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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.1. Second-Order Quasilinear Parabolic Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial t}+w\frac{\partial w}{\partial x}-
a\frac{\partial^2 w}{\partial x^2}=-f(x,t)$.\quad \ The generalized Burgers equation.
Solution(s),
Transformation(s),
Integral(s)
:
The generalized Burgers equation is connected with the linear equation
$$
\frac{\partial \zeta}{\partial t}=a\frac{\partial^2 w}{\partial \zeta^2}+\frac 1{2a}F(x,t)\zeta,\quad \
F(x,t)=\int f(x,t)\,dx
$$
by the B\"acklund transformation
$$
\frac{\partial \zeta}{\partial x}+\frac 1{2a}\zeta w=0,\quad \
\frac{\partial \zeta}{\partial t}+\frac {a^2}2\frac{\partial (\zeta w)}{\partial x}=\frac 1{2a}F(x,t)\zeta.
$$
Remarks:For $f(x,t)=f(t)$, the transformation
$$
w=u(z,t)-\int^t_{t_0} f(\tau)\,d\tau,\quad z=x+\int^t_{t_0}(t-\tau)f(\tau)\,d\tau,
$$
where $t_0$ is any number, leads to the classical Burgers equation 
$$
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial z}-a\frac{\partial^2 u}{\partial z^2}=0.
$$

Reference:

Polyanin, A. D. and Zaitsev, V. F., {\it Handbook of Nonlinear Partial Differential Equations}, Chapman \& Hall/CRC Press, Boca Raton, 2004 (page~78).
Novelty:New equation(s) & transformation(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Wed 25 Feb 2009 13:31
Edits by author:0

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