izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2023-10-22-35izvuzmath-11Research ArticleСтатьиОбратная задача определения ядра интегро-дифференциального уравнения дробной диффузии в ограниченной областиInverse problem of determining the kernel of integro-differential fractional diffusion equation in bounded domainДурдиевД. К.DurdievD. K.

Дурдиев Дурдимурод Каландарович.

ул. Университетская, д. 46, Ташкент, 100170; ул. М. Икбол, д. 11, Бухара, 200118

Durdimurod K. Durdiev.

46 University str., Tashkent, 100170; 11 M. Ikbol str., Bukhara, 200118

durdiev65@mail.ruЖумаевЖ. Ж.JumaevJ. J.

Жумаев Жонибек Жамолович.

ул. Университетская, д. 46, Ташкент, 100170; ул. М. Икбол, д. 11, Бухара, 200118

Jonibek J. Jumaev.

46 University str., Tashkent, 100170; 11 M. Ikbol str., Bukhara, 200118

jonibekjj@mail.ruИнститут математики им. В.И. Романовского Академии наук Республики Узбекистан; Бухарский государственный университетУзбекистанInstitute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State UniversityUzbekistan2023261020230102235Copyright © Дурдиев Д.К., Жумаев Ж.Ж., 20232023Дурдиев Д.К., Жумаев Ж.Ж.Durdiev D.K., Jumaev J.J.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/11

Исследуется обратная задача определения ядра в одномерном интегро-дифференциальном уравнении диффузии с дробной производной по времени с начально-краевыми условиями и условиями переопределения. Сначала вводится эквивалентная этой задаче вспомогательная задача. Методом Фурье вспомогательная задача сводится к эквивалентным интегральным уравнениям. Затем, используя оценки функции Миттаг-Леффлера и метод последовательных приближений, находится оценка решения прямой задачи через норму неизвестного ядра, эта оценка будет использоваться при исследовании обратной задачи. Обратная задача сводится к эквивалентному интегральному уравнению. Для решения этого уравнения применяется принцип сжимающего отображения. Доказаны результаты о локальном существовании и глобальной единственности.

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.

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