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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):$u_t=[f(w/u)u_x]_x+ug_1(w/u)+wh_1(w/u)$, \\
$w_t=[f(w/u)w_x]_x+ug_2(w/u)+wh_2(w/u)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solution:
$$
\begin{array}[c]{ll} \vspace{0.1cm}
u=\exp\left\{\int [g_1(\varphi)+\varphi
h_1(\varphi)]\,dt\right\}\theta(x,\tau),\quad \tau=\int
f(\varphi)\,dt,\\ w=\varphi(t)\exp\left\{\int
[g_1(\varphi)+\varphi
h_1(\varphi)]\,dt\right\}\theta(x,\tau),\end{array}
$$
where
functions $\varphi=\varphi(t)$ is described by the ordinary
differential equation
$$\varphi'_t=g_2(\varphi)+\varphi
h_2(\varphi)-\varphi[g_1(\varphi)+\varphi h_1(\varphi)],
$$
and the function $\theta=\theta(x,t)$ satisfies the liner heat
equation
$$
\frac{\partial \theta}{\partial \tau}=\frac{\partial^2
\theta}{\partial x^2}.
$$
Remarks:Here $f$ $g_1$, $g_2$, $h_1$, and $h_2$ 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 12:25
Edits by author:0

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