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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):$$
\frac{\partial {w}}{\partial t} = \frac{\partial}{\partial
x}\left[\chi(t)\left(\frac{\alpha w+\gamma}{(aw+b)^2}
+\frac{\beta}{(aw+b)^2} \ln{\left|aw+b\right|} \right)\frac{\partial {w}}{\partial x}\right] +
(aw+b)^2\theta(t).
$$
Solution(s),
Transformation(s),
Integral(s)
:
Solution:
$$
w=\frac{\beta}{\alpha-1-\beta a^2 z}-\frac{b}{a}, \quad\ z =
\varphi(t)\,x + \psi(t),
$$
where $\varphi(t)$ and $\psi(t)$ are described by
$$
\begin{array}[c]{ll}
\vspace{0.1cm} \varphi(t) = \pm \left(C_1-
2a^2\beta\int{\chi(t)dt}\right)^{-1/2},
\\
\psi(t) = \varphi(t)
\left[\int{\cfrac{\chi(t)\varphi^2(t)+\theta(t)}{\varphi(t)}dt}+C_2\right].\end{array}
$$
Remarks:Here $\chi(t)$ and $\theta(t)$ are arbitrary functions.
Novelty:New equation(s) & solution(s) & transformation(s)
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 12:31
Edits by author:0

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