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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 319269, 17 pages
Research Article

Construction Solutions of PDE in Parametric Form

Department of Applied Mathematics, Faculty of Computer Sciences, Moscow State University of Railway Engineering, 141400 Himki, Moscow region, Russia

Received 22 July 2008; Accepted 31 July 2009

Academic Editor: Attila Gilanyi

Copyright © 2009 Alexandra K. Volosova and Konstantin Alexandrovich Volosov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The new important property of wide class PDE was found solely by K. A. Volosov. We make an arbitrary replacement of variables. In the case of two independent variables ๐‘ฅ , ๐‘ก , then it always gives the possibility of expressing all PDE second and more order as ๐ด ๐‘‹ = ๐‘ . This is a linear algebraic equations system with regards derivatives to old variables ๐‘ฅ ( ๐œ‰ , ๐›ฟ ) , ๐‘ก ( ๐œ‰ , ๐›ฟ ) on new variables ๐œ‰ , ๐›ฟ โˆถ ( ๐‘ฅ โ€ฒ ๐œ‰ , ๐‘ฅ โ€ฒ ๐›ฟ , ๐‘ก โ€ฒ ๐œ‰ , ๐‘ก โ€ฒ ๐›ฟ ) . This system has the unique solution. In the case of three and more independent variables ๐‘ฅ , ๐‘ฆ , ๐‘ก , โ€ฆ , then it gives the possibility of expressing PDE second order as ๐ด ๐‘‹ = ๐‘ , if we do same compliment proposes. In the present paper, we suggest a new method for constructing closed formulas for exact solutions of PDE, then support on this important new property.