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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:4.1. Second-Order Quasilinear Parabolic Equations
Equation(s):$$
\frac{\partial {w}}{\partial t} = \frac{\partial}{\partial
x}\left[f({w})\left( \frac{\partial {w}}{\partial
x}\right)^n\right] + g(w).
$$
Solution(s),
Transformation(s),
Integral(s)
:
1. Let the functions $f(w)$ and $g(w)$ are described implicitly by formulas
$$
\begin{array}[c]{ll} \vspace{0.1cm}
f(w)=A\dfrac{\zeta(z)}{\bigl[\zeta'_z(z)\bigr]^n}z^{n-1}, &
g(w)=B(n+1)\zeta'_z(z)z^{-\frac{n}{n+1}}+B\zeta(z)z^{-\frac{2n+1}{n+1}},
\end{array}
$$
where $\zeta(z)$ is an arbitrary function, and $A$ and $B$ are arbitrary constants.

Exact solution:
$$
w=\int{z^{-\frac{n}{n+1}}\zeta'_z(z)dz}+C_1,
$$
where
$$
z=\left[\frac{x^{n+1}}{C_2-(n+1)k^nAt}\right]^{\frac{1}{n}}+(n+1)Bt-
\frac{(n+1)BC_2}{nk^{n+1}A}, \quad\ k=\frac{n+1}{n},
$$
$C_1$ and $C_2$ are arbitrary constants.
\medskip

2. Let $f(w)$ and $g(w)$ are described implicitly by formulas
$$
\begin{array}[c]{ll} \vspace{0.1cm}
f(w)=\dfrac{1}{\lambda^n}\left(A-\dfrac{n\lambda}{n+1}w\right)
\bigl[\zeta'_w(w)\bigr]^n, &
g(w)=\lambda\dfrac{\zeta(w)}{\zeta'_w(w)}+\dfrac{n\lambda}{n+1}w-A,
\end{array}
$$
where $\zeta(z)$ is an arbitrary function, and $A$ and $\lambda$ are arbitrary constants.

Exact solution:
$$
\varphi(w)=C_1e^{\lambda
t}+\frac{n\lambda}{n+1}\left(x+C_2\right)^{\frac{n+1}{n}}.
$$
Novelty:New solution(s) & transformation(s)
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 12:34
Edits by author:0

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