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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:4.8. Systems of Three and More Equations
Equation(s):$$
\frac{\partial u_m}{\partial
t}=L[u_m]+\sum^n_{k=1}u_kf_{mk}\left(t,\frac
{u_1}{u_n},\ldots,\frac{u_{n-1}}{u_n}\right),\qquad m=1,\dots,n.
$$
Solution(s),
Transformation(s),
Integral(s)
:
Solution:
$$\begin{array}[c]{ll} \vspace{0.1cm}
u_m(x_1,\ldots,x_n,t)=\varphi_m(t)\,F(t)\,\theta(x_1,\ldots,x_n,t),\qquad
m=1,\ldots,n,\\ F(t)=\exp\left[\int{ \sum
\limits^n_{k=1}\varphi_k(t)f_{nk}(t,\varphi_1,\ldots,\varphi_{n-1})\,dt}
\right],\qquad \varphi_n(t)=1,\end{array}
$$
where $\varphi_m=\varphi_m(t)$ are described by the system of ordinary differential equations
$$
\varphi'_m=\sum^n_{k=1}\varphi_k
f_{mk}(t,\varphi_1,\ldots,\varphi_{n-1})
-\varphi_m\sum^n_{k=1}\varphi_k
f_{nk}(t,\varphi_1,\ldots,\varphi_{n-1}),\quad \ m=1,\ldots,n-1,
$$
and the function $\theta=\theta(x_1,\dots,x_n,t)$ satisfies the liner equation
$$
\frac{\partial \theta} {\partial t}=L[\theta].
$$
Remarks:Here $L$ is an arbitrary linear operator with respect to the spatial variables $x_1$, \dots, $x_n$ (of any order in derivatives) and $f_{mk}$ are arbitrary functions. It is assumed thad $L[{\rm const]=0$.
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 12:27
Edits by author:0

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