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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):$u_t=a_1u_{xx}+f(bu+cw)$,\\ 
$w_t=a_2w_{xx}+g(bu+cw)$.
Solution(s),
Transformation(s),
Integral(s)
:
Suppose $f$ and $g$ are arbitrary functions of linear combination of unknown quantities.

\medskip
1) Solution for any $a_1$ and $a_2$:\hfill\break
$\displaystyle u=c(\alpha x^2+\beta x+\gamma t)+y(\xi),\quad w=-b(\alpha
x^2+\beta x+\gamma t)+z(\xi),\quad \xi=kx-\lambda t,$\hfill\break
where $k$, $\alpha$, $\beta$, $\gamma$, $\lambda$ are arbitrary constants 
and the functions $y(\xi)$ and $z(\xi)$ are described by the autonomous 
system of ordinary differential equations \hfill\break
$\displaystyle \begin{array}[c]{ll}
\vspace{0.1cm} a_1k^2y''_{\xi\xi}+\lambda
y'_\xi+2a_1c\alpha-c\gamma+f(by+cz)=0,\\
a_2k^2z''_{\xi\xi}+\lambda
z'_\xi-2a_2b\alpha+b\gamma+g(by+cz)=0.\end{array}$\hfill\break
\medskip

2) Solution for $a_1=a_2=a$:\hfill\break
$\displaystyle u=c \theta(x,t)+y(\xi),\quad w=-b\theta(x,t)+z(\xi),\quad
\xi=kx-\lambda t,$\hfill\break
where functions $y(\xi)$ and $z(\xi)$ are described by the autonomous 
system of ordinary differential equations \hfill\break
$\displaystyle \begin{array}[c]{ll}
\vspace{0.1cm} ak^2y''_{\xi\xi}+\lambda y'_\xi+f(by+cz)=0, \\
ak^2z''_{\xi\xi}+\lambda z'_\xi+g(by+cz)=0,\end{array}$\hfill\break
and function $\theta=\theta(x,t)$ satisfies the linear heat transfer 
equation:\hfill\break
$\displaystyle \frac{\partial \theta}{\partial t}=a\frac{\partial^2
\theta}{\partial x^2}.$\hfill\break
Novelty:Material has been fully published elsewhere
References:1. A.D. Polyanin and E.A. Vyaz’mina. New Classes of Exact Solutions to Nonlinear Systems of Reaction-Diffusion Equations. Doklady Mathematics, 2006, Vol. 74, No. 1, pp. 597-602.
2. E.A. Vyaz’mina and A.D. Polyanin. New Classes of Exact Solutions to General Nonlinear Diffusion-Kinetic Equations. Theor. Found. Chem. Eng, 2006, Vol. 40, No. 6, pp. 555-563.
Author/Contributor's Details
Last name:Vyazmina
First name:Elena
Country:Russia
City:Moscow
Affiliation:Institute for Problems in Mechanics
Statistic information
Submission date:Sun 23 Sep 2007 11:50
Edits by author:0

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