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 (or kinematic viscosity, as in the original fluid mechanical context) , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:
When the diffusion term is absent (i.e. ), 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 advective form of the Burgers' equation. The conservative form is found to be more useful in numerical integration
Explanation of Terms
There are 4 terms in Burgers' equation: and . In a system consisting of a moving viscous fluid with one spatial () and one temporal () dimension, e.g. a thin ideal pipe with fluid running through it, Burgers' equation describes the speed of the fluid at each location along the pipe as time progresses. The terms of the equation represent the following quantities:
- : spatial coordinate
- : temporal coordinate
- : speed of fluid at the indicated spatial and temporal coordinates
- : viscosity of fluid
The viscosity is a constant physical property of the fluid, and the other terms represent the dynamics contingent on that viscosity.
Inviscid Burgers' equation
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 (or parameter) on the x-axis (t = 0) 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
Inviscid Burgers' equation for linear initial condition
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.[better source needed] Explicit solutions for other relevant initial conditions are, in general, not known.
Viscous Burgers' equation
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 (e.g. if the problem is to be solved on a periodic domain), 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:
Generalized Burgers' equation
The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e.,
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 .
- It relates to the Navier–Stokes momentum equation with the pressure term removed Burgers Equation (PDF): here the variable is the flow speed y=u
- It arises from Westervelt equation with an assumption of strictly forward propagating waves and the use of a coordinate transformation to a retarded time frame: here the variable is the pressure
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