KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY
This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variab...
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| Format: | Article |
| Language: | English |
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Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin.
2023-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/579 |
| _version_ | 1797374210617638912 |
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| author | Durdimurod K. Durdiev Javlon Z. Nuriddinov |
| author_facet | Durdimurod K. Durdiev Javlon Z. Nuriddinov |
| author_sort | Durdimurod K. Durdiev |
| collection | DOAJ |
| description | This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane \(x_n = 0\) is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem. |
| first_indexed | 2024-03-08T19:02:36Z |
| format | Article |
| id | doaj.art-3d3f5a3f6d6b4693b35f4eb7c0007ccc |
| institution | Directory Open Access Journal |
| issn | 2414-3952 |
| language | English |
| last_indexed | 2024-03-08T19:02:36Z |
| publishDate | 2023-12-01 |
| publisher | Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin. |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj.art-3d3f5a3f6d6b4693b35f4eb7c0007ccc2023-12-28T04:27:28ZengKrasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin.Ural Mathematical Journal2414-39522023-12-019210.15826/umj.2023.2.007186KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORYDurdimurod K. Durdiev0Javlon Z. Nuriddinov1Bukhara Branch of V.I. Romanovskiy Institute of Mathematics, Academy of Sciences of Uzbekistan, 11 M. Ikbal St., Bukhara, 200100Bukhara State University, 11 M. Ikbal St., Bukhara, 200100This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane \(x_n = 0\) is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem.https://umjuran.ru/index.php/umj/article/view/579inverse problem, resolvent, integral equation, fixed point theorem, existence, uniqueness |
| spellingShingle | Durdimurod K. Durdiev Javlon Z. Nuriddinov KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY Ural Mathematical Journal inverse problem, resolvent, integral equation, fixed point theorem, existence, uniqueness |
| title | KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY |
| title_full | KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY |
| title_fullStr | KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY |
| title_full_unstemmed | KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY |
| title_short | KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY |
| title_sort | kernel determination problem for one parabolic equation with memory |
| topic | inverse problem, resolvent, integral equation, fixed point theorem, existence, uniqueness |
| url | https://umjuran.ru/index.php/umj/article/view/579 |
| work_keys_str_mv | AT durdimurodkdurdiev kerneldeterminationproblemforoneparabolicequationwithmemory AT javlonznuriddinov kerneldeterminationproblemforoneparabolicequationwithmemory |