We consider the first boundary-value problem for a third-order equation with multiple characteristics u xxx − u yy = f(x, y) in a domain D = {(x, y):0 < x < p, 0 < y < l}. The unique solvability of the problem is proved by the method of energy integrals and its explicit solution is constructed by the method of Green functions.
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References
H. Block, “Sur les équations linéaires aux dérivées partielles a caractéristiques multiples,” Ark. Mat., Astron., Fys. Note 1, 7, No. 13, 1–34 (1912); Ark. Mat., Astron., Fys. Note 2, 7, No. 21, 1–30 (1912); Ark. Mat., Astron., Fys. Note 3, 8, No. 23, 1–51 (1912–1913).
E. Del Vecchio, “Sulle equazioni Z xxx − Z y + φ 1(x, y) = 0, Z xxx − Z yy + φ 2(x, y) = 0;” Mem. Real Acad. Cienc. Torino. Ser. 2, 66, 1–41 (1915).
E. Del Vecchio, “Sur deux problèmes d’intégration pour les équations paraboliques Z xxx −Z y =0, Z xxx −Z yy =0;” Ark. Mat., Astron., Fys., 11, 32–43 (1916).
L. Cattabriga, “Potenziali di linea e di dominio per equazioni non paraboliche in due variabili a caratteristiche multiple,” Rend. Semin. Mat. Univ. Padova, 31, 1–45 (1961).
T. D. Dzhuraev and Yu. P. Apakov, “On the self-similar solution of a third-order equation with multiple characteristics,” Vestn. Samar. Gos. Univ., Fiz.-Mat. Nauk., No. 2(15), 18–26 (2007).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1954).
M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, Appl. Math., Ser. 55 (1964).
O. S. Ryzhov, “Asymptotic picture of the sonic flow of a viscous heat-conducting gas around bodies of revolution,” Prikl. Mat. Mekh., 29, Issue 6, 1004–1014 (1965).
V. N. Diesperov, “On the Green function of a linearized viscous transonic equation,” Zh. Vych. Mat. Mat. Fiz., 12, No. 5, 1265–1279 (1972).
T. D. Dzhuraev and Yu. P. Apakov, “On the theory of the third-order equation with multiple characteristics containing the second time derivative,” Ukr. Mat. Zh., 62, No. 1, 40–51 (2010); English translation: Ukr. Math. J., 62, No. 1, 43–55 (2010).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1966).
M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko, Integral Equations [in Russian], Nauka, Moscow (1976).
L. D. Kudryavtsev, A Course of Mathematical Analysis [in Russian], Vol. 1, Vysshaya Shkola, Moscow (1989).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 1, pp. 3–13, January, 2012.
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Apakov, Y.P. On the solution of a boundary-value problem for a third-order equation with multiple characteristics. Ukr Math J 64, 1–12 (2012). https://doi.org/10.1007/s11253-012-0625-1
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DOI: https://doi.org/10.1007/s11253-012-0625-1