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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):$u_x=f(a_1u+b_1w)$,\\
$w_t=g(a_2u+b_2w)$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solution with additive separetion of variables at $\Delta=a_1b_2-a_2b_1\not=0$:\hfill\break
$\displaystyle u=\frac 1\Delta[b_2\varphi(x)-b_1\psi(t)],\quad w=\frac
1\Delta[a_1\psi(t)-a_2\varphi(x)],$\hfill\break
where $\varphi(x)$ and $\psi(t)$ are defined implicitly by the formulas:\hfill\break
$\displaystyle \frac{b_2}{\Delta}\int\frac{d\varphi}{f(\varphi)}=x+C_1,\quad \
\frac{a_1}{\Delta}\int\frac{d\psi}{g(\psi)}=t+C_2,$\hfill\break 
where $C_1$ and $C_2$ are arbitrary constants.
\medskip

2. Solution for $a_1=a_2=a$ and $b_1=b_2=b$:
\hfill\break
$\displaystyle u=b(k_1x-\lambda_1 t)+y(\xi),\quad w=-a(k_1x-\lambda_1
t)+z(\xi),\quad \xi=k_2 x-\lambda_2 t,$\hfill\break
where $k_1$, $k_2$, $\lambda_1$, and $\lambda_2$ are arbitrary constants and functions
$y(\xi)$ and $z(\xi)$ are described by the autonomous system of ordinary differential equations
\hfill\break
$\displaystyle k_2 y'_\xi+bk_1=f(ay+bz),\quad \ -\lambda_2
z'_\xi+a\lambda_1=g(ay+bz).$\hfill\break
Remarks:Here $f$ and $g$ are arbitrary functions depending on linear combinations of desired quantites.
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:Sat 22 Sep 2007 23:14
Edits by author:0

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