izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2023-11-3-14izvuzmath-19Research ArticleСтатьиЕдинственность задачи определения ядра в интегро-дифференциальном параболическом уравнении с переменными коэффициентамиUniqueness of the kernel determination problem in an integro-differential parabolic equation with variable coefficientДурдиевД. К.DurdievD. K.

Дурдимурод Каландарович Дурдиев

ул. Университетская, д. 46, г. Ташкент, 100170

ул. М. Икбола, д. 11, г. Бухара, 200117

Durdimurod Kalandarovich Durdiev

46 University str., Tashkent, 100170

11 M. Ikbol str., Bukhara, 200117

durdiev65@mail.ruНуриддиновЖ. З.NuriddinovJ. Z.

Жавлон Зафарович Нуриддинов

ул. М. Икбола, д. 11, г. Бухара, 200117

Javlon Zafarovich Nuriddinov

11 M. Ikbol str., Bukhara, 200117

j.zafarovich@mail.ruИнститут математики им. В.И. Романовского Академии наук Республики Узбекистан; Бухарский государственный университетУзбекистанV.I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan; Bukhara State UniversityUzbekistanБухарский государственный университетУзбекистанBukhara State UniversityUzbekistan202308122023011314Copyright © Дурдиев Д.К., Нуриддинов Ж.З., 20232023Дурдиев Д.К., Нуриддинов Ж.З.Durdiev D.K., Nuriddinov J.Z.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/19

Исследуется обратная задача определения зависящего от времени и положения в пространстве ядра интегрального члена в n-мерном интегро-дифференциальном уравнении теплопроводности по известному решению задачи Коши для этого уравнения. Во-первых, исходная задача заменяется эквивалентной задачей, в которой дополнительное условие содержит неизвестное ядро без интеграла. Изучается вопрос о единственности нахождения этого ядра. Далее, предполагая наличие двух решений k1(x, t) и k2(x, t) поставленной задачи, составляется уравнение на разность этих решений. Дальнейшие исследования ведутся для разности k1(x, t) - k2(x, t) решений задачи с использованием методов оценки интегральных уравнений. 

Abstract. We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the n-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions k1(x, t) and k2(x, t) of the stated problem, it is formed an equation for the difference of this solution. Further research is being conducted for the difference k1(x, t) - k2(x, t) of solutions of the problem and using the techniques of integral equations estimates.

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