Burgers' equation


Burgers' equation or Bateman–Burgers equation is a fundamental partial differential 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 and diffusion coefficient , the general form of Burgers' equation in one space dimension is the dissipative system:
When the diffusion term is absent, Burgers' equation becomes the inviscid Burgers' equation:
which is a prototype for conservation equations that can develop discontinuities. The previous equation is the advective form of the Burgers' equation. The conservative form is found to be more useful in numerical integration

Inviscid Burgers' equation

The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. The solution to the equation and along with the initial condition
can be constructed by the method of characteristics. The characteristic equations are
Integration of the second equation tells us that is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e.,
where is the point on the x-axis of the x-t plane from which the characteristic curve is drawn. Since at the point, the velocity is known from the initial condition and the fact that this value is unchanged as we move along the characteristic emanating from that point, we write on that characteristic. Therefore, the trajectory of that characteristic is
Thus, the solution is given by
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 and leads to the formation of a shock wave. In fact, the breaking time before a shock wave can be formed is given by

Complete integral

provided the explicit solution in 1943 when the initial condition is linear, i.e.,, where a and b are constants. The explicit solution is
This solution is also the complete integral of the inviscid Burgers' equation because it contains as many arbitrary constants as the number of independent variables appearing in the equation. Explicit solutions for other relevant initial conditions are, in general, not known.

Viscous Burgers' equation

The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation
which turns it into the equation
which can be integrated with respect to to obtain
where is a function that depends on boundary conditions. If identically, then we get the diffusion equation
The diffusion equation can be solved, and the Cole-Hopf transformation inverted, to obtain the solution to the Burgers' equation:

Other forms

Generalized Burgers' equation

The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e.,
where is any arbitrary function of u. The inviscid equation is still a quasilinear hyperbolic equation for and its solution can be constructed using method of characteristics as before.

Stochastic Burgers' equation

Added space-time noise forms a stochastic Burgers' equation
This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field upon substituting.