Burgers' equation

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 ${\displaystyle u(x,t)}$ 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 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 quickly 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 that is on the front side. The role of the right-side diffusive term is essentially to stop the gradient becoming infinite.