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Volume 16
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Article

A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation

by
Huilin Xu
,
Baoxia Wang
and
Duanmei Zhou
*
College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China
*
Author to whom correspondence should be addressed.
Symmetry 2024, 16(11), 1549; https://doi.org/10.3390/sym16111549
Submission received: 29 October 2024 / Revised: 17 November 2024 / Accepted: 18 November 2024 / Published: 19 November 2024
(This article belongs to the Section Mathematics)
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Abstract

This paper considers a Cauchy problem for the multi-dimensional modified Helmholtz equation with inhomogeneous Dirichlet and Neumann data. The Cauchy problem is severely ill-posed, and a general mollification method is introduced to solve the problem. Both the a priori and a posteriori choice strategies of the regularization parameter are proposed, and error estimations of the corresponding regularization solutions are also presented. Finally, two numerical examples are introduced to show the effectiveness of the general mollification regularization method.
Keywords: modified Helmholtz equation; Cauchy problem; mollification method; regularization parameter; regularization solution; error estimation modified Helmholtz equation; Cauchy problem; mollification method; regularization parameter; regularization solution; error estimation

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MDPI and ACS Style

Xu, H.; Wang, B.; Zhou, D. A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation. Symmetry 2024, 16, 1549. https://doi.org/10.3390/sym16111549

AMA Style

Xu H, Wang B, Zhou D. A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation. Symmetry. 2024; 16(11):1549. https://doi.org/10.3390/sym16111549

Chicago/Turabian Style

Xu, Huilin, Baoxia Wang, and Duanmei Zhou. 2024. "A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation" Symmetry 16, no. 11: 1549. https://doi.org/10.3390/sym16111549

APA Style

Xu, H., Wang, B., & Zhou, D. (2024). A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation. Symmetry, 16(11), 1549. https://doi.org/10.3390/sym16111549

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