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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):$u_x=u^kf(u^nw^m)$,\\ 
$w_t=w^sg(u^nw^m)$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solution with $s\neq1$, $n\neq0$:
$$
u=t^{\frac{m}{n(s-1)}}y(\xi), \quad\ w=t^{-\frac{1}{s-1}}z(\xi),
\quad\ \xi=xt^{\frac{m(k-1)}{n(s-1)}},
$$
where functions $y(\xi)$ and $z(\xi)$ are described by the system
of ordinary differential equations
$$
y'_{\xi}=y^kf(y^nz^m), \qquad\ m(k-1)\xi
z'_{\xi}-nz=n(s-1)z^sg(y^nz^m).
$$
\medskip

2. Solution with $s=1$:
$$
u=e^{mt}y(\xi), \quad\ w=e^{-nt}z(\xi), \quad\ \xi=e^{m(k-1)t}x,
$$
where functions $y(\xi)$ and $z(\xi)$ are described by the system
of ordinary differential equations
$$
y'_{\xi}=y^kf(y^nz^m), \qquad\ m(k-1)\xi z'_{\xi}-nz=zg(y^nz^m).
$$
\medskip

3. Solution with $k=1$, $s=1$:
$$
u=e^{m(px-\lambda t)}y(\xi), \quad\ w=e^{-n(px-\lambda t)}z(\xi),
\quad\ \xi=\alpha x- \beta t,
$$
where functions $y(\xi)$ and $z(\xi)$ are described by the system
of autonomous ordinary differential equations
$$
\alpha y'_{\xi} + mpy=yf(y^nz^m), \qquad\ -\beta z'_{\xi} +
n\lambda z=zg(y^nz^m).
$$
Remarks:Here $f$ and $g$ are arbitrary functions of the composite argument.
Novelty:Material has been fully published elsewhere
References:E. A. Vyazmina, P. G. Bedrikovetskii, and A. D. Polyanin. New classes of exact solutions to nonlinear sets of equations in the theory of filtration and convective mass transfer. Theoretical Foundations of Chemical Engineering, 2007, Vol. 41, No. 5, pp. 556-564.
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 12:21
Edits by author:0

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