On some new solutions of the multi-dimensional first order partial differential equation with power-law non-linearities

I. V. Rakhmelevich

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2015), pp. 18-25

Résumé

Investigations of nonlinear partial differential equations of the first order with an arbitrary number of independent variables are an important part of up-to-date mathematical physics. For many equations of this class, only solutions of the simplest kind are known, in particular, solutions of the travelling wave type. The present work is devoted to finding solutions of a more complex form for the multi-dimensional equation of the first order with power-law non-linearity in derivatives. To solve this problem, in this paper we propose a new variant of the method of separation of variables — the method of two-level functional separation of variables. The characteristic feature of this method is that the desired function depends on a superposition of functions of the first and second levels of one variable, and these functions are determined as the result of solving some ordinary differential equations. Based on the method proposed in the paper, new exact solutions of the considered equation are obtained in an implicit form. The solutions contain some generalized polynomials of independent variables. Conditions of the existence of these solutions are specified. The results of this work can be generalized to other non-linear first order equations and equations of higher orders with many independent variables.

Détail

Keywords: partial differential equation, functional separation of variables, power-law non-linearity.

Bibliographie

[1] Zaytsev V. F., Polyanin A. D., Spravochnik po differentsial'nym uravneniyam s chastnymi proizvodnymi pervogo poryadka, Fizmatlit Publ., M., 2003 (in Russian)

[2] Kamke E., Spravochnik po differentsial'nym uravneniyam v chastnykh proizvodnykh pervogo poryadka, Nauka Publ., M., 1966, 260 pp. (in Russian)

[3] Polyanin A. D., Zaytsev V. F., Spravochnik po nelineynym uravneniyam matematicheskoy fiziki: tochnye resheniya, Fizmatlit Publ., M., 2002, 432 pp. (in Russian) | MR

[4] Polyanin A. D., Zhurov A. I., “Obobshchennoe i funktsional'noe razdelenie peremennykh v matematicheskoy fizike i mekhanike”, Doklady RAN, 382:5 (2002), 606–611 (in Russian) | MR | Zbl

[5] Rakhmelevich I. V., “O primenenii metoda razdeleniya peremennykh k uravneniyam matematicheskoy fiziki, soderzhashchim odnorodnye funktsii ot proizvodnykh”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2013, no. 3(23), 37–44 (in Russian)

[6] Rakhmelevich I. V., “Ob uravneniyakh matematicheskoy fiziki, soderzhashchikh mul'tiodnorodnye funktsii ot proizvodnykh”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2014, no. 1(27), 42–50 (in Russian)

[7] Rakhmelevich I. V., “O resheniyakh mnogomernogo uravneniya Klero s mul'tiodnorodnoy funktsiey ot proizvodnykh”, Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika, mekhanika, informatika, 14:4-1 (2014), 374–381 (in Russian) | Zbl

[8] Rakhmelevich I. V., “O dvumernykh giperbolicheskikh uravneniyakh so stepennoy nelineynost'yu po proizvodnym”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2015, no. 1(33), 12–19 (in Russian)

[9] Miller J. (Jr.), Rubel L. A., “Functional separation of variables for Laplace equations in two dimensions”, Journal of Physics A, 26 (1993), 1901–1913 | MR | Zbl

[10] Zhdanov R. Z., “Separation of variables in the non-linear wave equation”, Journal of Physics A, 27 (1994), L291–L297 | MR | Zbl

[11] Grundland A. M., Infeld E., “A family of non-linear Klein–Gordon equations and their solutions”, Journal of Mathematical Physics, 33:7 (1992), 2498–2503 | MR | Zbl