Burgers' equation
The Burgers equation is a simple non-linear partial differential equation that occurs in various fields of applied mathematics. The equation is named after the Dutch physicist Johannes Martinus Burgers.
The equation is a nonlinear partial differential equation of second order for a function of two variables.
In general form, the equation looks like this (also viscous Burgers equation called ):
This is obviously equivalent to
The parameter can be interpreted as the viscosity parameter here.
Often also the above equation for the case called Burger equation, some authors call this special case the nonviscous Burgers equation ( engl: inviscid Burgers' equation ):
Or
Although both formal representations are equivalent, but the second form is more advantageous for numerical calculations. The reason for this is the " conservation form" of the differential equation.
See also: conservation equation, finite volume method.
Application
Burgers equation is the simplest example of a non-linear hyperbolic differential equations, and is therefore often used as a test case of numerical algorithms for this type of equation.
It can also be seen as a simple model of a one-dimensional flow. As an example, the density of traffic is often placed on the road, their time course can be modeled with the help of the Burger equation.
The interpretation of a one-dimensional flow is due to the similarity of the equation by the non-linear part of the Navier -Stokes equation forth.
Solutions
The nonviscous equation can be solved using the method of characteristics. However, the equation has not necessarily a unique solution. With appropriately chosen starting values shocks can be observed. The viscous equation motivated then also for the Euler equations, the concept of the solution with vanishing viscosity. That is the one solution to the unviskosen Burger equation, which corresponds to a solution of the viscous equation with vanishing viscosity.
For the viscous Burgers equation performs a Hopf -Cole transformation to the target.