- Burgers' equation
Burgers' equation is a fundamental
partial differential equation fromfluid mechanics . It occurs in various areas ofapplied mathematics , such as modeling ofgas dynamics andtraffic flow . It is named forJohannes Martinus Burgers (1895-1981).For a given
velocity "u" andviscosity coefficient u , the general form of Burgers' equation is::frac{partial u}{partial t} + u frac{partial u}{partial x} = u frac{partial^2 u}{partial x^2}.
When u = 0, Burgers' equation becomes the inviscid Burgers' equation:
:frac{partial u}{partial t} + u frac{partial u}{partial x} = 0,
which is a prototype for equations for which the solution can develop discontinuities (
shock wave s).Solution
The inviscid Burgers' equation is a first order partial differential equation. Its solution can be constructed by the
method of characteristics . This method yields that if X(t) is a solution of theordinary differential equation :frac{dX(t)}{dt} = u [X(t),t]
then U(t) := u [X(t),t] is constant as a function of t. Hence X(t),U(t)] is a solution of the system of ordinary equations
:frac{dX}{dt}=U
:frac{dU}{dt}=0.
The solutions of this system are given in terms of the initial values by
:displaystyle X(t)=X(0)+tU(0)
:displaystyle U(t)=U(0).
Substitute X(0)= eta, then U(0)=u [X(0),0] =u(eta,0). Now the system becomes
:displaystyle X(t)=eta+tu(eta,0)
:displaystyle U(t)=U(0).
Conclusion:
:displaystyle u(eta,0)=U(0)=U(t)=u [X(t),t] =u [eta+tu(eta,0),t] .
This is an implicit relation that determines the solution of the inviscid Burgers' equation.
The viscous Burgers equation can be linearized by the Cole-Hopf substitution :u=-2 u frac{1}{phi}frac{partialphi}{partial x},which turns it into the
diffusion equation:frac{partialphi}{partial t}= ufrac{partial^2phi}{partial x^2}.That allows one to solve an initial value problem::u(x,t)=-2 ufrac{partial}{partial x}lnBigl{(4pi u t)^{-1/2}int_{-infty}^inftyexpBigl [-frac{(x-x')^2}{4 u t} -frac{1}{2 u}int_0^{x'}u(x",0)dx"Bigr] dx'Bigr}.External links
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde1301.pdf Burgers' Equation] at EqWorld: The World of Mathematical Equations.
Wikimedia Foundation. 2010.