A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain
Abstract The Cauchy problem of the Laplace equation is investigated for both exact and perturbed data on a doubly connected domain, i.e., the numerical reconstruction of the function value and the normal derivative value on a part of the boundary from the knowledge of exact or noisy Cauchy data on t...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-04-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-023-01717-2 |
| _version_ | 1797849939055738880 |
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| author | Xingtian Gong Shuwei Yang |
| author_facet | Xingtian Gong Shuwei Yang |
| author_sort | Xingtian Gong |
| collection | DOAJ |
| description | Abstract The Cauchy problem of the Laplace equation is investigated for both exact and perturbed data on a doubly connected domain, i.e., the numerical reconstruction of the function value and the normal derivative value on a part of the boundary from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary, which is completely different from the Cauchy problem on a simply connected bounded region. We first establish the existence of a solution through the potential theory. By expressing the solution as a sum of single-layer potentials using boundary value condition, we get the integral equation systems about the density function on the boundary, and by applying local regularization scheme to the obtained integral equation systems, we get the regularization solution of the original problem. Some numerical results are presented to validate the applicability and effectiveness of the proposed method. |
| first_indexed | 2024-04-09T18:53:23Z |
| format | Article |
| id | doaj.art-fb6888dce22c4939ad68b753700fed61 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-04-09T18:53:23Z |
| publishDate | 2023-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-fb6888dce22c4939ad68b753700fed612023-04-09T11:22:50ZengSpringerOpenBoundary Value Problems1687-27702023-04-012023111210.1186/s13661-023-01717-2A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domainXingtian Gong0Shuwei Yang1Lanzhou Technology and Business CollegeSchool of Science, Lanzhou University of TechnologyAbstract The Cauchy problem of the Laplace equation is investigated for both exact and perturbed data on a doubly connected domain, i.e., the numerical reconstruction of the function value and the normal derivative value on a part of the boundary from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary, which is completely different from the Cauchy problem on a simply connected bounded region. We first establish the existence of a solution through the potential theory. By expressing the solution as a sum of single-layer potentials using boundary value condition, we get the integral equation systems about the density function on the boundary, and by applying local regularization scheme to the obtained integral equation systems, we get the regularization solution of the original problem. Some numerical results are presented to validate the applicability and effectiveness of the proposed method.https://doi.org/10.1186/s13661-023-01717-2Laplace equationCauchy problemIntegral equationsNumericsLocal regularizationDoubly connected domain |
| spellingShingle | Xingtian Gong Shuwei Yang A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain Boundary Value Problems Laplace equation Cauchy problem Integral equations Numerics Local regularization Doubly connected domain |
| title | A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain |
| title_full | A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain |
| title_fullStr | A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain |
| title_full_unstemmed | A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain |
| title_short | A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain |
| title_sort | local regularization scheme of cauchy problem for the laplace equation on a doubly connected domain |
| topic | Laplace equation Cauchy problem Integral equations Numerics Local regularization Doubly connected domain |
| url | https://doi.org/10.1186/s13661-023-01717-2 |
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