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:4.1. Second-Order Quasilinear Parabolic Equations
Equation(s):$\displaystyle
\frac{\partial {w}}{\partial t} = \frac{\partial}{\partial
x}\left[\chi(t)f({w})\frac{\partial {w}}{\partial x}\right] +
g(w)\theta(t).
$
Solution(s),
Transformation(s),
Integral(s)
:
1. Let
$$
f(w) = e^{\int{\frac{\lambda-g'_w}{g(w)}dw}}\left[aw+
b\int{e^{\lambda\int{\frac{dw}{g(w)}}}dw}+c\right].
$$
Solution:
$$
w = w(z), \quad    z = \varphi(t)\,x + \psi(t),
$$
where $w(z)$, $\varphi(t)$, and $\psi(t)$ are described by
$$
\begin{array}[c]{ll}
\vspace{0.1cm} z
=\frac{1}{\lambda}\left(C_1e^{\lambda\int{\frac{dw}{g(w)}}}-a\cfrac{\lambda^2-\lambda}{b}\right), \\
\varphi(t) = \pm
e^{\lambda\int{\theta(t)dt}}\left[C_2-\cfrac{2b}{a\lambda}
\int{e^{2\lambda\int{\theta(t)dt}}\chi(t)dt}\right]^{-1/2},
\\
\psi(t) = \varphi(t)
\left[\int{\frac{b\chi(t)\varphi^2(t)+a(\lambda^2-1)\theta(t)}{b\varphi(t)}dt}+C_3\right].\end{array}
$$
\medskip

2. Let
$$
f(w) = \frac{1}{g(w)}\left[aw+
b\int{\left(\int{\frac{dw}{g(w)}}\right)dw}+c\right].
$$
Solution:
$$
w = w(z), \quad    z = \varphi(t)\,x + \psi(t),
$$
where $w(z)$, $\varphi(t)$, and $\psi(t)$ are described by
$$
\begin{array}[c]{ll}
\vspace{0.1cm} z =\int{\frac{dw}{g(w)}}, \\
\varphi(t) = \pm \left(C_1- 2b\int{\chi(t)dt}\right)^{-1/2},
\\
\psi(t) = \varphi(t)
\left[\int{\frac{a\chi(t)\varphi^2(t)+b\theta(t)}{a\varphi(t)}dt}+C_2\right].\end{array}
$$
Remarks:Here $\chi(t)$ and $\theta(t)$ are arbitrary functions.
Novelty:New equation(s) & solution(s) & transformation(s)
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 12:28
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