Abstract
In the work Tricomi problem was investigated for a parabolic-hyperbolic type equation in a mixed domain. If the parabolic degeneration line is a characteristic of a hyperbolic equation, then Tricomi problem for considered equation will not be uniquely solvable. Therefore, another formulation of Tricomi problem was proposed which the gluing condition is given as in [1, 2]. To study Tricomi problem in the hyperbolic part of the domain, \(R_{00}^{\lambda}\) class of the regular solutions of the view changed Cauchy problem for the equation of the hyperbolic part are introduced. An explicit form of the solution is found for the Cauchy problem from this class. The solution of the Tricomi problem in the hyperbolic part of the domain is found as a regular solution from the class \(R_{00}^{\lambda}\) of the view changed Cauchy problem, and in the parabolic part of the domain as the solution of the first boundary value problem. For proving the existence of the solution of the problem, the theory of second kind Volterra integral equations is used.
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REFERENCES
V. A. Eleev, ‘‘Generalized Tricomi problem for a mixed hyperbolic-parabolic equation with simultaneous degeneration of type and order,’’ Dokl. Akad. Nauk SSSR 253, 796–799 (1980).
N. K. Mamadaliev, ‘‘Tricomi problem for strongly degenerate equations of parabolic-hyperbolic typ,’’ Math. Notes 66, 310–315 (1999).
I. M. Gel’fand, ‘‘Some questions of analysis and differential equations,’’ Am. Math. Soc., Transl. 2 (26), 201–219 (1963).
L. A. Zolina, ‘‘On a boundary value problem for a model equation of hyperbolo-parabolic type,’’ USSR Comput. Math. Math. Phys. 6 (6), 63–78 (1966).
T. D. Dzhuraev, A. S. Sopuev, and M. Mamazhanov, Boundary-Value Problem for Equations of Hyperbolic-Parabolic Type (Fan, Tashkent, 1986) [in Russian].
V. A. Eleev, ‘‘The general Tricomi problem for a mixed equation of hyperbolic-parabolic typ with discountinuous coefficients,’’ Differ. Uravn. 17 (1), 58–72 (1981).
K. B. Sabitov, ‘‘Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation,’’ Math. Notes 102, 378–395 (2017).
K. B. Sabitov, ‘‘Tricomi problem for a mixed parabolic-hyperbolic equation in a rectangular domain,’’ Math. Notes 86, 249–254 (2009).
S. N. Sidorov, ‘‘Nonlocal problems for an equation of mixed parabolic-hyperbolic type with power degeneration,’’ Russ. Math. (Iz. VUZ) 59 (12), 46–55 (2015).
T. Sh. Kal’menov and M. A. Sadybekov, ‘‘On a Frankl-type problem for a mixed parabolic-hyperbolic equation,’’ Siber. Math. J. 58, 227–231 (2017).
M. S. Salakhitdinov and A. K. Urinov, Boundary-Value Problems for the Mixed Type Equations with Spectral Parameter (Fan, Tashkent, 1997) [in Russian].
M. M. Smirnov, Mixed Type Equation (Vysshaya Shkola, Moscow, 1985) [in Russian].
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Nauka, Moscow, 1981; Gordon and Breach Science, New York, 1986).
M. S. Salakhiddinov and N. K. Mamadaliev, ‘‘Tricomi problem for the elliptic-hyperbolic equation of the second kind,’’ Korean J. Math. 19, 111–127 (2011).
A. K. Urinov and A. B. Okboev, ‘‘A view-changed Cauchy Problem for a second kind degenerated hyperbolic equation,’’ in Differential Equations and Adjacent Problems, Proceedings of the International Conference, Sterlitamak, June 25–29, 2018, pp. 134–137.
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(Submitted by A. M. Elizarov)
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Okboev, A. Tricomi Problem for Second Kind Parabolic Hyperbolic Type Equation. Lobachevskii J Math 41, 58–70 (2020). https://doi.org/10.1134/S1995080220010096
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DOI: https://doi.org/10.1134/S1995080220010096