Abstract:
We study the inverse problem on determining the energy-temperature relation χ(t) and the heat conduction relation k(t) functions in the one-dimensional integro-differential heat equation. The direct problem is an initial-boundary value problem for this equation with the Dirichlet boundary conditions. The integral terms involve the time convolution of unknown kernels and a direct problem solution. As an additional information for solving inverse problem, the solution of the direct problem for x=x0 and x=x1 is given. We first introduce an auxiliary problem equivalent to the original one. Then the auxiliary problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to the unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, we prove the main result of the article, which a local existence and uniqueness theorem for the inverse problem.
Citation:
D. K. Durdiev, J. J. Jumaev, D. D. Atoev, “Inverse problem on determining two kernels in integro-differential equation of heat flow”, Ufa Math. J., 15:2 (2023), 119–134
\Bibitem{DurJumAto23}
\by D.~K.~Durdiev, J.~J.~Jumaev, D.~D.~Atoev
\paper Inverse problem on determining two kernels in integro-differential equation of heat flow
\jour Ufa Math. J.
\yr 2023
\vol 15
\issue 2
\pages 119--134
\mathnet{http://mi.mathnet.ru/eng/ufa658}
\crossref{https://doi.org/10.13108/2023-15-2-119}
Linking options:
https://www.mathnet.ru/eng/ufa658
https://doi.org/10.13108/2023-15-2-119
https://www.mathnet.ru/eng/ufa/v15/i2/p120
This publication is cited in the following 5 articles:
Jonibek J. Jumaev, “NUMERICAL ANALYSIS OF INVERSE PROBLEMS FOR THE DIFFUSION EQUATION WITH INITIAL-BOUNDARY AND OVERDETERMINATION CONDITIONS”, J Math Sci, 2025
D. K. Durdiev, H. B. Elmuradova, A. A. Rakhmonov, “Inverse kernel determination problem for a class of pseudo-parabolic integro-differential equations”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 29:1 (2025) (to appear)
D. K. Durdiev, J. J. Jumaev, “Recovering Source Function and Kernel for a Time-fractional Diffusion Equation in the Bounded Domain”, Lobachevskii J Math, 45:4 (2024), 1691
Durdimurod K. Durdiev, Zhavlon Z. Nuriddinov, “Kernel determination problem for one parabolic equation with memory”, Ural Math. J., 9:2 (2023), 86–98
D. K. Durdiev, “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa s kharakteristicheskoi liniei izmeneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 27:4 (2023), 607–620