{"id":2077,"date":"2025-05-13T10:43:23","date_gmt":"2025-05-13T09:43:23","guid":{"rendered":"http:\/\/newskbncran.ru\/?page_id=2077"},"modified":"2026-03-27T12:35:01","modified_gmt":"2026-03-27T12:35:01","slug":"26-4-12-en","status":"publish","type":"page","link":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/26-4-12-en\/","title":{"rendered":"26.4.12 en"},"content":{"rendered":"\n

Boundary value problem for a differential-difference equation with a fractional derivative<\/strong><\/h1>\n\n\n\n

L.M. Vidzizheva, D.A. Kanametova<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/p>\n\n\n\n

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Abstract<\/em><\/strong>.<\/em> The work is devoted to the study of a differential-difference equation with a fractional derivative of order not exceeding one. For the equation under consideration, a boundary value problem is posed and solved on a manifold that is a countable union of intervals. To solve the problem, we used an analogue of the Green function method, adapted for differential-difference equations. A general representation of the solution to the equation under study has been found, a fundamental solution has been constructed in terms of the Prabhakar function, its properties have been studied, and a theorem on the existence and uniqueness of a solution to the problem under study has been proven.<\/p>\n\n\n\n

Keywords:<\/em><\/strong> <\/em>fractional derivative, McKendrick \u2013 Von Foerster equation, fractional integration operator, fractional differentiation operator, differential-difference equation, Riemann \u2013 Liouville integral, difference operators, Prabhakar function, Mittag-Leffler function<\/p>\n\n\n\n

For citation<\/strong>. Vidzizheva L.M., Kanametova D.A.Boundary value problem for a differential-difference equation with a fractional derivative. News of the Kabardino-Balkarian Scientific Center of RAS.<\/em>2024. Vol. 26. No. 4. Pp. 130\u2013144. DOI: 10.35330\/<\/strong>1991-6639-2024-26-4-130-144<\/p>\n\n\n\n

References<\/strong><\/summary>\n
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  1. Nakhushev A.M. Drobnoye ischisleniye i yego primeneniye<\/em> [Fractional calculus and its application]. Moscow: FIZMATLIT, 2003. 272 p. ISBN: 5-9221-0440-3. EDN: UGLEPD. (In Russian)<\/li>\n\n\n\n
  2. Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M. Chislennyye metody<\/em> [Numerical methods]. Moscow: BINOM. Knowledge Laboratory, 2011. 636 p. ISBN: 978-5-9963-0449-3. EDN: QJXMXL. (In Russian)<\/li>\n\n\n\n
  3. Kenetova R. O., Losanova F. M. On a nonlocal boundary value problem for the generalized McKendrick \u2013 von F\u00f6rster equation. News of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences. 2017. No. 2(76). Pp. 49\u201353. EDN: ORSLWH. (In Russian)<\/li>\n\n\n\n
  4. Losanova F.M. About one mathematical model with the generalized McKendrick \u2013 von F\u00f6rster equation. Vestnik KRAUNC. <\/em>Phys.-math. Sciences. 2020. Vol. 33. No. 4. Pp. 71\u201377. DOI: 10.26117\/2079-6641-2020-33-4-71-77. (In Russian)<\/li>\n\n\n\n
  5. Bogatyreva F.T. Boundary value problems for first order partial differential equations with Dzhrbashyan \u2013 Nersesyan operators. Chelyab. Phys.-Math. zhur. 2021. Vol. 6. No. 4. Pp. 403\u2013416. DOI: 10.47475\/2500-0101-2021-16401. (In Russian)<\/li>\n\n\n\n
  6. Lavrentyev M.A., Shabat B.V. Metody teorii funktsiy kompleksnoy peremennoy<\/em> [Methods of the theory of functions of a complex variable]. Moscow: FIZMATGIZ, 1958. 749 p. (In Russian)<\/li>\n\n\n\n
  7. Prabhakar T.R., A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 1971. No. 19. Pp. 7\u201315.<\/li>\n\n\n\n
  8. Garra R., Garrappa R. The Prabhakar or three parameter Mittag-Leffler function: Theory and application. Commun Nonlinear Sci Numer Simulat. 2018. No. 56. Pp. 314\u2013329.<\/li>\n\n\n\n
  9. Shukla A.K., Prajapati J.C. On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 2007. No. 336. Pp. 797\u2013811.<\/li>\n\n\n\n
  10. Bogatyreva F.T., Gadzova L.Kh., Efendiev B.I. Osnovy drobnogo integrirovaniya i differentsirovaniya: metodicheskoye posobiye<\/em> [Fundamentals of fractional integration and differentiation: Methodical manual]. Nalchik: Izdatel’stvo KBNTS RAN, 2020. 46 p. ISBN 978-5-6045584-2-3. EDN: UJQESX. (In Russian)<\/li>\n<\/ol>\n\n\n\n

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

    Information about the authors<\/strong><\/summary>\n

    Leila M. Vidzizheva<\/strong>, Post-graduate Student, Scientific and Educational Center, Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
    360010, Russia, Nalchik, 2 Balkarov street;
    Dana A. Kanametova<\/strong>, Candidate of Economic Sciences, Researcher, Institute of Applied
    Mathematics and Automation \u2013 branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
    360000, Russia, Nalchik, 89 A Shortanov street;
    danocha_999@mail.ru, SPIN-code: 6070-1196<\/p>\n\n\n\n

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

    Boundary value problem for a differential-difference equation with a fractional derivative L.M. Vidzizheva, D.A. Kanametova Upload the full text Abstract. The work is devoted to the study of a differential-difference equation with a fractional derivative of order not exceeding one. For the equation under consideration, a boundary value problem is posed and solved on a manifold […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2077","page","type-page","status-publish","hentry"],"yoast_head":"\n26.4.12 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\/26-4-12-en\/\" \/>\n<meta property=\"og:locale\" content=\"ru_RU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"26.4.12 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=\"Boundary value problem for a differential-difference equation with a fractional derivative L.M. 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