Poincare–Tricomi problem for the equation of a mixed elliptico-hyperbolic type of second kind

A. A. Abdullayev   ; T. G. Ergashev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 65 (2020), pp. 5-21

Résumé

It is known, that if the partial differential equation of the second order belongs to the elliptic type in one part of the domain and to the hyperbolic type in the other part, then such equation is called an equation of mixed type; both parts of the domain are separated by a transition line on which the equation either degenerates into parabolic or is not defined. Equations of mixed elliptic-hyperbolic type are divided into equations of the first and second kind. For equations of the first kind, the line of parabolic degeneracy is the return point of the family of characteristics of the corresponding hyperbolic equation. The equation whose degeneration line is simultaneously the envelope of a family of characteristics, i.e. is itself a characteristic, is an equation of the second kind. Therefore, equations of a mixed type of the second kind in all respects are relatively little studied. For example, the Poincare–Tricomi problem and its various generalizations for equations of the first kind have long been studied. For equations of mixed type of the second kind, depending on the degree of degeneracy, the limiting values of the desired solution and its derivative on the line of change of the type of equation can have singularities. To ensure the necessary smoothness of the desired solution outside the line of characteristic degeneracy, one has to require increased smoothness of the given limit functions. In order to weaken this requirement, in the present paper we introduce a class of generalized solutions. In the hyperbolic part of the mixed domain, we seek a generalized solution; in the elliptic part, a regular solution. This paper is devoted to the study of the Poincare–Tricomi problem for one equation of the mixed elliptic-hyperbolic type of the second kind. The conditions under which the problem has a unique solution are identified. In the elliptic part of the mixed domain, a classical solution is sought and a similar second functional relationship brought from the ellipticity domain of the equation is derived. Then, after exclusion of one of the two unknown functions from these two functional relationships, the solution of the posed problem is reduced to solving a singular integral equation for the limit value of the sought function on the line separating the types of the equation. Under certain restrictions on the given functions and parameters of the Poincare–Tricomi type problem, this singular integral equation can be reduced to a Fredholm integral equation of the second kind by the Carleman method. The unique solvability of this equation follows from the Fredholm alternative and uniqueness theorem for the posed problem.

Détail

Keywords: generalized solution, Poincare-Tricomi problem, equation of the second kind, integral equation, energy integral method, Green's function.

Bibliographie

[1] M. B. Kapilevich, “Confluent hypergeometric Horn functions”, Differentsialnye Uravneniya–Differential Equations, 2:9 (1966), 1239–1254 | MR | Zbl

[2] A. K. Urinov, T. G. Ergashev, “Confluent hypergeometric functions of many variables and their application to the finding of fundamental solutions of the generalized Helmholtz equation with singular coefficients”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika–Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 55, 45–56 | DOI

[3] H. M. Srivastava, A. Hasanov, J. Choi, “Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation”, Sohag J. Math., 2:1 (2015), 1–10 | DOI

[4] Berdyshev A.S, A. Hasanov, T. G. Ergashev, “Double-layer potentials for a generalized biaxially symmetric Helmholtz equation. II”, Complex Variables and Elliptic Equations, 65:2 (2020), 316–332 | DOI | MR | Zbl

[5] T. G. Ergashev, “Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation”, Ufa Mathematical Journal, 10:4 (2018), 111–121 | DOI | MR | Zbl

[6] T. G. Ergashev, “The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika–Tomsk State University Journal of Mathematics and Mechanics, 2017, no. 50, 45–56 | DOI

[7] M. S. Salakhitdinov, T. G. Ergashev, “Integral representation of a generalized solution of the Cauchy problem in the class $R_{2k}^\lambda$ for a hyperbolic equation of the second kind”, Uzbekskiy matematicheskiy zhurnal–Uzbek Mathematical Journal, 1995, no. 1, 67–75

[8] I. L. Karol, “A boundary value problem for an equation of mixed elliptichyperbolic type”, Doklady AN SSSR–Reports of the Academy of Sciences of the USSR, 88:2 (1953), 197–200 | MR | Zbl

[9] T. G. Ergashev, “Generalized solutions of the degenerate hyperbolic equation of the second kind with a spectral parameter”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika–Tomsk State University Journal of Mathematics and Mechanics, 2017, no. 46, 41–49 | DOI

[10] K. B. Sabitov, S. L. Bibakova, “Construction of eigenfunctions of the Tricomi-Neumann problem for equations of mixed type with characteristic degeneration and their application”, Mathematical Notes, 74:1 (2003), 70–80 | DOI | MR | Zbl

[11] I. T. Tojiboev, “Boundary problems in a special domain for an equation of mixed type”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika–Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 56, 17–28 | DOI | MR

[12] R. S. Khairullin, The Tricomi problem for an equation of the second kind with strong degeneracy, Kazan University, Kazan, 2015

[13] N. K. Mamadaliev, “On representation of a solution to a modified Cauchy problem”, Siberian Mathematical Journal, 41:5 (2000), 889–899 | DOI | MR | Zbl

[14] V. M. Dolgopolov, M. V. Dolgopolov, I. N. Rodionova, “Construction of a special class of solutions for some differential equations of hyperbolic type”, Doklady Mathematics, 80 (2009), 860–866 | DOI | MR | Zbl

[15] M. M. Smirnov, Equations of the mixed type, Vysshaya shkola, M., 1985 | MR

[16] M. M. Smirnov, “Mixed problem for the equation $y^mu_{xx}+u_{yy}=0$”, Siberian Mathematical Journal, 4:5 (1963), 1150–1161 | MR | Zbl

[17] T. G. Ergashev, N. M. Safarbayeva, “Dirichlet problem for the multidimensional Helmholtz equation with one singular coefficient”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika–Tomsk State University Journal of Mathematics and Mechanics, 2019, no. 6, 55–67 | DOI

[18] M. S. Salakhitdinov, A. A. Abdullayev, “Nonlocal boundary value problem for a mixed type equation of the second kind”, Doklady AN Respubliki Uzbekistan–Reports of the Academy of Sciences of the Republic of Uzbekistan, 2013, no. 1, 3–5

[19] B. I. Islomov, A. A. Abdullayev, “On a problem for an elliptic type equation of the second kind with a conormal and integral condition”, Journal Nanosystems: Physics, Chemistry, Mathematics, 9:3 (2018), 307–318 | DOI

[20] M. S. Salokhitdinov, B. I. Islomov, The equations of the mixed type with two lines of degeneration, Mumtoz so'z, Tashkent, 2009

[21] M. M. Smirnov, Equations of the mixed type, Nauka, M., 1970

[22] N. I. Muskhelishvili, Singular integral equations, Nauka, M., 1968