Burgers' equation

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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 $\nu$ , the general form of Burgers' equation (also known as viscous Burgers' equation) is:

${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}$ .

When $\nu =0$ , 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 waves). The previous equation is the 'advection form' of the Burgers' equation. The 'conservation form' is:

${\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial }{\partial x}}{\big (}u^{2}{\big )}=0.$

Solution

Inviscid Burgers' equation

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 $X(t)$ is a solution of the ordinary 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 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

$u=-2\nu {\frac {1}{\phi }}{\frac {\partial \phi }{\partial x}},$

which turns it into the equation

${\frac {\partial }{\partial x}}{\Bigl (}{\frac {1}{\phi }}{\frac {\partial \phi }{\partial t}}{\Bigr )}=\nu {\frac {\partial }{\partial x}}{\Bigl (}{\frac {1}{\phi }}{\frac {\partial ^{2}\phi }{\partial x^{2}}}{\Bigr )}$

which can be rewritten as

${\frac {\partial \phi }{\partial t}}=\nu {\frac {\partial ^{2}\phi }{\partial x^{2}}}+f(t)\phi$

with f(t) an arbitrary function. Assuming it vanishes, we get the diffusion equation

${\frac {\partial \phi }{\partial t}}=\nu {\frac {\partial ^{2}\phi }{\partial x^{2}}}.$

This allows one to solve an initial value problem:

$u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln {\Bigl \{}(4\pi \nu t)^{{-1/2}}\int _{{-\infty }}^{\infty }\exp {\Bigl [}-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{{x'}}u(x'',0)dx''{\Bigr ]}dx'{\Bigr \}}.$

References

External links

Burgers' Equation at EqWorld: The World of Mathematical Equations.
Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.
Burgers shock-waves and sound in a 2D microfluidic droplets ensemble Phys. Rev. Lett. 103, 114502 (2009).

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