Unique solvability of a Bitsadze–Samarskiy type problem for equations with discontinuous coefficient

A. G. Ezaova

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 50-58

Résumé

In this paper, unique solvability of a Bitsadze–Samarsky type problem for a third-order equation with discontinuous coefficients in a simply connected domain is investigated. The boundary condition of the problem contains the fractional integro-differentiation operator with the Gauss hypergeometric function. Under certain inequality type constraints on given functions and orders of fractional derivatives in the boundary condition, the energy integrals method enables one to proved the uniqueness of the solution of the problem. The functional relations between the trace of the desired solution and its derivative are obtained, which are brought to the degeneration line from the hyperbolic and parabolic parts of the mixed region. Under the conditions of the uniqueness theorem the existence of a solution to the problem is proved by equivalent reduction to the second kind Fredholm integral equations with the derivative of the sought function as an unknown, the unconditional solvability of which is deduced from the uniqueness of the solution of the problem. The limits of the change of orders of fractional integro-differential operators in which the solution of the problem exists and is unique are also determined. The effect of the coefficient of the lowest derivative in the equation on the solvability of the problem is established.

Bibliographie

[1] Tricomi F., On Linear Equations of Mixed Type, Gostekhizdat, M., 1947 (in Russian)

[2] Gellerstedt S., Sur un Probleme aux Limits Pour One Equation Linear aux Derives Partielles du Second Order de Type Mixed, Thesis, Uppsala, 1935

[3] Frankl F. I., “On the Problems of Chaplygin for Mixed Before and Supersonic Flows”, Izv. Akad. Nauk SSSR Ser. Mat., 9:2 (1945), 121–142 (in Russian)

[4] Frankl F. I., “Generalization of the Tricomi Problem and Its Application to the Solution of the Direct Problem of Laval Nozzle Theory”, Mat. Sb. (N.S.), 54 (96):2 (1961), 225–236 (in Russian) | Zbl

[5] Bitsadze A. V., Some Classes of Partial Differential Equations, Nauka Publ., M., 1981, 488 pp. (in Russian)

[6] Nakhushev A. M., “On Some Boundary Value Problems for Hyperbolic Equations and Equations of Mixed Type”, Differential Equations, 5:1 (1969), 44–59 (in Russian) | MR | Zbl

[7] Nakhushev A. M., “New Boundary Value Problem for a Degenerate Hyperbolic Equation”, Dokl. Akad. Nauk SSSR, 187:4 (1969), 736–739 (in Russian) | Zbl

[8] Saigo M., “A Certain boundary value problem for the Euler–Dorboux equation. III”, Mathematical Japonica, 26:1 (1981), 103–119 | MR | Zbl

[9] Repin O. A., “On a Nonlocal Boundary Value Problem with M. Saigo Operators for the Generalized Equation of Euler–Poisson–Darboux Equations”, Collection of Scientific Works. A Institute of Mathematics of Ukraine, Integral Transforms and Boundary Value Problems, 13, 1996, 175–181 (in Russian)

[10] Repin O. A., Kumykova S. K., “On the Problem with Generalized Operators of Fractional Differentiation for Degenerate Hyperbolic Equation Within the Domain”, Vestnik of Samara State University. Natural Science Series, 2012, no. 9(100), 52–60 (in Russian) | Zbl

[11] Nakhushev A. M., Displacement Problems for Partial Differential Equations, Nauka Publ., M., 2006, 287 pp. (in Russian)

[12] Ezaova A. G., “On One Nonlocal Problem for Mixed Type Equation of the Third Order”, Proceedings of the Kabardino-Balkarian State University, 1:4 (2011), 26–31 (in Russian)

[13] Repin O. A., Kumykova S. K., “Task with the Operators of Riemann–Liouville for the Mixed Type Equation of the Third Order”, Vestnik of Samara State Technical University. Ser. Physical and Mathematical Sciences, 20:1 (2016), 43–53 (in Russian) | DOI | Zbl

[14] Ezaova A. G., Dumaeva L. V., “On One Inner-Boundary Value Problem for the Equation of the Third Order with a Group of Younger Members”, Fundamental Research, 2015, no. 2(27), 6032–6036 (in Russian)

[15] Bitsadze A. V., “On Mixed-Type Equations in Three-Dimensional Domains”, Dokl. Akad. Nauk SSSR, 143:5 (1962), 1017–1019 (in Russian) | Zbl

[16] Smirnov M. M., Equations of Mixed Type, Vysshaya shkola, M., 1985, 304 pp. (in Russian) | MR

[17] Smirnov M. M., Degenerate Elliptic and Hyperbolic Equations, Nauka Publ., M., 1966, 292 pp. (in Russian)

[18] Samko S. G., Kilbas A. A., Marichev O. I., Integrals and Derivatives of FractiOnal Order and Some of Their Applications, Nauka i tekhnika, Minsk, 1987, 688 pp. (in Russian)