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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):$u_x=uf(w)$,\\
$w_t=u^kg(w).$
Solution(s),
Transformation(s),
Integral(s)
:
1. In general case the system is reduced to the first-order equation\hfill\break
$\displaystyle \frac{\partial w}{\partial
x}=kg(w)\int\frac{f(w)}{g(w)}\,dw+\theta(x)g(w)$,\hfill\break
where $\theta(x)$ is an arbitrary function.
\medskip

2. The special case $\theta(x)={\rm const}$ corresponds to the following special 
solution: \hfill\break
$\displaystyle w=w(z),\quad u=[\psi'(t)]^{1/k}v(z),\quad z=x+\psi(t),$
\hfill\break where $w(z)$ and $v(z)$ are described by 
\hfill\break
$\displaystyle \int\frac{dw}{g(w)[kF(w)+C_1]}=z+C_2,\quad \
v=[kF(w)+C_1]^{1/k},\quad \ F(w)=\int\frac{f(w)}{g(w)}\,dw.$
\hfill\break
Novelty:Material has been fully published elsewhere
References: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.
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 10:14
Edits by author:0

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