Zaytsev M.L., Akkerman V.B. Reduction of Overdetermined Differential Equations of Mathematical Physics

https://doi.org/10.15688/mpcm.jvolsu.2017.4.5

Maxim  Leonidovich  Zaytsev
Postgraduate Student,
Nuclear Safety Institute of Russian Academy of Sciences
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Bolshaya Tulskaya St., 52, 115191 Moscow, Russian Federation

Vyacheslav  Borisovich Akkerman
Candidate of Physical and Mathematical Sciences, Assistant Professor,
West Virginia University
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WV 26506-6106 Morgantown, USA

Abstract. 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.

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

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Reduction of Overdetermined Differential Equations of  Mathematical  Physics by Zaytsev M.L., Akkerman V.B. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation . Vol. 20 No. 4 2017 pp. 43-67

 

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