Burgers equation
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Burgers' equation or Bateman–Burgers equation is a fundamental
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
and
convection–diffusion equation The convection–diffusion equation is a combination of the diffusion equation, diffusion and convection (advection equation, advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferr ...
occurring in various areas of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, such as
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
,
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 gas ...
,
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 r ...
, and
traffic flow In mathematics and 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 devi ...
. The equation was first introduced by
Harry Bateman Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician with a specialty in differential equations of mathematical physics. With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poincare ...
in 1915 and later studied by
Johannes Martinus Burgers Johannes (Jan) Martinus Burgers (January 13, 1895 – June 7, 1981) was a Dutch physicist and the brother of the physicist Wilhelm G. Burgers. Burgers studied in Leiden under Paul Ehrenfest, where he obtained his PhD in 1918. He is credited to ...
in 1948. For a given field u(x,t) and
diffusion coefficient Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enco ...
(or ''kinematic viscosity'', as in the original fluid mechanical context) \nu, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the
dissipative system A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dis ...
: \frac + u \frac = \nu\frac. When the diffusion term is absent (i.e. \nu=0), Burgers' equation becomes the inviscid Burgers' equation: \frac + u \frac = 0, which is a prototype for
conservation equations Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws. Conservation may also refer to: Environment and natural resources * Nature conservation, the protection and manageme ...
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 med ...
s). The previous equation is the ''advective form'' of the Burgers' equation. The ''conservative form'' is found to be more useful in numerical integration \frac + \frac\frac = 0.


Terms

There are 4 parameters in Burgers' equation: u, x, t and \nu. In a system consisting of a moving viscous fluid with one spatial (x) and one temporal (t) 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: * x: spatial coordinate * t: temporal coordinate * u(x,t): speed of fluid at the indicated spatial and temporal coordinates * \nu: viscosity of fluid The viscosity is a constant physical property of the fluid, and the other parameters represent the dynamics contingent on that viscosity.


Inviscid Burgers' equation

The inviscid Burgers' equation is a
conservation equation In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...
, more generally a first order quasilinear
hyperbolic equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
. The solution to the equation and along with the initial condition \frac + u \frac = 0, \quad u(x,0) = f(x) can be constructed by the
method of characteristics In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial d ...
. The characteristic equations are \frac = u, \quad \frac=0. Integration of the second equation tells us that u is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e., u=c, \quad x = ut + \xi where \xi is the point (or parameter) on the ''x''-axis (''t'' = 0) of the ''x''-''t'' plane from which the characteristic curve is drawn. Since u at x-axis is known from the initial condition and the fact that u is unchanged as we move along the characteristic emanating from each point x=\xi, we write u=c=f(\xi) on each characteristic. Therefore, the family of trajectories of characteristics parametrized by \xi is x=f(\xi) t+ \xi. Thus, the solution is given by u(x,t) = f(\xi) = f(x-ut), \quad \xi = x - f(\xi) t. 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 med ...
. 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 t_b = \inf_x\left(\frac\right)


Inviscid Burgers' equation for linear initial condition

Subrahmanyan Chandrasekhar Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for "... ...
provided the explicit solution in 1943 when the initial condition is linear, i.e., f(x) = ax + b, where a and b are constants. The explicit solution is u(x,t) = \frac. 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. Using this complete integral, Chandrasekhar obtained the general solution described for arbitrary initial conditions from the ''
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...
'' of the complete integral.


Viscous Burgers' equation

The viscous Burgers' equation can be converted to a linear equation by the
Cole–Hopf transformation The Cole–Hopf transformation is a method of solving parabolic partial differential equations (PDEs) with a quadratic nonlinearity of the form:u_ - a\Delta u + b\, \nabla u\, ^ = 0, \quad u(0,x) = g(x) where x\in \mathbb^, a,b are constants, \Del ...
, u = -2\nu \frac\frac, which turns it into the equation \frac \left( \frac\frac\right) = \nu \frac \left( \frac\frac\right) which can be integrated with respect to x to obtain \frac = \nu \frac + g(t) \phi where g(t) is a function that depends on boundary conditions. If g(t)=0 identically (e.g. if the problem is to be solved on a periodic domain), then we get the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
\frac=\nu\frac. The diffusion equation can be solved, and the Cole–Hopf transformation inverted, to obtain the solution to the Burgers' equation: u(x,t)=-2\nu\frac\ln\left\.


Other forms


Generalized Burgers' equation

The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e., \frac + c(u) \frac = \nu\frac. where c(u) is any arbitrary function of u. The inviscid \nu=0 equation is still a quasilinear hyperbolic equation for c(u)>0 and its solution can be constructed using
method of characteristics In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial d ...
as before.


Stochastic Burgers' equation

Added space-time noise \eta(x,t) = \dot W(x,t), where W is an L^2(\mathbb R) Wiener process, forms a stochastic Burgers' equation \frac + u \frac = \nu \frac-\lambda\frac This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field h(x,t) upon substituting u(x,t)=-\lambda\partial h/\partial x.


See also

*
Euler–Tricomi equation In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi. : u_+xu_=0. \, It is elliptic in the ha ...
*
Chaplygin's equation In gas dynamics, Chaplygin's equation, named after Sergei Alekseevich Chaplygin (1902), is a partial differential equation useful in the study of transonic flow. It is : \frac + \frac\frac+v \frac=0. Here, c=c(v) is the speed of sound, determi ...
*
Conservation equation In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...
*
Fokker–Planck equation In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...


References

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External links


Burgers' Equation
at EqWorld: The World of Mathematical Equations.
Burgers' Equation
at NEQwiki, the nonlinear equations encyclopedia. Conservation equations Equations of fluid dynamics Fluid dynamics