Geodesic
Cauchy Problem For the System Of the General Hyperbolic Differential Equations
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 4 (2014), pp. 7-15.

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We consider the Cauchy problem for the hyperbolic differential equation of the forth order with nonmultiple characteristics. We generalize this problem from the similar Cauchy problem for the hyperbolic differential equation of the third order with nonmultiple characteristics which solution was constructed as an analogue of D'Alembert formula. We obtain the regular solution of the Cauchy problem for the hyperbolic differential equation of the forth order with nonmultiple characteristics in an explicit form. This solution is also an analogue of D'Alembert formula. The existence and uniqueness theorem for the regular solution of the Cauchy problem for the hyperbolic differential equation of the forth order with nonmultiple characteristics is formulated as the result of the research. In the paper we consider the Cauchy problem for the system of the general hyperbolic differential equations of the forth order with nonmultiple characteristics.
Keywords: hyperbolic differential equation of the forth order, nonmultiple characteristics, Cauchy problem, system of general hyperbolic differential equations of the forth order.
Mots-clés : D'Alembert formula 
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A. A. Andreev; J. O. Yakovleva. Cauchy Problem For the System Of the General Hyperbolic Differential Equations. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 4 (2014), pp. 7-15. http://geodesic.mathdoc.fr/item/VSGTU_2014_4_a0/

[1] Rieman B., “Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite (Aus dem achten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 1860.)”, Bernard Riemann's Gesammelte mathematische Werke und wissenschaftlicher Nachlass, eds. R. Dedekind, H. M. Weber, BiblioLife, United States, 2009, 145–164 (In German) | DOI

[2] Ali Raeisian S. M., “Effective Solution of Riemann Problem for Fifth Order Improperly Elliptic Equation on a Rectangle”, AJCM, 2:4 (2012), 282–286 | DOI | MR

[3] “Sur les systèmes linéaires aux dérivées partielles du premier ordre deux variables indépendantes à caractéristiques réelles et distinetes”, Arkiv f. Mat., Astr. och Fys., 5:1 (1909), 13 pp. (In Swedish) | Zbl

[4] Petrovskiy I. G., Izbrannye trudy. Sistemy uravnenii s chastnymi proizvodnymi. Algebraicheskaia geometriia [Selected works. Systems of partial differential equations. Algebraic geometry], Nauka, Moscow, 1986, 500 pp. (In Russian) | MR | Zbl

[5] Yakovleva J. O., “The analogue of D'Alembert formula for hyperbolic differential equation of the third order with nonmultiple characteristics”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 1(26), 247–250 (In Russian) | DOI

[6] Nikolov A., Popivanov N., “Singular solutions to Protter's problem for (3+1)-D degenerate wave equation” (8–13 June 2012; Sozopol, Bulgaria), AIP Conf. Proc., 1497, 2012, 233–238 | DOI

[7] Korzyuk V. I., Cheb E. S., Le Thi Thu, “Solution of the mixed problem for the biwave equation by the method of characteristics”, Tr. Inst. Mat., 18:2 (2010), 36–54 (In Russian) | MR | Zbl

[8] Yakovleva J. O., “One characteristic problem for the general hyperbolic differential equation of the third order with nonmultiple characteristics”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 3(28), 180–183 (In Russian) | DOI | MR | Zbl

[9] Andreev A. A., Yakovleva J. O., “The Characteristic Problem for one Hyperbolic Differentional Equation of the Third Order with Nonmultiple Characteristics”, Izv. Saratov. Univ. (N.S.), Ser. Math. Mech. Inform., 13:1(2) (2013), 3–6 (In Russian) | Zbl

[10] Tikhonov A. N., Samarskii A. A., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Nauka, Moscow, 1972, 736 pp. (In Russian) | MR | Zbl

[11] Bitsadze A. V., Uravneniia matematicheskoi fiziki [Equations of mathematical physics], Nauka, Moscow, 1982, 336 pp. (In Russian) | MR | Zbl

[12] Bellman R., Introduction to matrix analysis, 2nd ed., Reprint of the 1970 Orig., Classics in Applied Mathematics, 19, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997, xxviii+403 pp. | MR | Zbl

[13] Andreev A. A., Yakovleva Yu. O., “Kharakteristicheskaya zadacha dlya sistemy giperbolicheskikh differentsialnykh uravnenii tretego poryadka obschego vida s nekratnymi kharakteristikami”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2013, no. 1(30), 31–36 | DOI

[14] Gantmakher F. R., Teoriia matrits [Theory of matrices], Nauka, Moscow, 1988, 549 pp. (In Russian) | MR | Zbl