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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 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
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, such as
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,
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, and
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. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. For a given field u(x,t) and diffusion coefficient (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: :\frac + u \frac = \nu\frac. The term u\partial u/\partial x can also be rewritten as \partial(u^2/2)/\partial x. 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 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 \nu becomes intuitively clear when one examines the left-hand side of the equation. The term \partial/\partial t + u \partial/\partial x is evidently a wave operator describing a wave propagating in the positive x-direction with a speed u. Since the wave speed is u, regions exhibiting large values of u will be propagated rightwards quicker than regions exhibiting smaller values of u; in other words, if u is decreasing in the x-direction, initially, then larger u's that lie in the backside will catch up with smaller u'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 :\frac + u \frac = 0, \quad u(x,0) = f(x) 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 t be the parameter characterising any given characteristics in the x-t plane, then the characteristic equations are given by :\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 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 :t_b = \frac.


Complete integral of the inviscid Burgers' equation

The implicit solution described above containing an arbitrary function f 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 :u(x,t) = \frac. where a and b are arbitrary constants. The complete integral satisfies a linear initial condition, i.e., f(x) = ax + b. 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, :u(x,t) = -2 \nu \frac\ln \varphi(x,t), which turns it into the equation :2 \nu \frac\left frac\left(\frac- \nu \frac\right)\right0, which can be integrated with respect to x to obtain :\frac- \nu \frac=\varphi \frac, where df/dt is an arbitrary function of time. Introducing the transformation \varphi\to \varphi e^f (which does not affect the function u(x,t)), 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. :\frac= \nu \frac. The diffusion equation can be solved. That is, if \varphi(x,0)=\varphi_0(x), then :\varphi(x,t) = \frac\int_^\infty \varphi_0(x') \exp \left \frac\rightx'. The initial function \varphi_0(x) is related to the initial function u(x,0)=f(x) by :\ln \varphi_0(x) = - \frac\int_0^x f(x') dx', where the lower limit is chosen arbitrarily. Inverting the Cole–Hopf transformation, we have :u(x,t)=-2\nu\frac\ln\left\ which simplifies, by getting rid of the time-dependent prefactor in the argument of the logarithm, to :u(x,t)=-2\nu\frac\ln\left\. This solution is derived from the solution of the heat equation for \varphi that decays to zero as x\to\pm\infty; other solutions for u can be obtained starting from solutions of \varphi 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 u(x,0)=f(x) is such that f(-\infty)=f^+ and f(+\infty)=f^- and f'(x)<0, then we have a traveling-wave solution (with a constant speed c=(f^++f^-)/2) given by :u(x,t) = c - \frac\tanh\left frac(x-ct)\right 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 f^+=2 and f^-=0 this simplifies to :u(x,t)=\frac with c=1.


Delta function as an initial condition

If u(x,0) = 2\nu Re \delta(x), where Re (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 :u(x,t)= \sqrt \left frac\right In the limit Re\to 0, the limiting behaviour is a diffusional spreading of a source and therefore is given by :u(x,t) = \frac \exp\left(-\frac\right). On the other hand, In the limit Re\to \infty, the solution approaches that of the aforementioned Chandrasekhar's shock-wave solution of the inviscid Burgers' equation and is given by :u(x,t) = \begin\frac, \quad 0 The shock wave location and its speed are given by x=\sqrt and \sqrt.


N-wave solution

The N-wave solution comprises a compression wave followed by a rarefaction wave. A solution of this type is given by :u(x,t) = \frac\left + \frac\sqrt\exp\left(-\frac\right)\right where Re_0 may be regarded as an initial Reynolds number at time t=t_0 and Re(t) = (1/2\nu) \int_0^\infty udx=\ln (1+\sqrt) with \tau = t_0 \sqrt, may be regarded as the time-varying Reynold number.


Other forms


Multi-dimensional Burgers' equation

In two or more dimensions, the Burgers' equation becomes :\frac + u \cdot \nabla u = \nu \nabla^2 u. One can also extend the equation for the vector field \mathbf u, as in :\frac + \mathbf u \cdot \nabla \mathbf u = \nu \nabla^2 \mathbf u.


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 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 ...
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 In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one o ...
, 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

* Chaplygin's equation * Conservation equation *
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 ...
*
Fokker–Planck equation In statistical mechanics and information theory, 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 (physi ...
* KdV-Burgers equation * Euler–Arnold equation


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