# Burgers' equation

Burgers equation
Order2nd
SolvableExactly solvable
The Cauchy problemSolvable by the Cole-Hopf transform
HierarchyBurgers' hierarchy
First introducedForsyth, 1906

## Definition

${u}_{t}+\alpha u{u}_{x}=\mu {u}_{xx}$

It includes nonlinearity and dissipation together in the simplest possible way and may be thought of as a nonlinear version of the heat equation.

## History

The equation was known to Forsyth (1906) and had been discussed by Bateman (1915). Due to extensive works of Burgers (1948) it is known as Burgers’ equation.

## The Cole-Hopf transformation

The Burgers' equation then can be linearized by the Cole-Hopf transformation

$u\left(x,t\right)=-\frac{2\mu }{\alpha }\frac{{z}_{x}}{z},$

$z=z\left(x,t\right).$

Substituting it to the Burgers' equation one'll get the linear heat equation

${z}_{t}=\mu {z}_{xx}.$

## The Cauchy problem

Let us suppose that function $u\left(x,t\right)$ satisfies the Burgers' equation

${u}_{t}+\alpha u{u}_{x}=\mu {u}_{xx}$

and that in the inital moment

$u\left(x,t=0\right)=\phi \left(x\right).$

Finally one will obtain the solution of the Cauchy problem for the Burgers' equation in the form

$u\left(x,t\right)=\frac{\underset{-\infty }{\overset{+\infty }{\int }}\frac{x-\theta }{t}\mathrm{exp}\left\{-\frac{\alpha G\left(\theta ,x,t\right)}{2\mu }\right\}d\theta }{\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{exp}\left\{-\frac{\alpha G\left(\theta ,x,t\right)}{2\mu }\right\}d\theta }$

with

$G\left(\theta ,x,t\right)=\frac{{\left(x-\theta \right)}^{2}}{2t}+\underset{0}{\overset{\theta }{\int }}\phi \left(\xi \right)d\xi .$

## Solutions

Let us change the scale by introducing the new variables $x\text{'}=-\frac{\mu }{\alpha }x,t\text{'}=\frac{\mu }{{\alpha }^{2}}t$. Then the Burgers' equation will take the form (primes are omitted)

${u}_{t}-u{u}_{x}={u}_{xx}.$

### Rational solutions

One can obtain the following rational solutions of the Burgers' equation

$u\left(x,t\right)=\frac{A-x}{B+t},$

$\begin{array}{l}u\left(x,t\right)=\lambda +\frac{2}{x+\lambda t+A},\\ u\left(x,t\right)=\frac{4x+2A}{{x}^{2}+Ax+2t+B},\\ u\left(x,t\right)=\frac{6\left({x}^{2}+2t+A\right)}{{x}^{3}+6xt+3Ax+B},\end{array}$

where $A,B$ and $\lambda$ are arbitrary constants.

### Other solutions

One can also obtain the following solutions of the Burgers' equation

$\begin{array}{l}u\left(x,t\right)=\frac{2\lambda }{1+A\mathrm{exp}\left(-{\lambda }^{2}t-\lambda x\right)},\\ u\left(x,t\right)=-\lambda +A\frac{\mathrm{exp}\left[A\left(x-\lambda t\right)\right]-B}{\mathrm{exp}\left[A\left(x-\lambda t\right)\right]+B},\end{array}$

$u\left(x,t\right)=-\lambda +2Ath\left[A\left(x-\lambda t\right)+B\right],$

$u\left(x,t\right)=\frac{\lambda }{{\lambda }^{2}t+A}\left[2th\left(\frac{\lambda x+B}{{\lambda }^{2}t+A}\right)-\lambda x-B\right],$

$u\left(x,t\right)=-\lambda +2Atg\left[A\left(\lambda t-x\right)+B\right],$

$u\left(x,t\right)=\frac{2\lambda \mathrm{cos}\left(\lambda x+A\right)}{B\mathrm{exp}\left({\lambda }^{2}t\right)+\mathrm{sin}\left(\lambda x+A\right)},$

$u\left(x,t\right)=\frac{2}{\sqrt{\pi \left(t+\lambda \right)}}\mathrm{exp}\left[-\frac{{\left(x+A\right)}^{2}}{4\left(t+\lambda \right)}\right]{\left[B+Aerf\left(\frac{x+A}{2\sqrt{t+\lambda }}\right)\right]}^{-1},$

where $A,B$ and $\lambda$ are arbitrary constants and $erf\left(z\right)\equiv \frac{2}{\sqrt{\pi }}\underset{0}{\overset{z}{\int }}{e}^{-{t}^{2}}dt$ is the "error function".

Other solutions of the Burgers' equation can be obtained using the Cole-Hopf transformation.

## Applications and connections

The Burgers' equation is used as a model in fields as wide as
• acoustics
• continues stochastic processes
• dispersive water waves
• gas dynamics
• heat conduction
• longitudinal elastic waves in an isotropic solid
• number theory
• shock waves
• turbulence
and so forth.

## Generalizations

There are some nonlinear partial differential equations that may be thought of as a formal generalization of the Burgers' equation:

the Burgers-Huxley equation

${u}_{t}+\alpha u{u}_{x}=\mu {u}_{xx}+\mu u+\eta {u}^{2}-\delta {u}^{3}$

the Kolmogorov-Petrovsky-Piskunov equation (Fisher equation)

${u}_{t}=\mu {u}_{xx}+\mu u+\eta {u}^{2}$

the Korteweg-de Vries-Burgers equation

${u}_{t}+\alpha u{u}_{x}+{u}_{xxx}=\mu {u}_{xx}$

the Kuramoto-Sivashinsky equation

${u}_{t}+u{u}_{x}+\alpha {u}_{xx}+\beta {u}_{xxx}+\gamma {u}_{xxxx}=0$

Burgers' hierarchy

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