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'''Burgers' equation''' is a fundamental [[partial differential equation]] from [[fluid mechanics]]. It occurs in various areas of [[applied mathematics]], such as modeling of [[gas dynamics]] and [[traffic flow]]. It is named for [[Johannes Martinus Burgers]] (1895–1981).


For a given [[velocity]] ''u'' and [[viscosity]] coefficient <math>\nu </math>, the general form of Burgers' equation (also known as '''viscous Burgers' equation''') is:
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:<math>\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}</math>.
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When <math>\nu = 0</math>, Burgers' equation becomes the '''inviscid Burgers' equation''':
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:<math>\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,</math>
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which is a prototype for equations for which the solution can develop discontinuities ([[shock wave]]s). The previous equation is the 'advection form' of the Burgers' equation. The 'conservation form' is:
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:<math>\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial}{\partial x}\big(u^2\big) = 0.</math>
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== Solution ==
 
===Inviscid Burgers' equation===
 
[[Image:Inviscid Burgers Equation in Two Dimensions.gif|right|frame|This is a numerical simulation of the inviscid Burgers Equation in two space variables up until the time of shock formation.]]
 
The inviscid Burgers' equation is a first order partial differential equation (PDE). Its solution can be constructed by the [[method of characteristics]]. This method yields that if <math>X(t)</math> is a solution of the [[ordinary differential equation]]
 
:<math>\frac{dX(t)}{dt} = u[X(t),t]</math>
 
then <math>U(t) := u[X(t),t]</math> is constant as a function of <math>t</math>. Hence <math>[X(t),U(t)]</math> is a solution of the system of ordinary equations:
 
:<math>\frac{dX}{dt}=U,</math>
 
:<math>\frac{dU}{dt}=0.</math>
 
The solutions of this system are given in terms of the initial values by:
 
:<math>\displaystyle X(t)=X(0)+tU(0),</math>
 
:<math>\displaystyle U(t)=U(0).</math>
 
Substitute <math>X(0)= \eta</math>, then <math>U(0)=u[X(0),0]=u(\eta,0)</math>. Now the system becomes
 
:<math>\displaystyle X(t)=\eta+tu(\eta,0)</math>
 
:<math>\displaystyle U(t)=U(0).</math>
 
Conclusion:
 
:<math>\displaystyle
u(\eta,0)=U(0)=U(t)=u[X(t),t]=u[\eta+tu(\eta,0),t].
</math>
 
This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect.  If the characteristics do intersect, then a classical solution to the PDE does not exist.
 
===Viscous Burgers' equation===
 
The viscous Burgers' equation can be linearized by the Cole–Hopf transformation
:<math>u=-2\nu \frac{1}{\phi}\frac{\partial\phi}{\partial x},</math>
 
which turns it into the  equation
 
:<math> \frac{\partial}{\partial x} \Bigl( \frac{1}{\phi}\frac{\partial\phi}{\partial t}\Bigr)
= \nu  \frac{\partial}{\partial x} \Bigl( \frac{1}{\phi}\frac{\partial^2\phi}{\partial x^2}\Bigr) </math>
 
which can be rewritten as
 
:<math> \frac{\partial\phi}{\partial t} = \nu\frac{\partial^2\phi}{\partial x^2} + f(t) \phi </math>
 
with f(t) an arbitrary function. Assuming it vanishes, we get the [[diffusion equation]]
 
:<math>\frac{\partial\phi}{\partial t}=\nu\frac{\partial^2\phi}{\partial x^2}.</math>
 
This allows one to solve an initial value problem:
 
:<math>u(x,t)=-2\nu\frac{\partial}{\partial x}\ln\Bigl\{(4\pi\nu t)^{-1/2}\int_{-\infty}^\infty\exp\Bigl[-\frac{(x-x')^2}{4\nu t}  -\frac{1}{2\nu}\int_0^{x'}u(x'',0)dx''\Bigr]dx'\Bigr\}.</math>
 
==References==
<references/>
 
==External links==
 
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde1301.pdf Burgers' Equation] at EqWorld: The World of Mathematical Equations.
* [http://www.primat.mephi.ru/wiki/ow.asp?Burgers%27_equation Burgers' Equation] at NEQwiki, the nonlinear equations encyclopedia.
* Burgers shock-waves and sound in a 2D microfluidic droplets ensemble [http://www.weizmann.ac.il/complex/tlusty/papers/PhysRevLett2009.pdf Phys. Rev. Lett. 103, 114502 (2009)].
[[Category:Partial differential equations]]
[[Category:Equations of fluid dynamics]]

Latest revision as of 04:24, 2 December 2014


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