MATHEMATICS

Inverse analysis of a loaded heat conduction equation

U. I. Baltaeva^a, P. Agarwal^b, B. Khasanov^c, H. Hayitbayev^d, F. Hubert<sup>e

a</sup> Department of Applied Mathematics and Mathematical Physics, Urgench State University, Urgench-220100, Uzbekistan
^b Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India
^c Department of Exact sciences, Khorezm Mamun Academy, Khiva, Uzbekistan
^d Department of Accounting and General Professional Sciences, Mamun University, Khiva, Uzbekistan
^e Aix-Marseille Université Marseille, France

Abstract: 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.

Keywords: heat conduction, inverse problem, fractional calculus, kernel identification, fixed-point method.

Received: 07.10.2025
Revised: 05.11.2025
Accepted: 06.11.2025

Language: English

DOI: 10.17586/2220-8054-2025-16-6-727-736