izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2023-10-46-59izvuzmath-13Research ArticleСтатьиОбратные коэффициентные задачи для временно-дробного волнового уравнения с обобщенной производной Римана-Лиувилля по времениInverse coefficient problems for a time-fractional wave equation with the generalized Riemann-Liouville time derivativeТурдиевХ. Х.TurdievH. H.

Турдиев Халим Хамроевич.

ул. М. Икбол, д. 11, Бухара, 200118

Halim H. Turdiev.

11 M. Ikbol str., Bukhara 200118

hturdiev@mail.ruБухарское отделение Института Математики Академии наук Республики Узбекистан; Бухарский государственный университетУзбекистанBukhara branch of the institute of Mathematics named after V.I. Romanovskiy at the Academy of sciences of the Republic of Uzbekistan; Bukhara State UniversityUzbekistan2023271020230104659Copyright © Турдиев Х.Х., 20232023Турдиев Х.Х.Turdiev H.H.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://izvuzmath.elpub.ru/jour/article/view/13

В работе рассматривается обратная задача определения нестационарного коэффициента в волновом уравнении дробного порядка с производной Гильфера. В этом случае прямая задача является начально-краевой задачей для этого уравнения с начальными и нелокальными краевыми условиями типа Коши. В качестве условия переопределенности дается нелокальное интегральное условие относительно решения прямой задачи. Методом Фурье  эта задача сводится к эквивалентным интегральным уравнениям. Затем, используя функцию Миттаг-Леффлера и обобщенное сингулярное неравенство Гронуолла, получаем априорную оценку решения через неизвестный коэффициент, эта оценка понадобится нам для исследования обратной задачи. Обратная задача сводится к эквивалентному интегральному уравнению типа Вольтерра. Для решения этого уравнения используется принцип сжимающего отображения. Доказаны результаты о локальном существовании и глобальной единственности.

This paper considers the inverse problem of determining the time-dependent coeffiicient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

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The authors declare that there are no conflicts of interest present.