Short Communication
Differential Equations and Mathematical Physics

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

J. O. Yakovleva

Samara State Technical University, Samara, 443100, Russian Federation

Abstract: 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 $n$.

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

UDC: 517.956.3

MSC: 35K70, 35K35

Received: May 13, 2024
Revised: October 29, 2024
Accepted: November 1, 2024
First online: December 25, 2024

DOI: 10.14498/vsgtu2092

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