Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical SciencesВестник Самарского государственного технического университета. Серия «Физико-математические науки»1991-86152310-7081Samara State Technical University4198710.14498/vsgtu1694ArticlesСтатьиResearch ArticleOn a boundary value problem for a third-order parabolic-hyperbolic type equation with a displacement boundary condition in its hyperbolicity domainОб одной краевой задаче для уравнения параболического-гиперболического типа третьего порядка с граничным условием смещения в области его гиперболичностиBalkizovZhiraslan AnatolievichБалкизовЖираслан Анатольевич

Candidate of physico-mathematical sciences, no status

кандидат физико-математических наук, без звания

Giraslan@yandex.ruInstitute of Applied Mathematics and AutomationИнститут прикладной математики и автоматизации – филиал Федерального государственного бюджетного научного учреждения "Федеральный научный центр "Кабардино-Балкарский научный центр Российской академии наук"31072020242VOL 24, NO2 (2020)ТОМ 24, №2 (2020)21122504082020Copyright ©; 2020, Samara State Technical UniversityCopyright ©; 2020, Самарский государственный технический университет2020Samara State Technical UniversityСамарский государственный технический университетhttps://creativecommons.org/licenses/by/4.0https://journals.eco-vector.com/1991-8615/article/view/41987

In the article, we investigate a boundary-value problem with a third-order inhomogeneous parabolic-hyperbolic equation with a wave operator in a hyperbolicity domain. A linear combination with variable coefficients in terms of derivatives of the sought function on independent characteristics, as well as on the line of type and order changing is specified as a boundary condition. We have established necessary and sufficient conditions that guarantee existence and uniqueness of a regular solution to the problem under study. In some cases, a solution representation is written out explicitly.

Исследована краевая задача со смещением для неоднородного уравнения параболо-гиперболического типа третьего порядка с волновым оператором в области гиперболичности, когда в качестве одного из граничных условий задана линейная комбинация с переменными коэффициентами производных от значений искомой функции на независимых характеристиках, а также на линии изменения типа и порядка. Найдены необходимые и достаточные условия существования и единственности регулярного решения задачи. В некоторых частных случаях представление решения исследуемой задачи выписано в явном виде.

