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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:4.5. Third-Order Equations
Equation(s):\noindent
$\displaystyle \frac{\partial w}{\partial t}=
a\frac{\partial^3 w}{\partial x^3}+(b\ln w+c)\frac{\partial w}{\partial x}$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
Solution:\hfill\break
$$
w(x,t)=e^{\lambda t}u(z),\quad \ z=x+\tfrac12b\lambda t^2+kt,
$$
where $k$ and $\lambda$ are arbitrary constants, and the function $u(z)$ 
is determined by the autonomous ordinary differential equation
$$
au'''_{zzz}+(b\ln u+c-k)u'_z-\lambda u=0.
$$
Remarks:\noindent
For other solutions see:
\smallskip

\noindent
1. W. I. Fushchich, N. I. Serov, T. K. Ahmerov, 
{\it On the conditional symmetry of the generalized KdV equation},
Rep. Ukr. Acad. Sci., No. 12, pp.~15--18, 1991.
\smallskip

\noindent
2. A. D. Polyanin and V. F. Zaitsev, {\it Handbook of Nonlinear
Partial Differential Equations}, Chapman \& Hall/CRC Press, 
Boca Raton, 2004 (p.~525).
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Mon 11 Dec 2006 14:52
Edits by author:0

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