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
There are 4 parameters 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 parameters represent the dynamics contingent on that viscosity.
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 -axis is known from the initial condition and the fact that is unchanged as we move along the characteristic emanating from each point , we write on each characteristic. Therefore, the family of trajectories of characteristics parametrized by 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. Whether characteristics can intersect or not depends on the initial condition. In fact, the breaking time before a shock wave can be formed is given by
Inviscid Burgers' equation for linear initial condition
Subrahmanyan Chandrasekhar 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.[better source needed] Using this complete integral, Chandrasekhar obtained the general solution described for arbitrary initial conditions from the envelope of the complete integral.
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^Wang, W.; Roberts, A. J. (2015). "Diffusion Approximation for Self-similarity of Stochastic Advection in Burgers' Equation". Communications in Mathematical Physics. 333: 1287–1316. arXiv:1203.0463. doi:10.1007/s00220-014-2117-7.