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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):$\displaystyle \frac{\partial w}{\partial t}=\frac12\frac{\partial^2w}{\partial x^2}+w^2(1-w)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solution in implicit form:
$$
x=F(w,t),
$$
where the function $F=F(w,t)$ is determined by the system of first-order
partial differential equations
$$
\frac{\partial F}{\partial t}=\frac12 +\frac{3wR}{2(w^2-w-R)},\quad \
\frac{\partial F}{\partial w}=\frac{1}{w^2-w-R},
\eqno(1)
$$
where the function $R=R(w,t)$ is determined by the implicit formula
$$
\frac{1-w}{R}+\ln\frac{w+R}{R}=\frac32t.
\eqno(2)
$$
Relation (2) is the consistency condition for system (1).
Novelty:New solution(s) / integral(s)
References:K.A. Volosov, A method of analysis for evolution systems with distributed parameters, DSc thesis, Moscow, 2007.
Author/Contributor's Details
Last name:Volosov
First name:Konstantin
Middle(s) name:Alexandrovich
Country:Russia
City:Moscow
Statistic information
Submission date:Tue 18 Sep 2007 11:50
Edits by author:0

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