Abstract:
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.
Keywords:
generalized solution, Poincare-Tricomi problem, equation of the second kind, integral equation, energy integral method, Green's function.
Citation:
A. A. Abdullayev, T. G. Ergashev, “Poincare–Tricomi problem for the equation of a mixed elliptico-hyperbolic type of second kind”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2020, no. 65, 5–21
\Bibitem{AbdErg20}
\by A.~A.~Abdullayev, T.~G.~Ergashev
\paper Poincare--Tricomi problem for the equation of a mixed elliptico-hyperbolic type of second kind
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2020
\issue 65
\pages 5--21
\mathnet{http://mi.mathnet.ru/vtgu773}
\crossref{https://doi.org/10.17223/19988621/65/1}
Linking options:
https://www.mathnet.ru/eng/vtgu773
https://www.mathnet.ru/eng/vtgu/y2020/i65/p5
This publication is cited in the following 15 articles:
Alimardon Elmurodov, Muyassar Hidoyatova, Zardila Shakhobiddiova, NOVEL TRENDS IN RHEOLOGY IX, 2997, NOVEL TRENDS IN RHEOLOGY IX, 2023, 020042
V. Vakhabov, A. A. Fayziev, D. Bazarov, “Statistical analysis and forecasting of cotton yield dynamics in Bukhara region”, E3S Web of Conf., 401 (2023), 03051
V. V. Vakhabov, M. A. Hidoyatova, D. Bazarov, “The method of correlation analysis in agriculture”, E3S Web of Conf., 401 (2023), 05053
A. A. Abdullaev, N. M. Safarbayeva, B. Z. Usmonov, D. Bazarov, “On the unique solvability of a nonlocal boundary value problem with the poincaré condition”, E3S Web of Conf., 401 (2023), 03048
Akmaljon Abdullayev, Nigora Safarbayeva, Salohiddin Shamsitdinov, VII INTERNATIONAL CONFERENCE “SAFETY PROBLEMS OF CIVIL ENGINEERING CRITICAL INFRASTRUCTURES” (SPCECI2021), 2701, VII INTERNATIONAL CONFERENCE “SAFETY PROBLEMS OF CIVIL ENGINEERING CRITICAL INFRASTRUCTURES” (SPCECI2021), 2023, 050003
Akmal Abdullayev, Muyassar Hidoyatova, Nigora Safarbayeva, D. Bazarov, “About one boundary-value problem arising in modeling dynamics of groundwater”, E3S Web of Conf., 365 (2023), 01016
Anvar Hasanov, Valijon Vahobov, NOVEL TRENDS IN RHEOLOGY IX, 2997, NOVEL TRENDS IN RHEOLOGY IX, 2023, 020061
B. I. Islomov, A. A. Abdullayev, “A boundary value problem with a conormal derivative for the mixed type equation of second kind with a conjugation condition of the Frankl type”, Russian Math. (Iz. VUZ), 66:9 (2022), 11–25
B. A. Khudayarov, F. Zh. Turaev, K. Ruzmetov, A. A. Tukhtaboev, PROCEEDINGS OF THE III INTERNATIONAL CONFERENCE ON ADVANCED TECHNOLOGIES IN MATERIALS SCIENCE, MECHANICAL AND AUTOMATION ENGINEERING: MIP: Engineering-III – 2021, 2402, PROCEEDINGS OF THE III INTERNATIONAL CONFERENCE ON ADVANCED TECHNOLOGIES IN MATERIALS SCIENCE, MECHANICAL AND AUTOMATION ENGINEERING: MIP: Engineering-III – 2021, 2021, 050005
A Abdullayev, K Zhuvanov, K Ruzmetov, “A generalized solution of a modified Cauchy problem of class R
2 for a hyperbolic equation of the second kind”, J. Phys.: Conf. Ser., 1889:2 (2021), 022121
Akmaljon Abdullayev, Kholsaid Kholturayev, Nigora Safarbayeva, D. Bazarov, “Exact method to solve of linear heat transfer problems”, E3S Web Conf., 264 (2021), 02059
Olim Kucharov, Fozil Turaev, Sergey Leonov, Kholida Komilova, D. Bazarov, “Numerical study of nonlinear problems in the dynamics of thin-walled structural elements”, E3S Web Conf., 264 (2021), 05056
Andriy A. Verlan, O. Kucharov, F. Turaev, E. Yusupov, PROCEEDINGS OF THE III INTERNATIONAL CONFERENCE ON ADVANCED TECHNOLOGIES IN MATERIALS SCIENCE, MECHANICAL AND AUTOMATION ENGINEERING: MIP: Engineering-III – 2021, 2402, PROCEEDINGS OF THE III INTERNATIONAL CONFERENCE ON ADVANCED TECHNOLOGIES IN MATERIALS SCIENCE, MECHANICAL AND AUTOMATION ENGINEERING: MIP: Engineering-III – 2021, 2021, 040009
Akmaljon Abdullayev, Muyassar Hidoyatova, PROCEEDINGS OF THE III INTERNATIONAL CONFERENCE ON ADVANCED TECHNOLOGIES IN MATERIALS SCIENCE, MECHANICAL AND AUTOMATION ENGINEERING: MIP: Engineering-III – 2021, 2402, PROCEEDINGS OF THE III INTERNATIONAL CONFERENCE ON ADVANCED TECHNOLOGIES IN MATERIALS SCIENCE, MECHANICAL AND AUTOMATION ENGINEERING: MIP: Engineering-III – 2021, 2021, 070021
A Tukhtaboev, Sergey Leonov, Fozil Turaev, Kudrat Ruzmetov, D. Bazarov, “Vibrations of dam–plate of a hydro-technical structure under seismic load”, E3S Web Conf., 264 (2021), 05057
Citation in format AMSBIB
Contact us: