Burgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. It is named for Johannes Martinus Burgers (1895–1981). It relates to the Navier-Stokes equation for incompressible flow with the pressure term removed.
Added space-time noise forms a stochastic Burgers' equation
This stochastic PDE is equivalent to the Kardar-Parisi-Zhang equation in a field upon substituting . But whereas Burgers' equation only applies in one spatial dimension, the Kardar-Parisi-Zhang equation generalises to multiple dimensions.
When the viscosity , Burgers' equation becomes the inviscid Burgers' equation:
Inviscid Burgers' equation
it can be put into the current density homogeneous form:
then is constant as a function of . For Burgers equation in particular is a solution of the system of ordinary equations:
The solutions of this system are given in terms of the initial values by
Substitute , then . Now the system becomes
This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist.
Viscous Burgers' equation
which turns it into the equation
which can be rewritten as
with f(t) an arbitrary function. Assuming it vanishes, we get the diffusion equation
This allows one to solve an initial value problem:
in Burgers' equation brings to:
that brings to:
where f(t) is an arbitrary function of time. With the transformation we can finally convert the latter to:
This is the searched heat equation, α being the diffusivity parameter. The initial condition is analogously transformed as:
where the fixed point of integration here is 0, but in general it can be set arbitrarily.
- Burgers Equation (PDF)
- W. Wang and A. J. Roberts. Diffusion approximation for self-similarity of stochastic advection in Burgers’ equation. Communications in Mathematical Physics, July 2014.
- Eberhard Hopf (September 1950). "The partial differential equation ut + uux = μxx". Communications on Pure and Applied Mathematics 3 (3): 201–230. doi:10.1002/cpa.3160030302.
- Landau, Lifshits, 'Fluid Mechanics', par. 93, Problem 2