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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
$\displaystyle \frac{\partial w}{\partial t}=a\ln w\,\frac{\partial w}{\partial x}+
wF\left(t,\,\frac 1w\frac{\partial w}{\partial x},\,\frac 1w\frac{\partial^2 w}{\partial x^2},\,\ldots,\,\frac 1w\frac{\partial^n w}{\partial x^n}\right)$.
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.
\smallskip

\noindent
$2^\circ$. 
Generalized traveling-wave solution:
$$
w(x,t)=\exp\left[-\frac{x}{at}+
\frac 1t\int tF\left(t,\,-\frac 1{at},\,\ldots,\,\frac 1{(-at)^n}\right)\,dt\right].
$$
Remarks:\noindent Here, $F(t,u_1,u_2,\dots,u_n)$ is an arbitrary function.
Novelty:New equation(s) & 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 10:27
Edits by author:0

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