MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):\begin{multline*} \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+
\frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial x^2}-
\left[g(y)+\frac{\partial w}{\partial x}+h(w)\frac{\partial w}{\partial y}\right]\frac{\partial^2 w}{\partial x\partial y}+\\
+\left[f(x)+h(w)\frac{\partial w}{\partial x}\right]\frac{\partial^2 w}{\partial y^2}-
f(x)\frac{\partial w}{\partial y}+g(y)\frac{\partial w}{\partial x}=0.
\end{multline*}
Solution(s),
Transformation(s),
Integral(s)
:
The transformation
$$
U=\frac{\partial w}{\partial x}+\int f(x)\,dx+\int g(y)\,dy+\int h(w)\,dw, \quad
V=w+\frac{\partial w}{\partial y},
\quad T=\Psi_1\left(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}\right),
$$
leads to the equation
$$
\frac{\partial U}{\partial T}=0.
$$
Its solution is $\,\displaystyle{U(V,T)=\Psi(V)}$.
Hence, we have a first order PDE:
$$
\frac{\partial w}{\partial x}+\int f(x)\,dx+\int g(y)\,dy+\int h(w)\,dw=
\Phi\left(w+\frac{\partial w}{\partial y}\right),
$$
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 (http://eqworld.ipmnet.ru\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
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Wed 09 Jan 2008 13:35
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin