Abstract
The existence and uniqueness of the solution of the Darboux problem are proved. The solution of the Darboux problem is constructed in terms of a function similar to the Riemann–Hadamard function.
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References
A. V. Bitsadze, Some Classes of Partial Differential Equations (Nauka, Moscow, 1981) [in Russian].
E. I. Moiseev, “An integral representation of the solution of the Darboux problem,” Mat. Zametki 32 (2), 175–186 (1982) [Math. Notes 32 (1–2), 568–573 (1982)].
E. I. Moiseev, “Approximation of the classical solution of the Darboux problem by smooth solutions,” Differ. Uravn. 20 (1), 73–87 (1984).
E. I. Moiseev, Equations of Mixed Type with a Spectral Parameter (Izd. Moskov. Univ., Moscow, 1988) [in Russian].
K. B. Sabitov, “Construction in explicit form of solutions of the Darboux problems for the telegraph equation and its application in the inversion of integral equations. I.,” Differ. Uravn. 26 (6), 1023–1032 (1990) [Differ. Equations 26 (6), 747–755 (1990)].
K. B. Sabitov and G. G. Sharafutdinova, “Cauchy–Goursat problems for a degenerate hyperbolic equation,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 21–29 (2003) [Russian Math. (Iz. VUZ) 47 (5), 19–27 (2003)].
O. M. Dzhokhadze and S. S. Kharibegashvili, “Some properties of the Riemann and Green-Hadamard functions for second-order linear hyperbolic equations and their applications,” Differ.Uravn. 47 (4), 477–492 (2011) [Differ. Equations 47 (4), 471–487 (2011)].
V. I. Zhegalov, “A three-dimensional analogue of the Goursat problem,” in Nonclassical Equations and Equations of Mixed Type (Inst. Mat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1990), pp. 94–98 [in Russian].
V. F. Volkodavov, N. Ya. Nikolaev, O. K. Bystrova, and V. N. Zakharov, Riemann Functions for Some Differential Equations in n-Dimensional Euclidean Space and Their Applications (Samara Univ., Samara, 1995) [in Russian].
V. A. Sevast’yanov, “The Riemann method for a third-order three-dimensional hyperbolic equation,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 69–73 (1997) [Russian Math. (Iz. VUZ) 41 (5), 66–70 (1997)].
O. M. Dzhokhadze, “A Darboux-type problem in the trihedral angle for a third-order equation of hyperbolic type,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 22–30 (1999) [Russian Math. (Iz. VUZ) 43 (3), 20–28 (1999)].
V. I. Zhegalov and A. N. Mironov, Differential Equations with Higher-Order Derivatives, in Monographs in Mathematics (Kazan) (Izd. Kazan. Mat. Obshch., Kazan, 2001), Vol. 5 [in Russian].
A. N. Mironov, “Classes of Bianchi equations of third order,” Mat. Zametki 94 (3), 389–400 (2013) [Math. Notes 94 (3–4), 369–378 (2013)].
M. K. Fage, “Cauchy problem for the Bianchi equation,” Mat. Sb. 45 (87) (3), 281–322 (1958).
V. A. Zorich, Mathematical Analysis (Nauka, Moscow, 1984), Pt. 2 [in Russian].
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Original Russian Text © A. N. Mironov, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 1, pp. 64–71.
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Mironov, A.N. Darboux problem for the third-order Bianchi equation. Math Notes 102, 53–59 (2017). https://doi.org/10.1134/S0001434617070069
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DOI: https://doi.org/10.1134/S0001434617070069