MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

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):$$
\begin{array}[c]{ll} \vspace{0.1cm}
F_1\left(w,\cfrac{\partial w}{\partial x},\dots,\cfrac{\partial^m
w}{\partial x^m},\cfrac{1}{u^k}\,\cfrac{\partial w}{\partial t}
,\cfrac 1u\,\cfrac{\partial u}{\partial x} ,\dots,\cfrac
1u\,\cfrac{\partial^n w}{\partial x^n}\right)=0,\cr
F_2\left(w,\cfrac{\partial w}{\partial x},\dots,\cfrac{\partial^m
w}{\partial x^m},\cfrac{1}{u^k}\,\cfrac{\partial w}{\partial
t},\cfrac 1u\,\cfrac{\partial u}{\partial x},\dots,\cfrac
1u\,\cfrac{\partial^n u}{\partial x^n}\right)=0.
\end{array}
$$
Solution(s),
Transformation(s),
Integral(s)
:
Solution:
$$
w=W(z),\quad u=[\varphi'(t)]^{1/k}U(z),\quad z=x+\varphi(t),
$$
 where $\varphi(t)$ is an arbitrary function and $W(z)$ and $U(z)$ are described
by the system of ordinary differential equations
$$\begin{array}[c]{ll} \vspace{0.1cm}
&F_1\left(W,W'_z,\dots,W^{(m)}_z,W'_z/U^k,U'_z/U,\dots,U^{(n)}_z/U\right)=0,\cr
&F_2\left(W,W'_z,\dots,W^{(m)}_z,W'_z/U^k,U'_z/U,\dots,U^{(n)}_z/U\right)=0.\end{array}
$$
Remarks:Here $F_1$ and $F_2$ are arbitrary functions.
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 12:26
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin