Hypothesis on unification of solution of the Cauchy problem for overdetermined systems of differential equations

M. L. Zaytsev   ; V. B. Akkerman

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 4, pp. 12-25

Résumé

In this paper, we study the possibility of the existence of a universal solution of the Cauchy problem for the partial differential equation (PDE) systems in the case if this system is overdetermined so that the new overdetermined system of PDE contains all solutions of the initial PDE system and, in addition, reduces to the ordinary differential equation (ODE) systems, whose solution is then found. To do this, the article discusses the modification of the method of finding particular solutions for any overdetermined systems of differential equations by reduction to overdetermined systems of implicit equations. In the previous papers of the authors, a method was proposed for finding particular solutions for overdetermined PDE systems. In this method, in order to find solutions it is necessary to solve systems of ordinary implicit equations. In this case, it can be shown that the solutions that we need cannot depend on a continuous parameter, i.e. they are no more than countable. In advance, there is a need for such an overriding of the systems of differential equations, so that their general solutions are no more than countable. Such an initial overdetermination is rather difficult to achieve. However, the proposed method also allows to reduce the overdetermined systems of differential equations not only up to systems of implicit equations, but also up to the PDE systems of dimension less than that of the initial systems of PDE. In particular, under certain conditions, reduction to the ODE systems is possible. It is proposed to choose solutions for the overdetermined PDE systems using the parameterized Cauchy problem, which is posed for parameterized ODE systems under certain conditions. The solution of this Cauchy problem is some function of the initial data and their derivatives. In order to find the solution of any corresponding Cauchy problem for the initial system of PDE, it is sufficient to calculate the universal solver for the reduced ODE system once. In this case, the solution will not only exist and be unique, but will also depend continuously on the initial data, since this holds for ODE systems. The purpose of this paper is to study the Cauchy problem with the possibility of its universalization and the parameterized Cauchy problem as a whole for arbitrary PDE systems.

Détail

Keywords: overdetermined systems of differential equations, partial differential equation, ordinary differential equations, dimension of differential equations, Cauchy problem, parametric solutions of the systems of differential equations.

Bibliographie

[1] Ya. B. Zel'dovich, G. I. Barenblatt, V. B. Librovich, G. M. Makhviladze, The Mathematical Theory of Combustion and Explosion, Nauka, M., 1980, 478 pp. (in Russ.) | MR

[2] G.I. Sivashinsky, “Nonlinear Analysis of Hydrodynamic Instability in Laminar Flames I. Derivation of basic equations”, Acta Astronautica, 4:11–12 (1977), 1177–1206 | DOI | MR | Zbl

[3] E. Ott, “Nonlinear Evolution of the Rayleigh-Taylor Instability of a Thin Layer”, Phys. Rev. Lett., 29:21 (1972), 1429–1432 | DOI

[4] V. Bychkov, M. Zaytsev, V. Akkerman, “Coordinate-Free Description of Corrugated Flames with Realistic Density Drop at the Front”, Phys. Rev. E, 68:2 (2003), 026312–026324 | DOI | MR

[5] G. Joulin, H. El-Rabii, K. Kazakov, “On-Shell Description of Unsteady Flames”, J. Fluid Mech., 608 (2008), 217–242 | DOI | MR | Zbl

[6] V. B. Akkerman, M. L. Zaytsev, “Dimension Reduction in Fluid Dynamics Equations”, Computational Mathematics and Mathematical Physics, 51:8 (2011), 1418–1430 | DOI | MR | Zbl

[7] M. L. Zaytsev, V. B. Akkerman, “Hypothesis on reduction of overdetermined systems of differential equations and its application to equations of hydrodynamics”, Proceedings of Voronezh State University. Series: Physics. Mathematics, 2015, no. 2, 5–27 (in Russ.)

[8] M. L. Zaytsev, V. B. Akkerman, “Another Method for Finding Particular Solutions of Equations of Mathematical Physics”, Science Journal of VolSU. Mathematics. Physics, 2016, no. 6(37), 119–127 (in Russ.) | DOI

[9] R. Courant, D. Hilbert, Methods of Mathematical Physics, v. 2, Interscience Publ., New York–London, 1962, 830 pp. | DOI | MR | Zbl

[10] A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Nauka, M., 1966, 724 pp. (in Russ.) | MR

[11] M. V. Fedoryuk, Odinary Differential Equations, Lan' Publ., SPb., 2003, 447 pp. (in Russ.)

[12] M. L. Zaytsev, V. B. Akkerman, “Transformation of Systems of Partial Differential Equations to Systems of Quasilinear and Linear Differential Equations. Their Reduction and Unification”, Mathematical Physics and Computer Simulation, 21:1 (2018), 18–33 (in Russ.) | DOI | MR