najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-6-727-736najo-1611Research ArticleMATHEMATICSМАТЕМАТИКАInverse analysis of a loaded heat conduction equationОбратный анализ нагруженного уравнения теплопроводностиhttps://orcid.org/0000-0001-9687-5220БалтаеваУ.BaltaevaU.

Умида Балтаева

Umida Baltaeva – Department of Applied Mathematics and mathematical physics

Urgench-220100

umida_baltayeva@mail.ruhttps://orcid.org/0000-0001-7556-8942АгарвалП.AgarwalP.

Правин Агарвал

Praveen Agarwal – Department of Mathematics

Jaipur-303012

goyal.praveen2011@gmail.comhttps://orcid.org/0009-0003-5079-7150ХасановБ.KhasanovB.

Бобур Хасанов

Bobur Khasanov – Department of Exact sciences

Khiva

xasanovboburjon.1993@gmail.comhttps://orcid.org/0009-0002-0491-5104ХаитбаевХ.HayitbayevH.

Хамробек Хаитбаев

Hamrobek Hayitbayev – Department of Accounting and General Professional Sciences

Khiva

hamrohayitboyev073@gmail.comhttps://orcid.org/0000-0002-7553-9698ЮберФ.HubertF.

Флоренс Юбер

Florence Hubert – Aix-Marseille Universite

I2M, Marseille

florence.hubert@univ-amu.frUrgench State UniversityUzbekistanAnand International College of EngineeringIndiaKhorezm Mamun AcademyUzbekistanMamun universityUzbekistanAix-Marseille UniversiteFrance202506012026166727736Copyright © Baltaeva U., Agarwal P., Khasanov B., Hayitbayev H., Hubert F., 20262026Балтаева У., Агарвал П., Хасанов Б., Хаитбаев Х., Юбер Ф.Baltaeva U., Agarwal P., Khasanov B., Hayitbayev H., Hubert F.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1611

This work considers an inverse problem for a heat conduction equation that includes fractional loaded terms and coefficients varying with spatial coordinates. By reformulating the original equation into a system of equivalent loaded integro-differential equations, we establish sufficient conditions ensuring the existence and uniqueness of the solution. The study focuses on determining the multidimensional kernel associated with the fractional heat conduction operator. The approach is based on the contraction mapping principle and the use of Riemann-Liouville fractional integrals, providing a mathematical framework applicable to diffusion processes with spatial heterogeneity and memory effects.

В данной работе исследуется обратная задача для уравнения теплопроводности, содержащего дробные нагруженные члены и коэффициенты, зависящие от пространственных переменных. Путём сведения исходной задачи к эквивалентной системе нагруженных интегро-дифференциальных уравнений получены достаточные условия существования и единственности решения. Предлагаемый подход основан на принципе сжимающего отображения и использовании дробных интегралов Римана–Лиувилля. Разработанный математический аппарат может быть применён для описания диффузионных процессов в средах с пространственной неоднородностью и эффектами памяти.

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