Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University919410.18287/2541-7525-2020-26-4-7-14Research ArticleTRICOMI PROBLEM FOR MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATIONAldashevS. A.<p>Doctor of Physical and Mathematical Sciences, full professor</p>aldash51@mail.ruhttps://orcid.org/0000-0002-8223-6900Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, 13, Dostyk ave., Almaty, 050100, Republic of Kazakhstan.071220202647141708202117082021Copyright © 2021, Aldashev S.A.2021<p>It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the media. If the medium is non-conducting, then we obtain multidimensional hyperbolic equations. If the medium’s conductivity is higher, then we arrive at multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to multidimensional hyperbolic-parabolic equations. When studying these applications, one needs to obtain an explicit representation of solutions to the problems under study. Boundary-value problems for hyperbolic-parabolic equations on a plane are well studied; however, their multidimensional analogs have been analyzed very little. The Tricomi problem for the above equations has been previously investigated, but this problem in space has not been studied earlier. This article shows that the Tricomi problem is not uniquely solvable for a multidimensional mixed hyperbolic-parabolic equation. An explicit form of these solutions is given.</p>задача Трикомимногомерное уравнениеразрешимостьсферические функцииTricomi problemmultidimensional equationsolvabilityspherical functions[[1] Tikhonov A.N., Samarskii A.A. Equations of mathematical physics. Moscow: Nauka, 1977, 659 p. Available at: https://uch-lit.ru/matematika-2/dlya-studentov/tihonov-a-n-samarskiy-a-a-uravneniya-matematicheskoy-fiziki-onlayn (In Russ.)][[2] Bitsadze A.V. Some classes of partial differential equations. Moscow: Nauka, 1981, 448 p. Available at: https://www.studmed.ru/bicadze-av-nekotorye-klassy-uravneniy-v-chastnyh-proizvodnyh_5f371e781b6.html (In Russ.)][[3] Nakhushev A.M. Problems with displacement for partial differential equations. Moscow: Nauka, 2006, 287 p. Available at: https://www.elibrary.ru/item.asp?id=17962288. (In Russ.)][[4] Mikhlin S.G. Multidimensional singular integrals and integral equations. Moscow: Fizmatgiz, 1962, 254 p. Available at: https://booksee.org/book/578442. (In Russ.)][[5] Aldashev S.A. Nonuniqueness of the solution of the Tricomi problem for a multidimensional hyperbolic-parabolic equation. Differential Equations, 2014, vol. 50, no. 4, pp. 541–545. DOI: https://doi.org/10.1134/S0012266114040120. (English; Russian original)][[6] Bitsadze A.V. Mixed-type equations. Moscow: Izd. AN SSSR, 1959, 164 p. Available at: https://1lib.education/book/1289692/1f5275?id=1289692&secret=1f5275. (In Russ.)][[7] Aldashev S.A. Boundary value problems for multidimensional hyperbolic and mixed equations. Almaty: Gylym, 1994, 170 p. (In Russ.)][[8] Copson E.T. On the Riemann-Green function. (Archive for Rational Mechanics and Analysis), 1958, vol. 1, pp. 324–348. DOI: https://doi.org/10.1007/BF00298013.][[9] Samko S.G., Kilbas A.A., Marichev O.I. Integrals and derivatives of fractional order and some of their applications. Minsk: Nauka i tekhnika, 1987, 688 p. Available at: https://booksee.org/book/441860. (In Russ.)][[10] Kamke E. Handbook of ordinary differential equations. Moscow: Nauka, 1965, 703 p. Available at: https://booksee.org/book/567727. (In Russ.)][[11] Bateman H., Erdelyi A. Higher Transcendental Functions. Vol. 1. Moscow: Nauka, 1973, 294 p. Available at: http://ega-math.narod.ru/Books/Bateman.htm. (In Russ.)][[12] Bateman H., Erdelyi A. Higher Transcendental Functions. Vol. 2. Moscow: Nauka, 1974, 295 p. Available at: http://ega-math.narod.ru/Books/Bateman.htm. (In Russ.)]