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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):$$
\frac{\partial^2w}{\partial x^2}  = -\alpha \frac{\partial w}{\partial y}\frac{\partial^2w}{\partial y^2}.
$$
Solution(s),
Transformation(s),
Integral(s)
:
1. Solution:
$$
w(x,y) = \frac{{A^2 }}{{12}}x^4  + \frac{{AB}}{3}x^3  + \frac{{B^2
}}{2}x^2  + ax + b - \left( {Ax + B} \right) \left(\sqrt
{\frac{8}{{9\alpha }}} y^{3/2}  + C\right).
$$

2. Solution:
$$
w(x,y) = Ax + B + C_1\left( {C_2 - x }\right)^{ - 2} \left( {y +
c} \right)^3.
$$

3. Using the transformation
$$
w(x,y) + u(\xi ,\eta ) = x\xi  + y\eta ,\quad \xi  =
\frac{{\partial w}}{{\partial x}},\quad \eta  = \frac{{\partial
w}}{{\partial y}},
$$
the equation can be reduced to the linear equation
$$
\frac{{\partial ^2 u}}{{\partial \eta ^2 }} = \alpha \eta
\frac{{\partial ^2 u}}{{\partial \xi ^2 }},
$$
so the solution of the linear equation has the form
$$
u(\xi ,\eta ) = \Phi (\eta )\Psi (\xi ).
$$
where
$$
\Phi (\eta ) = C_1 {\rm{Ai}}(\lambda \eta ) + C_2
{\rm{Bi}}(\lambda \eta ),
$$
$$
\Psi (\xi ) = C_3 e^{\sqrt {\frac{\lambda }{\alpha }} \lambda \xi
}  + C_4 e^{ - \sqrt {\frac{\lambda }{\alpha }} \lambda \xi }.
$$
Here are ${\rm{Ai}}(\eta )$ and ${\rm{Bi}}(\eta )$ are Airy functions.
Novelty:New solution(s) & transformation(s)
Admin's Comment:The given results are known, see
A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004
(pp. 361-362, 262-264).
Author/Contributor's Details
Last name:Vyazmina
First name:Elena
Country:Russia
City:Moscow
Affiliation:Institute for Problems in Mechanics
Statistic information
Submission date:Fri 25 Jan 2008 12:39
Edits by author:0

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