On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order
In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy prob...
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MDPI AG
2022-07-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/7/968 |
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| author | Davron Aslonqulovich Juraev Ali Shokri Daniela Marian |
| author_facet | Davron Aslonqulovich Juraev Ali Shokri Daniela Marian |
| author_sort | Davron Aslonqulovich Juraev |
| collection | DOAJ |
| description | In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal. |
| first_indexed | 2024-03-09T11:55:37Z |
| format | Article |
| id | doaj.art-72acd60b0c584cb0b1cd7bdc2fd9dd99 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T11:55:37Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-72acd60b0c584cb0b1cd7bdc2fd9dd992023-11-30T23:09:37ZengMDPI AGEntropy1099-43002022-07-0124796810.3390/e24070968On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First OrderDavron Aslonqulovich Juraev0Ali Shokri1Daniela Marian2Department of Natural Science Disciplines, Higher Military Aviation School of the Republic of Uzbekistan, Karshi 180117, UzbekistanDepartment of Mathematics, Faculty of Basic Sciences, University of Maragheh, Maragheh 83111-55181, IranDepartment of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, RomaniaIn this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.https://www.mdpi.com/1099-4300/24/7/968integral formularegularization of the Cauchy problemapproximate solutionCarleman matrixfamily of vector functionsBessel and Hankel functions |
| spellingShingle | Davron Aslonqulovich Juraev Ali Shokri Daniela Marian On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order Entropy integral formula regularization of the Cauchy problem approximate solution Carleman matrix family of vector functions Bessel and Hankel functions |
| title | On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order |
| title_full | On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order |
| title_fullStr | On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order |
| title_full_unstemmed | On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order |
| title_short | On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order |
| title_sort | on an approximate solution of the cauchy problem for systems of equations of elliptic type of the first order |
| topic | integral formula regularization of the Cauchy problem approximate solution Carleman matrix family of vector functions Bessel and Hankel functions |
| url | https://www.mdpi.com/1099-4300/24/7/968 |
| work_keys_str_mv | AT davronaslonqulovichjuraev onanapproximatesolutionofthecauchyproblemforsystemsofequationsofelliptictypeofthefirstorder AT alishokri onanapproximatesolutionofthecauchyproblemforsystemsofequationsofelliptictypeofthefirstorder AT danielamarian onanapproximatesolutionofthecauchyproblemforsystemsofequationsofelliptictypeofthefirstorder |