Laplace方程柯西问题的B样条方法

doi:10.16088/j.issn.1001-6600.2022041401

Abstract

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

CLC Number: O241.82

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

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