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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:4.6. Higher-Order Equations
Equation(s):\noindent
$\displaystyle \frac{\partial w}{\partial t}=a\ln w\,\frac{\partial w}{\partial x}+
wF\left(\frac 1w\frac{\partial w}{\partial x},\,\frac 1w\frac{\partial^2 w}{\partial x^2},\,\ldots,\,\frac 1w\frac{\partial^n w}{\partial x^n}\right)$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
$1^\circ$. Let the function $w(x,t)$ is a solution of the equation in question. Then the function
$$
w_1=e^{C_1}w(x+aC_1t+C_2,t+C_3),
$$
where $C_1$, $C_2$, and $C_3$ are arbitrary constants,
is also a solution of the equation.
\medskip

\noindent
$2^\circ$. 
Generalized traveling-wave solution:
$$
w(x,t)=\exp\left[-\frac{x}{at}+
\frac 1t\int tF\left(-\frac 1{at},\,\ldots,\,\frac 1{(-at)^n}\right)\,dt\right].
$$
\medskip

\noindent
$3^\circ$. 
Solution:
$$
w(x,t)=e^{\lambda t}u(z),\quad \ z=x+\tfrac12a\lambda t^2+kt,
$$
where $k$ and $\lambda$ are arbitrary constants, and the function $u(z)$ 
is determined by the autonomous ordinary differential equation
$$
(a\ln u-k)u'_z-\lambda u+uF\left(\frac{u'_z}u,\frac{u''_{zz}}u,\ldots,\frac{u^{(n)}_z}u\right)=0.
$$
For the particular case $\lambda=0$, it is a traveling-wave solution.
Remarks:\noindent Here, $F(u_1,u_2,\dots,u_n)$ is an arbitrary function.
Novelty:New equation(s) & solution(s) & transformation(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Sun 10 Dec 2006 14:23
Edits by author:0

Edit (Only for author/contributor)


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