Construction Solutions of PDE in Parametric Form

Alexandra K. Volosova and Konstantin Alexandrovich Volosov

International Journal of Mathematics and Mathematical Sciences, 2009, vol. 2009, 1-17

Abstract:

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 ð ‘¥ , ð ‘¡ , then it always gives the possibility of expressing all PDE second and more order as ð ´ ð ‘‹ = ð ‘ . This is a linear algebraic equations system with regards derivatives to old variables ð ‘¥ ( 𠜉 , ð ›¿ ) , ð ‘¡ ( 𠜉 , ð ›¿ ) on new variables 𠜉 , ð ›¿ ∶ ( ð ‘¥ ′ 𠜉 , ð ‘¥ ′ ð ›¿ , ð ‘¡ ′ 𠜉 , ð ‘¡ ′ ð ›¿ ) . This system has the unique solution. In the case of three and more independent variables ð ‘¥ , 𠑦 , ð ‘¡ , … , then it gives the possibility of expressing PDE second order as ð ´ ð ‘‹ = ð ‘ , 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.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:319269

DOI: 10.1155/2009/319269

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