Journal of Mathematical Sciences, Vol. 264, No. 5, July, 2022 PARTIAL DECOMPOSITION OF A DOMAIN CONTAINING THIN TUBES FOR SOLVING THE DIFFUSION EQUATION A. A. Amosov ∗ National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia AmosovAA@mpei.ru G. P. Panasenko Institute Camille Jordan UMR CNRS 5208 and SFR MODMAD FED 4169 University of Lyon 23 rue P. Michelon, Saint-Etienne 42023, France grigory.panasenko@univ-st-etienne.fr UDC 517.95 In a domain containing thin cylindrical tubes, we consider the diffusion equation with the Neumann boundary condition on the lateral surface of the tubes. The problem is reduced to a problem of hybrid dimension so that the reduced problem has the original dimension outside the tubes, but is reduced to the one-dimensional diffusion equation inside the tubes. The docking of models of different dimensions is carried out according to the method of asymptotic partial decompositions of domains. We estimate the difference between the solutions to the initial problem and the problem of different dimension. Bibliography: 14 titles. Illustrations:3 figures. 1 Introduction Domains connected by cylindrical tubes appear in modeling of physical processes in industrial installations (for example, rod structures and pipelines) and also in some biological applications (for example, a blood flow in a network of vessels and organs). For unions of thin cylinders, called rod or tubular structures, such problems were studied in [1, 2], where asymptotic expansions were constructed and the method of partial asymptotical decomposition of a domain was developed and justified [3]. In the cited and other works [4]–[10] on asymptotic analysis of thin tubular structures, it is assumed that the domain depends on a small parameter ε, where ε is the ratio of the diameter of tubes to their length. However, for more general domains containing cylindrical (not necessarily thin) inclusions, this method was not tested and justified. The first result for the heat equation in such domains was obtained by the authors [11, 12]. The goal of this paper is to derive an error estimate for the diffusion equation − div (λ∇u)= f with the Neumann boundary condition. Moreover, we do not assume that the tube diameter is small. ∗ To whom the correspondence should be addressed. Translated from Problemy Matematicheskogo Analiza 116, 2022, pp. 25-33. 1072-3374/22/2645-0514 c  2022 Springer Science+Business Media, LLC 514 DOI 10.1007/s10958-022-06014-4