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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.1. Second-Order Quasilinear Parabolic Equations
Equation(s):$\displaystyle \frac{\partial {w}}{\partial t} = \frac{\partial}{\partial
x}\left(w^{m}\frac{\partial {w}}{\partial x}\right) +
aw-bw^{1+m}+ cw^{1-m}$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions:\hfill\break
$\displaystyle w=\left\{mc\left[\varphi(t)e^{\pm\lambda
x}+\psi(t)\right]\right\}^{\frac{1}{m}},\quad \ \lambda^2=\cfrac{m^2b}{2+m}$.\hfill\break
Here, functions $\psi(t)$ and $\varphi(t)$ have the form \hfill\break
$$
\psi(t)=\frac{(2+m)a}{2m(1+m)bc}+\frac{\sqrt{(2+m)^2a^2
-4(1+m)bc}}{2m(1+m)bc}\tanh{\left[\sqrt{\frac{1}{4}m^2a^2
-\frac{m^2(1+m)}{2+m}bc}\,\,t+C_1\right]},  \quad
\varphi(t)=C_2\exp{\left(bmt+m^2bc\int{\psi(t)dt}\right)},
$$
where $C_1$ and $C_2$ are arbitrary constants.
Novelty:Material has been fully published elsewhere
References:E. A. Vyazmina and A. D. Polyanin, Theoretical Foundations of Chemical Engineering, 2006, Vol. 40, No. 6, pp. 555563.
Author/Contributor's Details
Last name:Vyazmina
First name:Elena
Middle(s) name:Andreevna
Country:Russia
City:Moscow
Affiliation:Ph.D student
Statistic information
Submission date:Thu 21 Dec 2006 13:19
Edits by author:0

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