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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left[1+f\left(\frac{\partial w}{\partial x}\right)
\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\right]\left[\frac{\partial^2 w}{\partial x^2}
\frac{\partial^2 w}{\partial y^2}-\left(\frac{\partial^2 w}{\partial x\partial y}\right)^2\right]+
f\left(\frac{\partial w}{\partial x}\right)\left(\frac{\partial w}{\partial y}\right)^3
\frac{\partial^2 w}{\partial x\partial y}+\frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial y^2}
The transformation
U=-\left(\frac{\partial w}{\partial y}\right)^{-1}+
\int f\left(\frac{\partial w}{\partial x}\right)d\left(\frac{\partial w}{\partial x}\right), \quad
V=\frac{\partial w}{\partial x}\left(\frac{\partial w}{\partial y}\right)^{-1}, \quad T=x,
leads to the equation
\frac{\partial U}{\partial T}+\frac{\partial U}{\partial V}=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(T-V)}$.
Hence, we have a first order PDE:
-\left(\frac{\partial w}{\partial y}\right)^{-1}+
\int f\left(\frac{\partial w}{\partial x}\right)d\left(\frac{\partial w}{\partial x}\right)=
\Phi\left(x-\frac{\partial w}{\partial x}\left(\frac{\partial w}{\partial y}\right)^{-1}\right)
where $\Phi$ is an arbitrary function.
Remarks:This result was obtained in co-authorship with L.V. Linchuk,
using generalized group analysis. The main idea of the method 
is outlined at EqWorld (\slash 
ru/solutions/interesting.htm, in Russian).
Novelty:New equation(s) & solution(s) & transformation(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Statistic information
Submission date:Wed 09 Jan 2008 13:34
Edits by author:0

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