{"id":6006,"date":"2026-01-21T11:27:44","date_gmt":"2026-01-21T11:27:44","guid":{"rendered":"https:\/\/izvestiyakbncran.ru\/?page_id=6006"},"modified":"2026-06-02T10:19:45","modified_gmt":"2026-06-02T09:19:45","slug":"27-6-1-en","status":"publish","type":"page","link":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-1-en\/","title":{"rendered":"27.6.1 En"},"content":{"rendered":"\n

Mixed boundary value problem for one discontinuously loaded parabolic equation<\/strong><\/h1>\n\n\n\n

M.M. Karmokov, M.A. Kerefov, S.Kh. Gekkieva<\/strong><\/p>\n\n\n\n

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Abstract<\/strong>.<\/strong> <\/em>This article is devoted to current issues in the theory of partial differential equations related to the study of boundary value problems for loaded parabolic equations with a fractional integro-differentiation operator, which are of interest not only for the advancement of this specific theory, but also for their numerous applications.
Aim.<\/strong> The study is to prove the unique solvability of a mixed boundary value problem for a discontinuously loaded parabolic equation with the Riemann\u2013Liouville fractional derivative.
Research methods<\/strong>. The study employs the Green’s function method, simple layer potential theory, and fractional calculus theory.
Results<\/strong>. This paper demonstrates the unique solvability of a mixed boundary value problem for a loaded fractional-order parabolic equation.
Conclusion<\/strong>. The results obtained are significant for the development of the theory of boundary value problems for partial differential equations of fractional order, including loaded parabolic equations; they are also relevant for mathematical modeling of various processes and systems with distributed parameters and fractal structures.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/em><\/strong> boundary value problems, parabolic equations, fractional integro-differentiation operator, loaded equation, regular solution<\/p>\n\n\n\n

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For citation<\/strong>.<\/strong> Karmokov M.M., Kerefov M.A., Gekkieva S.Kh. Mixed boundary value problem for one discontinuously loaded parabolic equation. News of the Kabardino-Balkarian Scientific Center of RAS<\/em>. 2025. Vol. 27. No. 6. Pp. 13\u201323. DOI: 10.35330\/1991-6639-2025-27-6-13-23<\/p>\n\n\n\n

