Burgers' equation
[4] The equation was first introduced by Harry Bateman in 1915[5][6] and later studied by Johannes Martinus Burgers in 1948.
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: The term
The reason for the formation of sharp gradients for small values of
becomes intuitively clear when one examines the left-hand side of the equation.
will be propagated rightwards quicker than regions exhibiting smaller values of
's that lie in the backside will catch up with smaller
The role of the right-side diffusive term is essentially to stop the gradient becoming infinite.
The solution to the equation and along with the initial condition can be constructed by the method of characteristics.
is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn.
is unchanged as we move along the characteristic emanating from each point
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[8][9] The implicit solution described above containing an arbitrary function
However, the inviscid Burgers' equation, being a first-order partial differential equation, also has a complete integral which contains two arbitrary constants (for the two independent variables).
[10][better source needed] Subrahmanyan Chandrasekhar provided the complete integral in 1943,[11] which is given by where
The complete integral satisfies a linear initial condition, i.e.,
by where the lower limit is chosen arbitrarily.
Inverting the Cole–Hopf transformation, we have which simplifies, by getting rid of the time-dependent prefactor in the argument of the logarithm, to This solution is derived from the solution of the heat equation for
Explicit expressions for the viscous Burgers' equation are available.
, then we have a traveling-wave solution (with a constant speed
) given by This solution, that was originally derived by Harry Bateman in 1915,[5] is used to describe the variation of pressure across a weak shock wave[15].
(say, the Reynolds number) is a constant, then we have[17] In the limit
, the solution approaches that of the aforementioned Chandrasekhar's shock-wave solution of the inviscid Burgers' equation and is given by The shock wave location and its speed are given by
may be regarded as an initial Reynolds number at time
, may be regarded as the time-varying Reynold number.
and its solution can be constructed using method of characteristics as before.
Wiener process, forms a stochastic Burgers' equation[19] This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field
