{"id":2246,"date":"2025-05-13T18:59:03","date_gmt":"2025-05-13T17:59:03","guid":{"rendered":"http:\/\/newskbncran.ru\/?page_id=2246"},"modified":"2026-06-05T09:00:58","modified_gmt":"2026-06-05T08:00:58","slug":"26-6-2-en","status":"publish","type":"page","link":"http:\/\/izvestiyakbncran.ru\/index.php\/en\/26-6-2-en\/","title":{"rendered":"26.6.2 en"},"content":{"rendered":"\n

Initial value problem for a fractional order equation with the Gerasimov\u2013Caputo derivative with involution<\/strong><\/h2>\n\n\n\n

L.M. Eneeva<\/strong><\/strong><\/p>\n\n\n\n

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Abstract. <\/em><\/strong>The paper considers a linear ordinary differential equation with a fractional derivative in the Gerasimov\u2013Caputo sense. The equation under consideration belongs to the class of differential equations that arise, in particular, in the study of boundary value problems for differential equations containing a composition of left- and right-hand derivatives of fractional order, which, in turn, serve as the basis for modeling various physical and geophysical processes. In particular, such equations arise when describing dissipative oscillatory systems. In this work, the initial value problem in the unit interval is studied for the equation under consideration. A theorem for the existence and uniqueness of a solution to the problem under study is proven, and an explicit representation of the solution is constructed.<\/p>\n\n\n\n

Keywords:<\/em><\/strong> <\/strong>fractional order equation, Cauchy problem, Gerasimov\u2013Caputo derivative, involution, fundamental solution<\/p>\n\n\n\n

For citation<\/strong>. Eneeva L.M. Initial value problem for a fractional order equation with the Gerasimov\u2013Caputo derivative with involution.  News of the Kabardino-Balkarian Scientific Center of RAS.<\/em> <\/strong>2024. Vol. 26. No. 6. Pp. 19\u201325. DOI: 10.35330\/<\/strong>1991-6639-2024-26-6-19-25<\/p>\n\n\n\n

R<\/strong>eferences<\/summary>\n
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  1. Nakhushev A.M. Drobnoye ischisleniye i yego primeneniye [Fractional calculus and its application]. Moscow: FIZMATLIT, 2003. 272 p. (In Russian)<\/li>\n\n\n\n
  2. Eneeva L.M. On the question of solving a mixed boundary value problem for an equation with fractional derivatives with different origins. Reports of the Adyghe (Circassian) International Academy of Sciences. 2023. Vol. 23. No. 4. Pp. 62\u201368. DOI: 10.47928\/1726-9946-2023-23-4-62-68. (In Russian)<\/li>\n\n\n\n
  3. Stankovi\u00b4c B. An equation with left and right fractional derivatives. Publications de l\u2019institute math\u00b4ematique. Nouvelle serie. 2006. Vol. 80(94), Pp. 259\u2013272.<\/li>\n\n\n\n
  4. Atanackovic T.M., Stankovi\u00b4c B. On a differential equation with left and right fractional derivatives. Fractional Calculus and Applied Analysis. 2007. Vol. 10. No. 2. Pp. 139\u2013150.<\/li>\n\n\n\n
  5. Torres C. Existence of a solution for the fractional forced pendulum. Journal of Applied Mathematics and Computational Mechanics. 2014. Vol. 13. No. 1. Pp. 125\u2013142.<\/li>\n\n\n\n
  6. Eneeva L.M., Pskhu A.V., Potapov A.A. et al. Lyapunov inequality for a fractional differential equation modelling damped vibrations of thin film MEMS. Advances in Intelligent Systems and Computing. ICCD2019 (paper ID: E19100). 2021.<\/li>\n\n\n\n
  7. Rekhviashvili S.Sh., Pskhu A.V., Potapov A.A. et al. Modeling damped vibrations of thin film MEMS. Advances in Intelligent Systems and Computing. ICCD2019 (paper ID: E19101). 2021.<\/li>\n\n\n\n
  8. Eneeva L., Pskhu A., Rekhviashvili S. Ordinary differential equation with left and right fractional derivatives and modeling of oscillatory systems. Mathematics. 2020. Vol. 8(12). P. 2122. DOI: 10.3390\/math8122122<\/li>\n\n\n\n
  9. Eneeva L.M. Cauchy problem for fractional order equation with involution. Vestnik KRAUNC. Fiz.-mat. nauki. 2024. Vol. 48. No. 3. Pp. 43\u201355. DOI: 10.26117\/2079-6641-2024-48-3-43-55. (In Russian)<\/li>\n<\/ol>\n<\/details>\n\n\n\n
    Information about the author<\/strong><\/summary>\n
    \n

    Liana M. Eneeva<\/strong>, Candidate of Physical and Mathematical Sciences, Senior Researcher of Department of
    Mathematical Modeling of Geophysical Processes at 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;
    Eneeva72@list.ru, ORCID: https:\/\/orcid.org\/0000-0003-2530-5022, SPIN-code: 3403-8412<\/p>\n<\/div>\n<\/details>\n\n\n\n

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

    Initial value problem for a fractional order equation with the Gerasimov\u2013Caputo derivative with involution L.M. Eneeva Abstract. The paper considers a linear ordinary differential equation with a fractional derivative in the Gerasimov\u2013Caputo sense. The equation under consideration belongs to the class of differential equations that arise, in particular, in the study of boundary value problems […]<\/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-2246","page","type-page","status-publish","hentry"],"yoast_head":"\n26.6.2 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=\"http:\/\/izvestiyakbncran.ru\/index.php\/en\/26-6-2-en\/\" \/>\n<meta property=\"og:locale\" content=\"ru_RU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"26.6.2 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=\"Initial value problem for a fractional order equation with the Gerasimov\u2013Caputo derivative with involution L.M. Eneeva Abstract. The paper considers a linear ordinary differential equation with a fractional derivative in the Gerasimov\u2013Caputo sense. 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