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A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation

1
School of Mathematics and Statistics, NingXia University, Yinchuan 750021, China
2
College of Mathematics and Information Science and Technology, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(4), 360; https://doi.org/10.3390/math7040360
Received: 31 January 2019 / Revised: 6 April 2019 / Accepted: 16 April 2019 / Published: 20 April 2019
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems - Theory and Applications)
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Abstract

In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data. View Full-Text
Keywords: modified Helmholtz equation; ill-posed; de la Vallée Poussin kernel; mollification method; regularization solution; error estimate modified Helmholtz equation; ill-posed; de la Vallée Poussin kernel; mollification method; regularization solution; error estimate
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He, S.; Feng, X. A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation. Mathematics 2019, 7, 360.

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