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Burgers' equation
Partial differential equation / From Wikipedia, the free encyclopedia
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation[1] occurring in various areas of applied mathematics, such as fluid mechanics,[2] nonlinear acoustics,[3] gas dynamics, and traffic flow.[4] The equation was first introduced by Harry Bateman in 1915[5][6] and later studied by Johannes Martinus Burgers in 1948.[7] For a given field and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context)
, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Burgers_equation.gif/320px-Burgers_equation.gif)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Solution_of_the_Burgers_equation_-_N_wave.gif/320px-Solution_of_the_Burgers_equation_-_N_wave.gif)
The term can also rewritten as
. When the diffusion term is absent (i.e.
), Burgers' equation becomes the inviscid Burgers' equation:
which is a prototype for conservation equations that can develop discontinuities (shock waves).
The reason for the formation of sharp gradients for small values of becomes intuitively clear when one examines the left-hand side of the equation. The term
is evidently a wave operator describing a wave propagating in the positive
-direction with a speed
. Since the wave speed is
, regions exhibiting large values of
will be propagated rightwards quickly than regions exhibiting smaller values of
; in other words, if
is decreasing in the
-direction, initially, then larger
's that lie in the backside will catch up with smaller
's that is on the front side. The role of the right-side diffusive term is essentially to stop the gradient becoming infinite.