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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2021, Vol. 38 ›› Issue (6): 856-868.doi: 10.3969/j.issn.1005-3085.2021.06.008

• • 上一篇    下一篇

三维 Helmholtz 类方程柯西问题的一种基于修正核的数值解

何尚琴1,2,   冯秀芳2   

  1. 1. 北方民族大学数学与信息科学学院，银川 750021
    2. 宁夏大学数学统计学院，银川 750021
  • 出版日期:2021-12-15 发布日期:2022-02-15
  • 通讯作者: 冯秀芳 E-mail: xf_feng@nxu.edu.cn
  • 基金资助:
    国家自然科学基金 (11961054)；宁夏自然科学基金 (2020AAC03253)；北方民族大学中央高校基本科研业务费专项基金 (2020KYQD15).

Numerical Method for Modified Kernel to Solve the Cauchy Problem of Three-dimensional Helmholtz-type Equation

HE Shangqin1,2,   FENG Xiufang2   

  1. 1. School of Mathematics and Information Sciences, North Minzu University, Yinchuan 750021
    2. School of Mathematics and Statistics, Ningxia University, Yinchuan 750021
  • Online:2021-12-15 Published:2022-02-15
  • Contact: X. Feng. E-mail: xf_feng@nxu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11961054); the Natural Science Foundation of Ningxia (2020AAC03253); the Fundamental Research Funds for the Central Universities, North Minzu University (2020KYQD15).

PDF (PC)

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摘要：

针对 Helmholtz 类方程 Cauchy 问题的严重不适定性，提出了三维修正 Helmholtz 方程 Cauchy 问题基于精确解的修正核方法。通过构造软化算子，将不适定问题转化为适定问题，获得了稳定的数值逼近解。当波数 $k$ 和参数 $m$ 满足所需的条件时，分别给出了正则参数在先验选取规则之下的正则近似解与精确解之间的 $L^2$-误差估计和 Sobolev 型 $H^s$-误差估计，并通过数值算例对理论部分进行验证，结果表明所提出的正则化方法是稳定和有效性的。

关键词: Helmholtz 型方程, Cauchy 问题, 修正核, 逼近方法, 误差估计

Abstract:

In order to solve the severely ill-posed Cauchy problem of 3D Helmholtz-type equation, a modified kernel method based on exact solution is proposed. By constructing a mollification operator, the ill-posed problem is transformed into a well-posed problem, and stable numerical approximation solutions are obtained. When the wave-number $k$ and parameter $m$ meet the necessary conditions, the error estimates between the regularization approximation solution and exact solution are given in terms of $L^2$-estimate and $H^s$-estimate under the suitable choices of the regularization parameter. The theoretical part is verified by numerical examples. The results show that the proposed regularization method is stable and effective.

Key words: Helmholtz-type equation, Cauchy problem, modified kernel, approximation method, error estimate

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