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Burgers' equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow. It is named for Johannes Martinus Burgers (1895–1981).
Contents
For a given field
Added space-time noise
This stochastic PDE is equivalent to the Kardar–Parisi–Zhang equation in a field
When the diffusion term is absent (i.e. d = 0), Burgers' equation becomes the inviscid Burgers' equation:
which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the 'advection form' of the Burgers' equation. The 'conservation form' is
Inviscid Burgers' equation
The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. In fact by defining its current density as the kinetic energy density:
it can be put into the current density homogeneous form:
The solution of conservation equations can be constructed by the method of characteristics. This method yields that if
then
The solutions of this system are given in terms of the initial values by
Substitute
Conclusion:
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
The viscous Burgers' equation can be linearized by the Cole–Hopf transformation
which turns it into the equation
which can be integrated with respect to
where
The diffusion equation can be solved, and the Cole-Hopf transformation inverted, to obtain the solution to the Burgers' equation:
Heat equation
Burgers' equation can notably be converted to a heat equation through a nonlinear substitution, as suggested by E. Hopf in 1950. In fact substituting:
in Burgers' equation brings to:
that brings to:
where f(t) is an arbitrary function of time. With the transformation
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.