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Burgers' equation

Burgers equation
Order	2nd
Solvable	Exactly solvable
The Cauchy problem	Solvable by the Cole-Hopf transform
Hierarchy	Burgers' hierarchy
First introduced	Forsyth, 1906

Contents 1 Burgers' equation 1.1 Definition 1.2 History 1.3 The Cole-Hopf transformation 1.4 The Cauchy problem 1.5 Solutions 1.5.1 Rational solutions 1.5.2 Other solutions 1.6 Applications and connections 1.7 Generalizations 1.8 See also 1.9 References 1.10 External links

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

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

with

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

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

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

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

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

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

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

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

where $A,B$ and $\lambda$ are arbitrary constants and $erf(z)\equiv \frac{2}{\sqrt{\pi }}\underset{0}{\overset{z}{{\displaystyle \int }}}{e}^{-t^2}dt$ 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

$$u_t+u{u}_x+\alpha u_{xx}+\beta u_{xxx}+\gamma u_{xxxx}=0$$

See also

Burgers' hierarchy

References

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4. Benton E.R. Some new exact, viscous, nonsteady solutions of Burgers' equation // Phys. Fluids, 1966. 9. Pp.1247-1248. DOI:10.1063/1.1761828
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External links

1. Burgers' equation in Wikipedia, the free encyclopedia
2. Burgers' equation in Wolfram MathWorld
3. Burgers' equation in EqWorld, the world of mathematical equations