# On green’s function of second darboux problem for hyperbolic equation

### Abstract

A definition and justify a method for constructing the Green’s function of the second Darboux problem for a two-dimensional linear hyperbolic equation of the second order in a characteristic triangle is given. In contrast to the (well-developed) theory of the Green’s function for self-adjoint elliptic problems, this theory has not yet been developed. And for the case of asymmetric boundary value problems such studies have not been carried out. It is shown that the Green’s function for a hyperbolic equation of the general form can be constructed using the Riemann-Green function for some auxiliary hyperbolic equation. The notion of the Green’s function is more completely developed for Sturm-Liouville problems for an ordinary differential equation, for Dirichlet boundary value problems for Poisson equation, for initial boundary value problems for a heat equation. For many particular cases, the Greens’ function has been constructed explicitly. However, many more problems require their consideration. In this paper, the problem of constructing the Green’s function of the second Darboux problem for a hyperbolic equation was investigated. The Green’s function for the hyperbolic problems differs significantly from the Green’s function of problems for equations of elliptic and parabolic types.### References

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[16] Derbissaly B. O, Sadybekov M. A. On Green’s function of Darboux problem for hyperbolic equation // Bulletin of KazNU. Series of mathematics, mechanics, computer science. - 2021. - V. 111, No. 3. - P. 79-94.

[17] Riley K.F, Hobson M.P., Bence S.J. Mathematical methods for physics and engineering // Cambridge University Press, 2010.

[2] Kal’menov T. Sh. Spectrum of a boundary - value problem with translation for the wave equation // Differential equations. - 1983. - V. 19, No. 1. - P. 64 - 66. [in Russian]

[3] Sadybekov M. A., Orynbasarov E. M. Baseness of the system of the eigenfunctions and associated functions with displacement of Lavrentev-Bitsadze equation // Doklady Mathematics. - 1992. - V. 324, No. 6. - P. 1152-1154. [in Russian]

[4] Orynbasarov E. M., Sadybekov M. A. The basis property of the system of eigen- and associated functions of a boundary value problem with shift for the wave equation // Math. Notes. - 1992. - V. 51, No. 5. - P. 482-484. [in Russian]

[5] Yessirkegenov N. A., Sadybekov M. A. Spectral properties of boundary-value problem with a shift for wave equation // Russian Math. (Iz. VUZ). - 2016. - V. 60, No. 3. - P. 41-46.

[6] Kreith K. Symmetric Green’s functions for a class of CIV boundary value problems // Canad. Math. Bull. - 1988. - V. 31. - P. 272-279.

[7] Kreith K. Establishing hyperbolic Green’s functions via Leibniz’s rule // SIAM Rev. - 1991. - V. 33. - P. 101-105.

[8] Kreith K. A self-adjoint problem for the wave equation in higher dimensions // Comput. Math. Appl. - 1991. - V. 21. - P. 129-132.

[9] Kreith K. Mixed selfadjoint boundary conditions for the wave equation // Differential equations and its applications (Budapest), Colloq. Math. Soc. Janos Bolyai, 62, North-Holland, Amsterdam. - 1991. - P. 219-226.

[10] Iraniparast N. A method of solving a class of CIV boundary value problems // Canad. Math. Bull. - 1992. - V. 35, No. 3. - P. 371-375.

[11] Iraniparast N. A boundary value problem for the wave equation // Int. J. Math. Math. Sci. - 1999. - V. 22, No. 4. - P. 835-845.

[12] Iraniparast N. A CIV boundary value problem for the wave equation // Appl. Anal. - 2000. - V. 76, No. 3-4. - P. 261-271.

[13] Haws L. Symmetric Green’s functions for certain hyperbolic problems // Comput. Math. Appl. - 1991. - V. 21, No. 5. - P. 65-78.

[14] Iraniparast N. Boundary value problems for a two-dimensional wave equation // Journal of Computational and Applied Mathematics. - 1994. - V. 55. - P. 349-356.

[15] Iraniparast N. A selfadjoint hyperbolic boundary-value problem // Electronic Journal of Differential Equations, Conference. - 2003. - V. 10. - P. 153-161.

[16] Derbissaly B. O, Sadybekov M. A. On Green’s function of Darboux problem for hyperbolic equation // Bulletin of KazNU. Series of mathematics, mechanics, computer science. - 2021. - V. 111, No. 3. - P. 79-94.

[17] Riley K.F, Hobson M.P., Bence S.J. Mathematical methods for physics and engineering // Cambridge University Press, 2010.

How to Cite

DERBISSALY, B..
On green’s function of second darboux problem for hyperbolic equation.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 116, n. 4, dec. 2022. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1166>. Date accessed: 28 jan. 2023. doi: https://doi.org/10.26577/JMMCS.2022.v116.i4.01.
Section

Mathematics

Keywords
Hyperbolic equation, nitial-boundary value problem, second Darboux problem, boundary condition, Green function, a characteristic triangle, Riemann–Green function