
Burgers' equation or Bateman–Burgers equation is a fundamental
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
and
convection–diffusion equation occurring in various areas of
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, such as
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
,
nonlinear acoustics
Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics (for sound waves in liquids and gases) ...
,
gas dynamics
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ...
, and
traffic flow
In transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the ai ...
. 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:
:
The term
can also be rewritten as
. 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 wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
s).
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. The term
is evidently a wave operator describing a wave propagating in the positive
-direction with a speed
. Since the wave speed is
, regions exhibiting large values of
will be propagated rightwards quicker than regions exhibiting smaller values of
; in other words, if
is decreasing in the
-direction, initially, then larger
's that lie in the backside will catch up with smaller
's on the front side. The role of the right-side diffusive term is essentially to stop the gradient becoming infinite.
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
Method (, methodos, from μετά/meta "in pursuit or quest of" + ὁδός/hodos "a method, system; a way or manner" of doing, saying, etc.), literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In re ...
. Let
be the parameter characterising any given characteristics in the
-
plane, then the characteristic equations are given by
:
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
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
. 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
:
Complete integral of the inviscid Burgers' equation
The implicit solution described above containing an arbitrary function
is called the general integral. 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).
Subrahmanyan Chandrasekhar provided the complete integral in 1943, which is given by
:
where
and
are arbitrary constants. The complete integral satisfies a linear initial condition, i.e.,
. One can also construct the general integral using the above complete integral.
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 an arbitrary function of time. Introducing the transformation
(which does not affect the function
), the required equation reduces to that of the
heat equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
[Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. Page 352-354.]
:
The diffusion equation
can be solved. That is, if
, then
:
The initial function
is related to the initial function
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
that decays to zero as
; other solutions for
can be obtained starting from solutions of
that satisfies different boundary conditions.
Some explicit solutions of the viscous Burgers' equation
Explicit expressions for the viscous Burgers' equation are available. Some of the physically relevant solutions are given below:
Steadily propagating traveling wave
If
is such that
and
and
, then we have a traveling-wave solution (with a constant speed
) given by
:
This solution, that was originally derived by
Harry Bateman in 1915, is used to describe the variation of pressure across a
weak shock wave. When
and
this simplifies to
:
with
.
Delta function as an initial condition
If
, where
(say, the
Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
) is a constant, then we have
:
In the limit
, the limiting behaviour is a diffusional spreading of a source and therefore is given by
:
On the other hand, In the limit
, the solution approaches that of the aforementioned Chandrasekhar's shock-wave solution of the inviscid Burgers' equation and is given by
: