Reduction of overdetermined differential equations of mathematical physics

M. L. Zaytsev ; V. B. Akkerman

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 4, pp. 43-67

Résumé

A technical method of reducing the overdetermined systems of differential equations is further extended. Specifically, the fundamentals and validity limits of the method are identified, and the method is justified within its validity domain. Starting with an overview of the previous results, we subsequently employ them in deriving and justifying the new outcomes. In particular, overdetermined systems of ordinary differential equations (ODE) are studied as the simplest case. It is demonstrated that, if a determinant deviates from zero for an ODE system, then the solution to this system can be found. Based on this, we subsequently arrive to a more general statement for a system of partial differential equations (PDE). On a separate basis, a Cauchy problem for reduced overdetermined systems of differential equations is considered, and it is shown that such a problem cannot be with arbitrary initial conditions. It is also shown and substantiated how to employ a Cauchy problem to reduce the dimension of PDEs. A novel approach of how to transform ODE and PDE systems (such as Euler and Navies-Stokes equations as well as the analytical mechanics system of equations) into the overdetermined systems is presented. Finally, the results are generalized in such a manner that it is shown how to reduce an overdetermined system of deferential equations to that having a complete and explicit solution. The work also includes two appendices. The first appendix presents the algorithm of searching for a solution to an overdetermined system of differential equations, in particular, by means of the computational approaches. The second appendix is devoted to the study of the variety of the solutions to an overdetermined system of equations. In particular, it is shown that a certain condition for the determinant, associated with this system of equations, breaks the possibility, that such a variety of solutions can depend on a continuous factor (for instance, from Cauchy conditions). For instance, it could be not more than a countable set. The paper is concluded with a brief summary, where the major results of the work are listed again and discussed, including their potential practical applications such as developing and testing of new computer codes to solving systems of differential equations.

Détail

Keywords: overdetermined systems of differential equations, Euler and Navier- Stokes equations, differential equation on the surface, dimension of differential equations, Cauchy problem, partial differential equations.
Mots-clés : ODE