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R<\/strong>eferences<\/summary>\n
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  1. Nakhushev A.M. Nagruzhennye uravneniya i ih primenenie<\/em> [Loaded equations and their application]. Moscow: Nauka, 2012. 232 p. EDN: RPBPQZ. (In Russian)<\/li>\n\n\n\n
  2. Nakhushev A.M. Drobnoe ischislenie i ego primenenie<\/em> [Fractional calculus and its applications]. Moscow: Fizmatlit, 2003. 272 p. EDN: UGLEPD. (In Russian)<\/li>\n\n\n\n
  3. Nakhushev A.M. On the Darboux problem for a degenerate loaded second-order integrodifferential equation. Differential Equations. <\/em>1976. Vol. 12. No. 1. Pp. 103\u2013108. EDN: PDBUJB. (In Russian)<\/li>\n\n\n\n
  4. Dikinov Kh.B., Kerefov A.A., Nakhushev A.M. On a boundary value problem for a loaded heat conduction equation. Differential Equations. <\/em>1976. Vol. 12. No. 1. Pp. 177\u2013179. EDN:
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  5. Pskhu A.V. Uravneniya v chastnyh proizvodnyh drobnogo poryadka<\/em> [Fractional-order partial differential equations]. Moscow: Nauka, 2005. 199 p. EDN: QJPLZX. (In Russian)<\/li>\n\n\n\n
  6. Karmokov M.M., Nakhusheva F.M., Abregov M.Kh. Boundary value problem for loaded parabolic equations of fractional order. News of the Kabardino-Balkarian Scientific Center of RAS.<\/em> Vol. 26. No. 1. Pp. 69\u201377. DOI: 10.35330\/1991-6639-2024-26-1-69-77. (In Russian)<\/li>\n\n\n\n
  7. Karmokov M.M., Nakhusheva F.M., Gekkieva S.Kh. Boundary value problems for discontinuously loaded parabolic equations. Vestnik of Samara University. Natural Science
    Series. <\/em>2024. Vol. 30. No. 4. Pp. 7\u201317. DOI: 10.18287\/2541-7525-2024-30-4-7-17. (In Russian)<\/li>\n\n\n\n
  8. Gekkieva S.Kh., Kerefov M.A. Mixed boundary value problems for a loaded equation with a fractional derivative. Nelokal’nye kraevye zadachi i rodstvennye problemy matematicheskoy biologii, informatiki i fiziki: materialy III Mezhdunarodnoy konferencii<\/em> [Nonlocal boundary value problems and related problems in Mathematical Biology, Computer Science, and Physics: materials of the III International Conference]. Nalchik, 2006. EDN: QKREBL. (In Russian)<\/li>\n\n\n\n
  9. Kozhanov A.I. A non-local in time boundary value problem for linear parabolic equations. Sibirskii Zhurnal Industrial’noi Matematiki. <\/em>2004. Vol. 7. No. 1(17). Pp. 51\u201360. EDN: HZOGQL. (In Russian)<\/li>\n\n\n\n
  10. Kozhanov A.I. On the solvability of an edge problem with a non-local boundary condition for linear parabolic equations. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences<\/em>. 2004. No. 30. Pp. 63\u201369. EDN: HKZXBD. (In Russian)<\/li>\n\n\n\n
  11. Gekkieva S.Kh. Mixed boundary value problems for a loaded diffusion-wave equation. Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Seriya: Matematika. Fizika<\/em> [Scientific Bulletin of Belgorod State University. Series: Mathematics. Physics]. 2016. No. 6(227). Pp. 32\u201335. EDN: VUUBKR. (In Russian)<\/li>\n\n\n\n
  12. Gekkieva S.Kh. Boundary value problem for a generalized transport equation with a fractional derivative in a semi-infinite domain. Sovremennye metody v teorii kraevyh zadach.
    Materialy Voronezhskoy vesenney matematicheskoy shkoly \u201cPontryaginskie chteniya-XIII\u201d <\/em>[Modern methods in the theory of boundary value problems. Proceedings of the Voronezh Spring Mathematical School «Pontryagin Readings-XIII»]. Voronezh: VGU, 2002. P. 37. EDN: VNGVYT. (In Russian)<\/li>\n\n\n\n
  13. Beilin A.B., Bogatov A.V., Pulkina L.S. A problem with non-local conditions for a one-dimensional parabolic equation. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences.<\/em> 2022. Vol. 26. No. Pp. 380\u2013395. DOI: 10.14498\/vsgtu1904. (In Russian)<\/li>\n\n\n\n
  14. Kozhanov A.I., Ashurova G.R. Parabolic equations with degeneracy and unknown coefficient. Mathematical Notes of NEFU. <\/em>2024. Vol. 31. No. 1. Pp. 56\u201369. DOI: 10.25587\/2411-9326-2024-1-56-69. (In Russian)<\/li>\n\n\n\n
  15. Bogatov A.V., Pulkina L.S. Solvability of the inverse coefficient problem with integral redefinition for a one-dimensional parabolic equation. Vestnik of Samara University. Natural
    Science Series<\/em>. 2022. Vol. 28, No. 3-4. Pp. 7\u201317. DOI: 10.18287\/2541-7525-2022-28-3-4-7-17. (In Russian)<\/li>\n\n\n\n
  16. Fridman A. Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa<\/em> [Parabolic partial differential equations]. Moscow: Mir, 1968. 427 p. (In Russian)<\/li>\n<\/ol>\n<\/details>\n\n\n\n
    Information about the author<\/strong>s<\/summary>\n
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    Mukhamed M. Karmokov<\/strong>, Candidate of Physics and Mathematics, Associate Professor, Department of Applied Mathematics and Computer Science, Kabardino-Balkarian State University named after Kh.M. Berbekov;
    173, Chernyshevsky street, Nalchik, 360004, Russia;
    mkarmokov@yandex.ru, ORCID: https:\/\/orcid.org\/0000-0001-5189-6538, SPIN-code: 1771-6984
    Marat A. Kerefov<\/strong>, Candidate of Physics and Mathematics, Associate Professor, Department of Applied Mathematics and Computer Science, Kabardino-Balkarian State University named after Kh.M. Berbekov;
    173, Chernyshevsky street, Nalchik, 360004, Russia;
    kerefov@mail.ru, ORCID: https:\/\/orcid.org\/0000-0002-7442-5402, SPIN-code: 1424-6720
    Sakinat Kh. Gekkieva<\/strong>, Candidate of Physical and Mathematical Sciences, Leading Researcher, Department of Computational Methods, Institute of Applied Mathematics and Automation \u2013 branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
    89 A, Shortanov street, Nalchik, 360000, Russia;
    gekkieva_s@mail.ru, ORCID: https:\/\/orcid.org\/0000-0002-2135-2115, SPIN-code: 6711-3471<\/p>\n\n\n\n

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

    Mixed boundary value problem for one discontinuously loaded parabolic equation M.M. Karmokov, M.A. Kerefov, S.Kh. Gekkieva Abstract. This article is devoted to current issues in the theory of partial differential equations related to the study of boundary value problems for loaded parabolic equations with a fractional integro-differentiation operator, which are of interest not only for […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"wp-custom-template-home","meta":{"footnotes":""},"class_list":["post-6006","page","type-page","status-publish","hentry"],"yoast_head":"\n27.6.1 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-1-en\/\" \/>\n<meta property=\"og:locale\" content=\"ru_RU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"27.6.1 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\" \/>\n<meta property=\"og:description\" content=\"Mixed boundary value problem for one discontinuously loaded parabolic equation M.M. 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Karmokov, M.A. Kerefov, S.Kh. Gekkieva Abstract. This article is devoted to current issues in the theory of partial differential equations related to the study of boundary value problems for loaded parabolic equations with a fractional integro-differentiation operator, which are of interest not only for […]","og_url":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-1-en\/","og_site_name":"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","article_modified_time":"2026-06-02T09:19:45+00:00","twitter_card":"summary_large_image","twitter_misc":{"\u041f\u0440\u0438\u043c\u0435\u0440\u043d\u043e\u0435 \u0432\u0440\u0435\u043c\u044f \u0434\u043b\u044f \u0447\u0442\u0435\u043d\u0438\u044f":"5 \u043c\u0438\u043d\u0443\u0442"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-1-en\/","url":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-1-en\/","name":"27.6.1 En - 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