{"id":2243,"date":"2025-05-13T18:55:52","date_gmt":"2025-05-13T17:55:52","guid":{"rendered":"http:\/\/newskbncran.ru\/?page_id=2243"},"modified":"2026-03-27T10:31:56","modified_gmt":"2026-03-27T10:31:56","slug":"26-6-1-en","status":"publish","type":"page","link":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/26-6-1-en\/","title":{"rendered":"26.6.1 en"},"content":{"rendered":"\n

Problem with data on parallel characteristics for the loaded wave equation<\/strong><\/h1>\n\n\n\n

A.Kh. Attaev<\/strong><\/strong><\/p>\n\n\n\n

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Upload the full text<\/strong><\/p>\n\n\n\n

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Abstract<\/em><\/strong>.<\/strong> The theory of loaded equations is very relevant both in theoretical terms and in its numerous practical applications in various fields of modern natural science. This explains a huge number of works devoted to research and application of loaded equations in the last fifty years. The main objective of the study is to present the loaded equations as a method for setting new correct boundary value problems. The proof for the correctness of the problem is based on the d’Alembert’s formula obtained for the studied wave equation. The paper considers a loaded hyperbolic equation with two loaded terms. The load traces are related to different characteristic manifolds of a one-dimensional wave operator. Our goal is to study the problem with data on non-intersecting characteristics. We prove the existence and uniqueness of the solution, which is presented in an explicit form. The distinguishing feature of the problem is that it is ill-posed in the absence of loaded terms.<\/p>\n\n\n\n

Keywords:<\/em><\/strong> loaded equation, one-dimensional wave operator, characteristic, ill-posed problem, d’Alembert formula<\/p>\n\n\n\n

For citation<\/strong>. Attaev A.Kh. Problem with data on parallel characteristics for the loaded wave equation.  News of the Kabardino-Balkarian Scientific Center of RAS.<\/em> <\/strong>2024. Vol. 26. No. 6. Pp. 13\u201318 . DOI: 10.35330\/<\/strong>1991-6639-2024-26-6-13-18<\/p>\n\n\n\n

R<\/strong>eferences<\/summary>\n
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  1. Nakhushev A.M. The Darboux problem for a certain degenerate second order loaded integrodifferential equation. Differential Equations. 1976. Vol. 12. No. 1. Pp. 103\u2013108.
    EDN: PDBUJB. (In Russian)<\/li>\n\n\n\n
  2. Kaziev V.M. Gursa problem for one loaded integro-differential equation. Differential Equations. 1981. Vol. 17. No. 2. Pp. 313\u2013319. http:\/\/mi.mathnet.ru\/de4195. (In Russian)<\/li>\n\n\n\n
  3. Gogunokov Z.G. The Gursa problem for a loaded second-order hyperbolic equation. Reports of the Adyghe (Circassian) International Academy of Sciences. 2000. Vol. 5. No. 1. Pp. 20\u201323. (In Russian)<\/li>\n\n\n\n
  4. Ogorodnikov E.M. Some characteristic problems for loaded systems of differential equations and their relationship with non-local boundary value problems. J. Samara State Tech. Univ. Ser. Phys. math. Sci. 2003. Vol. 19. Pp. 22\u201328. EDN: EBRVHH. (In Russian)<\/li>\n\n\n\n
  5. Attaev A.Kh. Problem with data on parallel characteristics for a loaded wave equation. Reports of the Adyghe (Circassian) International Academy of Sciences. 2013. Vol. 15. No. 2. Pp. 25\u201327. EDN: RWCJIL. (In Russian)<\/li>\n\n\n\n
  6. Lomov I.S. Loaded differential operators: Convergence of spectral expansions. Differential Equations. 2014. Vol. 50. No. 8. Pp. 1070\u20131079. DOI: 10.1134\/S0374064114080068. (In Russian)<\/li>\n\n\n\n
  7. Attaev A.K. Boundary control problem for a loaded string vibration equation. Differential Equations. 2020. Vol. 56. No. 5. Pp. 635\u2013640. DOI: 10.1134\/S0374064120050088. (In Russian)<\/li>\n\n\n\n
  8. Attaev A.Kh. Characteristic Cauchy problem for a linear loaded hyperbolic equation. Reports of the Adyghe (Circassian) International Academy of Sciences. 2010. Vol. 12. No. 1. Pp. 9\u201310. EDN: OGYECJ. (In Russian)<\/li>\n\n\n\n
  9. Attaev A.Kh. Boundary value problems with inner shift for the string equation. Reports of the Adyghe (Circassian) International Academy of Sciences. 2014. Vol. 16. No. 2. Pp. 17\u201319. EDN: SJTMNX. (In Russian)<\/li>\n\n\n\n
  10. Nakhushev A.M. Nagruzhennyye uravneniya i ikh primeneniye [Loaded equations and their applications]. Moscow: Nauka, 2012. 232 p. ISBN: 978-5-02-037977-0. (In Russian)<\/li>\n<\/ol>\n<\/details>\n\n\n\n
    Information about the author<\/strong><\/summary>\n
    \n

    Anatoly Kh. Attaev<\/strong>, Candidate of Physical and Mathematical Sciences, Associate Professor,
    Head of Department, Leading Researcher of the Department of Mixed-Type Equations, the Institute
    of Applied Mathematics and Automation of the Kabardino-Balkarian Scientific Center of the Russian
    Academy of Sciences;
    360000, Russia, Nalchik, 89 A Shortanov street;
    attaev.anatoly@yandex.ru, ORCID: https:\/\/orcid.org\/0000-0001-5864-6283, SPIN-code: 6389-3114<\/p>\n\n\n\n

    <\/p>\n\n\n\n

    <\/p>\n<\/div>\n<\/details>\n","protected":false},"excerpt":{"rendered":"

    Problem with data on parallel characteristics for the loaded wave equation A.Kh. Attaev Upload the full text Abstract. The theory of loaded equations is very relevant both in theoretical terms and in its numerous practical applications in various fields of modern natural science. 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