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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:4.7. Systems of Two Equations
Equation(s):$u_t=ax^{-n}(x^nu_x)_x+uf_1(w/u)+wg_1(w/u)$,\\
$w_t=ax^{-n}(x^nw_x)_x+uf_2(w/u)+wg_2(w/u)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solution:
$$
\begin{array}[c]{ll} \vspace{0.1cm} u=\exp\bigl\{\int
[f_1(\varphi)+\varphi g_1(\varphi)]\,dt\bigr\}\theta(x,t),\\
w=\varphi(t)\exp\left\{\int [f_1(\varphi)+\varphi
g_1(\varphi)]\,dt\right\}\theta(x,t), \end{array}
$$
where functions $\varphi=\varphi(t)$ is described by the ordinary
differential equation
$$\varphi'_t=f_2(\varphi)+\varphi
g_2(\varphi)-\varphi[f_1(\varphi)+\varphi g_1(\varphi)],
$$
and the function $\theta=\theta(x,t)$ satisfies the liner heat
equation
$$
\frac{\partial \theta}{\partial t}=\frac {a}{x^n} \,\frac{\partial
}{\partial x}\left(x^n\frac{\partial \theta }{\partial x}\right).
$$
Remarks:Here $f_1$, $f_2$, $g_1$, and $g_2$ are arbitrary functions of
the composite argument.
Novelty:Material has been fully published elsewhere
References:A. D. Polyanin and E. A. Vyazmina. New classes of exact solutions to nonlinear systems of reaction-diffusion equations. Doklady Mathematics, 2006, Vol. 74, No. 1, pp. 597-602.
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 12:24
Edits by author:0

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