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 Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2019, Volume 11, Issue 4, Pages 12–25 (Mi vyurm424)

Mathematics

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

M. L. Zaytseva, V. B. Akkermanb

a Moscow, Russian Federation
b West Virginia University, Morgantown, USA

Abstract: 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.

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.

DOI: https://doi.org/10.14529/mmph190402

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UDC: 519.635

Citation: M. L. Zaytsev, V. B. Akkerman, “Hypothesis on unification of solution of the Cauchy problem for overdetermined systems of differential equations”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 11:4 (2019), 12–25

Citation in format AMSBIB
\Bibitem{ZayAkk19} \by M.~L.~Zaytsev, V.~B.~Akkerman \paper Hypothesis on unification of solution of the Cauchy problem for overdetermined systems of differential equations \jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz. \yr 2019 \vol 11 \issue 4 \pages 12--25 \mathnet{http://mi.mathnet.ru/vyurm424} \crossref{https://doi.org/10.14529/mmph190402}