The Riemann matrix for some systems of the differential hyperbolic-type equations of the high order

J. O. Yakovleva

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 28 (2024) no. 4, pp. 799-808.

Résumé

Solutions to some boundary value problems for systems of hyperbolic partial differential equations can be constructed explicitly in terms of the Riemann matrix. In this regard, the question of explicitly constructing the Riemann matrix for high-order hyperbolic systems of equations is relevant. We consider a system of third-order hyperbolic partial differential equations with three independent variables. For the specified system, the Riemann matrix is constructed as a solution to a special Goursat problem. Furthermore, the Riemann matrix satisfies a Volterra integral equation. The Riemann matrix is expressed explicitly in terms of a hypergeometric function of a matrix argument. Similarly, a system of fourth-order hyperbolic partial differential equations with four independent variables is considered. These results are generalized for a system of hyperbolic partial differential equations of order $n$ that does not contain derivatives of order less than $n$.

Détail

Keywords: system of $n$-th order hyperbolic PDEs, hypergeometrical function of matrix argument
Mots-clés : Riemann matrix, Goursat problem

Bibliographie

[1] Bitsadze A. V., Some Classes of Partial Differential Equations, Advanced Studies in Contemporary Mathematics, 4, Gordon Breach Science Publ., New York, 1988, xi+504 pp. | Zbl | Zbl

[2] Soldatov A. P., Shkhanukov M. Kh., “Boundary value problems with Samarskiy general nonlocal condition for higher-order pseudoparabolic equations”, Soviet Math. Dokl., 36:3 (1988), 507–511 | MR | Zbl

[3] Zikirov O. S., “Local and nonlocal boundary-value problems for third-order hyperbolic equations”, J. Math. Sci., 175:1 (2011), 104–123 | DOI | Zbl

[4] Zeitsch P. J., “On the Riemann function”, Mathematics, 6:12 (2018), 316 | DOI

[5] Zhegalov V. I., Mironov A. N., “Cauchy problems for two partial differential equations”, Russian Math. (Iz. VUZ), 46:5 (2002), 21–28 | MR | Zbl

[6] Mironov A. N., Mironova L. B., Yakovleva J. O., “The Riemann method for equations with a dominant partial derivative (A Review)”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 25:2 (2021), 207–240 (In Russian) | DOI | Zbl

[7] 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. (In Russian)

[8] Scott E. J., “The Riemann function for a class of equations of the form $\dfrac{\partial^2 v}{\partial x \partial y} +\nu(x)\mu(y)v=0$”, Ganita, 26:1 (1975), 19–28 | Zbl

[9] Andreev A. A., Yakovleva J. O., “The Goursat-type problem for a hyperbolic equation and system of third order hyperbolic equations”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:1 (2019), 186–194 (In Russian) | DOI | Zbl

[10] Yakovleva J. O., Tarasenko A. V., “The solution of Cauchy problem for thehyperbolic differential equations of the fourth order by the Riman method”, Vestn. Samar. Univ. Estestvennon. Ser. [Vestnik of Samara University. Natural Science Series], 25:3, 33–38 (In Russian) | DOI

[11] Gantmakher F. R., Teoriia matrits [Theory of Matrices], Nauka, Moscow, 1988, 549 pp. (In Russian) | Zbl

[12] Volterra V., Theory of Functionals and of Integral and Integro-Differential Equations, Dover Publ., New York, 1959, 226 pp. | Zbl

[13] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions, v. I, ed. H. Bateman, McGraw-Hill Book Co, New York – Toronto – London, 1953, 302 pp. | MR | MR | Zbl