On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane
In this paper, approximate solutions of the Cauchy problem for the Helmholtz equation on a two-dimensional bounded region are found. The problem under consideration belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. When solv...
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| Format: | Article |
| Language: | English |
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REA Press
2024-09-01
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| Series: | Computational Algorithms and Numerical Dimensions |
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| Online Access: | https://www.journal-cand.com/article_202803_0fa6b01739f8aa2d14bbc7bb78219618.pdf |
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| author | Davron Juraev Nazira Mammadzada Praveen Agarwal Shilpi Jain |
| author_facet | Davron Juraev Nazira Mammadzada Praveen Agarwal Shilpi Jain |
| author_sort | Davron Juraev |
| collection | DOAJ |
| description | In this paper, approximate solutions of the Cauchy problem for the Helmholtz equation on a two-dimensional bounded region are found. The problem under consideration belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. When solving applied problems, it is necessary to find not only an approximate solution but also a derivative of the approximate solution. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution and its derivative is established, as well as a regularization formula for the case when, under the specified conditions, instead of the initial Cauchy data, their continuous approximations with a given error in the uniform metric are given. Stability estimates for the solution of the Cauchy problem in the classical sense are obtained. |
| format | Article |
| id | doaj-art-e1a5b08148c947838d2d95038464cf49 |
| institution | Kabale University |
| issn | 2980-7646 2980-9320 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | REA Press |
| record_format | Article |
| series | Computational Algorithms and Numerical Dimensions |
| spelling | doaj-art-e1a5b08148c947838d2d95038464cf492025-01-30T11:23:23ZengREA PressComputational Algorithms and Numerical Dimensions2980-76462980-93202024-09-013318720010.22105/cand.2024.202803202803On the approximate solution of the Cauchy problem for the Helmholtz equation on the planeDavron Juraev0Nazira Mammadzada1Praveen Agarwal2Shilpi Jain3Department of Scientific Research, Innovation and Training of Scientific and Pedagogical Staff, University of Economy and Pedagogy, Karshi 180100, Uzbekistan.State Oil Company of the Azerbaijan Republic, Oil, and Gas Scientific Research ProjectInstitute, Baku, AZ1122, Azerbaijan.Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India.Poornima College of Engineering, University of Rajasthan, Jaipur 3020222, India.In this paper, approximate solutions of the Cauchy problem for the Helmholtz equation on a two-dimensional bounded region are found. The problem under consideration belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. When solving applied problems, it is necessary to find not only an approximate solution but also a derivative of the approximate solution. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution and its derivative is established, as well as a regularization formula for the case when, under the specified conditions, instead of the initial Cauchy data, their continuous approximations with a given error in the uniform metric are given. Stability estimates for the solution of the Cauchy problem in the classical sense are obtained.https://www.journal-cand.com/article_202803_0fa6b01739f8aa2d14bbc7bb78219618.pdfgreen’s integral formulacarleman functioncauchy problemapproximate solutionsregular solutions |
| spellingShingle | Davron Juraev Nazira Mammadzada Praveen Agarwal Shilpi Jain On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane Computational Algorithms and Numerical Dimensions green’s integral formula carleman function cauchy problem approximate solutions regular solutions |
| title | On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane |
| title_full | On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane |
| title_fullStr | On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane |
| title_full_unstemmed | On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane |
| title_short | On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane |
| title_sort | on the approximate solution of the cauchy problem for the helmholtz equation on the plane |
| topic | green’s integral formula carleman function cauchy problem approximate solutions regular solutions |
| url | https://www.journal-cand.com/article_202803_0fa6b01739f8aa2d14bbc7bb78219618.pdf |
| work_keys_str_mv | AT davronjuraev ontheapproximatesolutionofthecauchyproblemforthehelmholtzequationontheplane AT naziramammadzada ontheapproximatesolutionofthecauchyproblemforthehelmholtzequationontheplane AT praveenagarwal ontheapproximatesolutionofthecauchyproblemforthehelmholtzequationontheplane AT shilpijain ontheapproximatesolutionofthecauchyproblemforthehelmholtzequationontheplane |