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 $u(x,t)$ and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) $\nu$ , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:

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

When the diffusion term is absent (i.e. $\nu =0$ ), Burgers' equation becomes the inviscid Burgers' equation:

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

which is a prototype for conservation equations that can develop discontinuities (shock waves).

In conservative form, the Burgers' equation becomes

{\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial u^{2}}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}.

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