Burgers' equation

Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. For a given field ${\displaystyle u(x,t)}$ and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) ${\displaystyle \nu }$, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:

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

The term ${\displaystyle u\partial u/\partial x}$ can also be rewritten as ${\displaystyle \partial (u^{2}/2)/\partial x}$. When the diffusion term is absent (i.e. ${\displaystyle \nu =0}$), Burgers' equation becomes the inviscid Burgers' equation:

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=0,}$

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 ${\displaystyle \nu }$ becomes intuitively clear when one examines the left-hand side of the equation. The term ${\displaystyle \partial /\partial t+u\partial /\partial x}$ is evidently a wave operator describing a wave propagating in the positive ${\displaystyle x}$-direction with a speed ${\displaystyle u}$. Since the wave speed is ${\displaystyle u}$, regions exhibiting large values of ${\displaystyle u}$ will be propagated rightwards quicker than regions exhibiting smaller values of ${\displaystyle u}$; in other words, if ${\displaystyle u}$ is decreasing in the ${\displaystyle x}$-direction, initially, then larger ${\displaystyle u}$'s that lie in the backside will catch up with smaller ${\displaystyle u}$'s on the front side. The role of the right-side diffusive term is essentially to stop the gradient becoming infinite.