Laplace方程柯西问题的B样条方法

doi:10.16088/j.issn.1001-6600.2022041401

广西师范大学学报（自然科学版） ›› 2023, Vol. 41 ›› Issue (3): 118-129.doi: 10.16088/j.issn.1001-6600.2022041401

Laplace方程柯西问题的B样条方法

赵婷婷, 杨凤莲^*

河海大学理学院, 江苏南京 210000

收稿日期:2022-04-14 修回日期:2022-05-25 出版日期:2023-05-25 发布日期:2023-06-01
通讯作者: 杨凤莲(1982—), 女, 福建三明人, 河海大学副教授, 博士。E-mail: yangfenglian@hhu.edu.cn
基金资助:
国家自然科学基金(11771120, 12271140); 河海大学中央高校基本科研业务费(B220202081)

B-Spline Method for the Cauchy Problem of the Laplace Equation

ZHAO Tingting, YANG Fenglian^*

College of Science, Hohai University, Nanjing Jiangsu 210000, China

Received:2022-04-14 Revised:2022-05-25 Online:2023-05-25 Published:2023-06-01

摘要/Abstract

摘要： Laplace方程柯西问题极其不适定,需要有效的数值算法进行求解,本文提出一种B样条方法求解此问题。首先在三次B样条函数生成的平移不变空间中给出柯西问题逼近解的表达形式;然后借助B样条基函数导数可用低阶样条基函数表示及方程的性质,写出问题的变分形式;接着,为了降低噪音的影响,提出Tikhonov正则化方法,以获得稳定的数值解;最后分别对矩形区域和含非光滑边界的区域进行数值实验,证明此方法的有效性。

关键词: 柯西问题, Laplace方程, 平移不变空间, 三次B样条函数, 正则化

Abstract: The Cauchy problem is extremely ill-posed and requires effective numerical algorithms to solve it. A B-spline method is proposed to solve the problem in this paper. Firstly, the approximate solution of the problem is given in the shift invariant space generated by cubic B-spline functions. Then, based on the properties of equation and B-spline basis function whose derivative can be expressed by low-order spline basis function, the variational form of the problem is obtained. Furthermore, Tikhonov regularization method is proposed to obtain stable numerical solutions in order to reduce the influence of noise. Finally, numerical experiments on rectangular region and the region with non-smooth boundary show that the proposed method is effective.

Key words: Cauchy problem, Laplace equation, shift invariant spaces, cubic B-spline function, regularization

中图分类号: O241.82

赵婷婷, 杨凤莲. Laplace方程柯西问题的B样条方法[J]. 广西师范大学学报（自然科学版）, 2023, 41(3): 118-129.

ZHAO Tingting, YANG Fenglian. B-Spline Method for the Cauchy Problem of the Laplace Equation[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(3): 118-129.

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Laplace方程柯西问题的B样条方法

B-Spline Method for the Cauchy Problem of the Laplace Equation

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