirwin@163.com
基金资助:
黑龙江省自然科学基金(LH2024A018);哈尔滨师范大学研究生创新科研(HSDSSCX2024-25)

Solving Singularly Perturbed Delay Differential Equations Based on the Reproducing Kernel Method

Yuqing Shan¹, Wenxin Yu¹, Cuiping Ran², Jing Niu¹^,^*()

¹ School of Mathematical Sciences, Harbin Normal University, Harbin 150000
² Physics Group, Shengli No.1 Middle School, Shandong Dongying 257100

Received:2025-02-20 Revised:2025-06-27 Online:2026-06-26 Published:2026-06-16
Contact: Jing Niu E-mail:njirwin@163.com
Supported by:
Natural Science Funds of Heilongjiang Province of China(LH2024A018);Graduate Student Innovation Project of Harbin Normal University(HSDSSCX2024-25)

摘要/Abstract

摘要：

该文提出了一种求解存在边界层的奇异摄动转折点问题的数值方法. 该方法基于渐近展开技术和再生核方法, 将原问题分解为边界层问题和规则区域问题. 规则区域问题通过渐近展开方法求解; 边界层问题通过变量拉伸法和基于配置法的再生核方法求解. 值得一提的是, 对于具有单边界层的奇异摄动延迟微分方程, 作者在 $W_2^4$ 空间中基于再生核函数构造基函数, 并在每个细分单元内, 构造 4 点的勒让德高斯节点作为配置点. 与原有的拟合网格 B 样条配点法相比, 作者的方法得到的精度和收敛阶更高. 该文提供了四个数值例子来说明该方法的有效性. 数值例子的结果表明, 该方法能够提供非常准确的近似解, 且能达到最优收敛阶.

关键词: 奇异摄动方程, 延迟问题, 最优收敛阶, 再生核方法.

Abstract:

In this paper, a numerical method is proposed for solving singularly perturbed turning point problems with boundary layers. This method is based on the asymptotic expansion technique and the reproducing kernel method, and it decomposes the original problem into a boundary layer problem and a regular region problem. The regular region problem is solved by the asymptotic expansion method, while the boundary layer problem is solved by the variable stretching method and the reproducing kernel method based on the collocation method. For singularly perturbed delay differential-difference equations with a single boundary layer, we construct basis functions based on the reproducing kernel function in the $W_2^4$ space. Inside each sub - division cell, the Gaussian-Legendre nodes with 4 points are selected as collocation points. Compared with the original fitted mesh B-spline collocation method, the accuracy and convergence order obtained by our method are higher. Four numerical examples are provided to illustrate the effectiveness of this method. The results of the numerical examples show that this method can provide very accurate approximate solutions and achieve the optimal convergence order.

Key words: singularly perturbed equations, delay problems, optimal convergence rate, reproducing kernel method.

中图分类号:

O175.8

单雨晴, 于文欣, 冉翠平, 牛晶. 基于再生核方法求解奇异摄动延迟微分方程[J]. 数学物理学报, 2026, 46(3): 1232-1245.

Yuqing Shan, Wenxin Yu, Cuiping Ran, Jing Niu. Solving Singularly Perturbed Delay Differential Equations Based on the Reproducing Kernel Method[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1232-1245.

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