matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/2411-9326-2025-4-101-111matmess-128Research ArticleМАТЕМАТИКАAn inverse problem for a mixed-type equation in spaceAn inverse problem for a mixed-type equation in spaceFayazovK. SFayazovK. S.

Kudratillo S. Fayazov

17 Kichik Khalka Yuli Street, 100195 Tashkent

kudratillo52@mail.ruJuraevaD. S.JuraevaD. S.

Dildora S. Juraeva

17 Kichik Khalka Yuli Street, 100195 Tashkent

jurayevadildora1998@gmail.comTurin Polytechnic University in TashkentUzbekistan202530122025324101111Copyright © Fayazov K.S., Juraeva D.S., 20252025Fayazov K.S., Juraeva D.S.Fayazov K.S., Juraeva D.S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/128

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The paper investigates an inverse problem for a second-order parabolic equation in a two-dimensional domain with variable time direction. The objective is to determine an unknown source function together with the solution, subject to boundary, initial, final, and gluing conditions. The separation of variables method reduces the problem to a spectral formulation involving eigenvalues and eigenfunctions. Using orthogonality, explicit series expansions for the solution and source are derived. Convergence of the series is shown, while existence and uniqueness of a classical solution are established via functional analysis and the Hilbert–Schmidt theorem.

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