degenerate hyperbolic equationequation with multiple characteristicsparabolic-hyperbolic equation of third orderproblems with a shiftвырождающиеся уравненияуравнения с кратными характеристикамипараболо-гиперболическое уравнение третьего порядказадачи со смещениемDzhuraev T. D., Kraevye zadachi dlia uravnenii smeshannogo i smeshanno-sostavnogo tipov [Boundary Value Problems for Equations of Mixed and Mixed-Composite Types], Fan, Tashkent, 1979, 240 pp. (In Russian)Nakhushev A. M., Uravneniia matematicheskoi biologii [Equations of Mathematical Biology], Vyssh. shkola, Moscow, 1995, 301 pp. (In Russian)Bitsadze A. V., "On the theory of a class of equations of mixed type", Nekotorye problemy matematiki i mekhaniki [Some Problems of Mathematics and Mechanics], Nauka, Leningrad, 1970, 112–119 (In Russian)Nakhushev A. M., Zadachi so smeshcheniem dlia uravnenii v chastnykh proizvodnykh [Problems with Displacement for Equations in Partial Derivatives], Nauka, Moscow, 2006, 287 pp. (In Russian)Zhegalov V. I., "A boundary-value problem for an equation of mixed type with boundary conditions on both characteristics and with discontinuities on the transition curve", Boundary value problems in the theory of analytic functions, Uchenye Zapiski Kazanskogo Universiteta, 122, no. 3, Kazan Univ., Kazan, 1962, 3-16 (In Russian)Nakhushev A. M., "Certain boundary value problems for hyperbolic equations and equations of mixed type", Differ. Uravn., 5:1 (1969), 44-59 (In Russian)Nakhushev A. M., "A new boundary value problem for a degenerate hyperbolic equation", Sov. Math., Dokl., 10 (1969), 935-938Bitsadze A. V., Samarskii A. A., "On some simple generalizations of linear elliptic boundary problems", Sov. Math., Dokl., 10 (1969), 398-400Islamov N. B., "Analogue of Bitsadze-Samarskii problem for a class of parabolic-hyperbolic equations of second kind", Ufa Math. J., 7:1 (2015), 31-45Khubiev K., "A problem of the Bitsadze-Samarskii type for a loaded hyperbolic-parabolic equation", Mathematical notes of NEFU, 26:2 (2019), 31-40 (In Russian)Dezin A. A., "On the solvable extensions of partial differential operators", Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, 1963, 65–66Sabitov K. B., Gushchina V. A., "A. A. Dezin's problem for inhomogeneous Lavrent'ev–Bitsadze equation", Russian Math. (Iz. VUZ), 61:3 (2017), 31-43Kirzhinov R. A., "On the solving of the A. A. Dezin problem analogue for a second-order mixed-type equation by the Green's function method", Vestnik KRAUNC. Fiz.-Mat. Nauki, 2018, no. 3(23), 36-41 (In Russian)Nakhusheva Z. A., Nelokal'nye kraevye zadachi dlia osnovnykh i smeshannogo tipov differentsial'nykh uravnenii [Nonlocal Boundary-Value Problems for Basic orMixed Type Differential Equations], KBNTz RAN, Nal'chik, 2011, 196 pp. (In Russian)Carleman T., "Sur la theorie des equations integrales et ses applications", Verhandl. Internat. Math. Kongr, v. 1, Orell Füssli, Zürich-Leipzig, 1932, 138–151Steklov V. A., Osnovnye problemy matematicheskoi fiziki [Basic Problems of Mathematical Physics], Nauka, Moscow, 1983, 432 pp. (In Russian)Frankl F. I., "Two gas-dynamical applications of the Lavrentiev–Bitsadze boundary value problem", Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom., 6:11 (1951), 3–7 (In Russian)Frankl F. I., "Subsonic flow past profiles with supersonic zone terminating with a normal shock", Prikl. Mat. Mekh., 20:2 (1956), 196-202 (In Russian)Frankl F. I., Selected Works on Gas Dynamics, Nauka, Moscow, 1973, 712 pp. (In Russian)Bitsadze A. V., "On a Frankl's problem", Dokl. Akad. Nauk SSSR, 109:6 (1956), 1091-1094 (In Russian)Zhegalov V. I., "The Frankl' problem with a shift", Soviet Math. (Iz. VUZ), 23:9 (1979), 10-19Pskhu A. V., "On the solution of the Frankl problem", Differ. Equ., 36:3 (2000), 474–476Pskhu A. V., "The Frankl' problem for a hyperbolic-parabolic equation", Differ. Equ., 39:1 (2003), 112-120Sabitov K. B., "On the theory of the Frankl problem for equations of mixed type", Izv. Math., 81:1 (2017), 99-136Bers L., Mathematical aspects of subsonic and transonic gas dynamics, Dover Publ., Mineola, NY, 2016, xv+278 pp.Salakhitdinov M. S., Uravneniia smeshanno-sostavnogo tipa [Equations of Mixed-Composite Type], Fan, Tashkent, 1974, 156 pp. (In Russian)Salakhitdinov M. S., Urinov A. K., K spektral'noi teorii uravnenii smeshannogo tipa [To Spectral Theory of Mixed Type Equations], Fan, Tashkent, 1997, 165 pp.Repin O. A., Kraevye zadachi so smeshcheniem dlia uravnenii giperbolicheskogo i smeshannogo tipov [Boundary-Value Problems for Hyperbolic and Mixed Type Equations], Saratov Univ., Samara branch, Samara, 1992, 162 pp.Kalmenov T. Sh., Kraevye zadachi dlia lineinykh uravnenii v chastnykh proizvodnykh giperbolicheskogo tipa [Boundary-Value Problems for Linear Partial Differential Equations of Hyperbolic Type], Gylaia, Shymkent, 1993, 328 pp.Zhegalov V. I., Mironov A. N., Differentsial'nye uravneniia so starshimi chastnymi proizvodnymi [Differential Equations with Higher Partial Derivatives], Kazan Math. Society, Kazan, 2001, 226 pp.Marichev O. I., Kilbas A. A., Repin O. A., Kraevye zadachi dlia uravnenii s chastnymi proizvodnymi s razryvnymi koeffitsientami [Boundary Value Problems for Partial Differential Equations with Discontinuous Coefficients], Samara State Univ. of Economics, Samara, 2008, 275 pp. (In Russian)Pulkina L. S., Zadachi s neklassicheskimi usloviiami dlia giperbolicheskikh uravnenii [Problems with Non-Classic Conditions for Hyperbolic Equations], Samara State Univ., Samara, 2012, 194 pp. (In Russian)Sabitov K. B., K teorii uravnenii smeshannogo tipa [On the Theory of Mixed-Type Equations], Fizmatlit, Moscow, 2014, 304 pp. (In Russian)Sabitov K. B., Priamye i obratnye zadachi dlia uravnenii smeshannogo parabolo-giperbolicheskogo tipa [Direct and Inverse Problems for Mixed Parabolic-Hyperbolic Type Equations], Gilem, Ufa, 2015, 236 pp. (In Russian)Balkizov Zh. A, "On a boundary value problem of the type of the Tricomi problem for a mixed second-order parabolic-hyperbolic equation with three displacements in the hyperbolic part of the domain", Belgorod State University Scientific Bulletin. Mathematics Physics, 51:1 (2019), 5-14 (In Russian)Repin O. A., "A nonlocal boundary value problem for a parabolic-hyperbolic equation with a characteristic line of change of type", Differ. Uravn., 28:1 (1992), 173-176 (In Russian)Kilbas A. A., Repin O. A., "A problem with a shift for a parabolic-hyperbolic equation", Differ. Equ., 34:6 (1998), 796-802Repin O. A., Efimova S. V., "A nonlocal boundary value problem for a parabolic-hyperbolic equation with a non-characteristic line of type changing", Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2002, no. 16, 10-14 (In Russian)"Efimova S. V., Repin O. A.", Differ. Equ., 40:10 (2004), 1498-1502Mirsaburov M., Khairullaev I. N., Bobomurodov U. E., "A generalization of Bitsadze–Samarskii problem for mixed type equation", Russian Math. (Iz. VUZ), 60:10 (2016), 29-32Efimova S. V., "A nonlocal problem for a hyperbolic equation degenerating inside a region", Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2005, no. 34, 194-196 (In Russian)Balkizov Zh. A., "A boundary value problem with displacement for a model equation of a parabolic-hyperbolic type of the third order", Vestnik KRAUNC. Fiz.-Mat. Nauki, 2018, no. 3(23), 19-26 (In Russian)Tikhonov A. N., Samarskii A. A., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Nauka, Moscow, 1972, 735 pp. (In Russian)