Abstract
The Darboux problem and the definition of the Riemann–Hadamard function are presented for the Bianchi equation of the fourth order—a linear fourth-order equation with four independent variables that has a dominant derivative not containing multiple differentiation with respect to any of the independent variables. Sufficient conditions on the coefficients of this equation under which its Riemann–Hadamard function allows explicit construction in terms of hypergeometric functions are obtained.
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REFERENCES
Bitsadze, A.V., Nekotorye klassy uravnenii v chastnykh proizvodnykh (Some Classes of Partial Differential Equations), Moscow: Nauka, 1981.
Moiseev, E.I., Uravneniya smeshannogo tipa so spektral’nym parametrom (Mixed Type Equations with a Spectral Parameter), Moscow: Izd. Mosk. Gos. Univ., 1988.
Sabitov, K.B. and Sharafutdinova, G.G., Cauchy–Goursat problems for a degenerate hyperbolic equation, Russ. Math., 2003, vol. 47, no. 5, pp. 19–27.
Dzhokhadze, O.M. and Kharibegashvili, S.S., Some properties and applications of the Riemann and Green–Hadamard functions for linear second-order hyperbolic equations, Differ. Equations, 2011, vol. 47, no. 4, pp. 471–487.
Mironov, A.N., Darboux problem for the third-order Bianchi equation, Math. Notes, 2017, vol. 102, no. 1, pp. 53–59.
Mironov, A.N., Construction of the Riemann–Hadamard function for the three-dimensional Bianchi equation, Russ. Math., 2021, vol. 65, no. 3, pp. 68–74.
Zhegalov, V.I. and Sevast’yanov, V.A., Goursat problem in a four-dimensional space, Differ. Uravn., 1996, vol. 32, no. 10, pp. 1429–1430.
Sevast’yanov, V.A., On one instance of Cauchy problem, Differ. Uravn., 1998, vol. 34, no. 12, pp. 1706–1707.
Mironov, A.N., The construction of the Riemann function for a fourth-order equation, Differ. Equations, 2001, vol. 37, no. 12, pp. 1787–1791.
Utkina, E.A., On the general case of the Goursat problem, Russ. Math., 2005, vol. 49, no. 8, pp. 53–58.
Mironov, A.N., The Riemann method for equations with leading partial derivative in \( \mathbb {R}^n\), Sib. Math. J., 2006, vol. 47, no. 3, pp. 481–490.
Mironov, A.N., On the Riemann method for solving one mixed problem, Vestn. Samar. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki, 2007, no. 2, pp. 27–32.
Koshcheeva, O.A., Construction of the Riemann function for the Bianchi equation in an \(n \)-dimensional space, Russ. Math., 2008, vol. 52, no. 9, pp. 35–40.
Mironov, A.N., Darboux Problem for the Fourth-Order Bianchi Equation, Differ. Equations, 2021, vol. 57, no. 3, pp. 328–341.
Ovsyannikov, L.V., Gruppovoi analiz differentsial’nykh uravnenii (Group Analysis of Differential Equations), Moscow: Nauka, 1978.
Mironov, A.N., On the Laplace invariants of a fourth-order equation, Differ. Equations, 2009, vol. 45, no. 8, pp. 1168–1173.
Mironov, A.N., On some classes of fourth-order Bianchi equations with constant ratios of Laplace invariants, Differ. Equations, 2013, vol. 49, no. 12, pp. 1524–1533.
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Higher Transcendental Functions (Bateman Manuscript Project), New York: McGraw-Hill, 1953. Translated under the title: Vysshie transtsendentnye funktsii. T. 1, Moscow: Nauka, 1973.
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Translated by V. Potapchouck
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Mironov, A.N., Yakovleva, Y.O. Constructing the Riemann–Hadamard Function for a Fourth-Order Bianchi Equation. Diff Equat 57, 1142–1149 (2021). https://doi.org/10.1134/S0012266121090032
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DOI: https://doi.org/10.1134/S0012266121090032