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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):$$
\frac{\partial {w}}{\partial t} = \frac{\partial}{\partial
x}\left[\chi(t)\left(\alpha w^2 +\beta w +
\gamma\right)\frac{\partial {w}}{\partial x}\right] + b\theta(t).
$$
Solution(s),
Transformation(s),
Integral(s)
:
Solution:
$$
w = bz-\frac{\beta b- 1}{2\alpha b^2}, \quad    z = \varphi(t)\,x
+ \psi(t),
$$
where the functions $\varphi(t)$ and $\psi(t)$ are described by
$$
\begin{array}[c]{ll}
\vspace{0.1cm} \varphi(t) = \pm \left(C_1-
4b^2\alpha\int{\chi(t)dt}\right)^{-1/2},
\\
\psi(t) = \varphi(t)
\left[\int{\cfrac{\chi(t)\varphi^2(t)+\theta(t)}{\varphi(t)}dt}+C_2\right].\end{array}
$$
Remarks:Here $\chi(t)$, and $\theta(t)$ are arbitrary functions.
Novelty:New equation(s) & solution(s) & transformation(s)
Admin's Comment:It is a degenerate solution.
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 12:29
Edits by author:0

Edit (Only for author/contributor)


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