Solutions of multidimensional partial differential equations representable as a one-dimensional flow
In: Theoretical and Mathematical Physics, Jg. 178 (2014-03-01), S. 299-313
Online
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Zugriff:
We propose an algorithm for reducing an (M+1)-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow ut + \(w_{x_1 } \) (u, ux uxx,…) = 0 (where w is an arbitrary local function of u and its xi derivatives, i = 1,…, M) to a family of M-dimensional nonlinear PDEs F(u,w) = 0, where F is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the M-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original (M+1)-dimensional equation. Moreover, a spectral parameter can be introduced in the function F, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.
Titel: |
Solutions of multidimensional partial differential equations representable as a one-dimensional flow
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Autor/in / Beteiligte Person: | Zenchuk, A. I. |
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Zeitschrift: | Theoretical and Mathematical Physics, Jg. 178 (2014-03-01), S. 299-313 |
Veröffentlichung: | Springer Science and Business Media LLC, 2014 |
Medientyp: | unknown |
ISSN: | 1573-9333 (print) ; 0040-5779 (print) |
DOI: | 10.1007/s11232-014-0144-3 |
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