Английская Википедия:Burgers' equation
Шаблон:Short description Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation[1] occurring in various areas of applied mathematics, such as fluid mechanics,[2] nonlinear acoustics,[3] gas dynamics, and traffic flow.[4] The equation was first introduced by Harry Bateman in 1915[5][6] and later studied by Johannes Martinus Burgers in 1948.[7]
For a given field <math>u(x,t)</math> and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) <math>\nu</math>, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system: <math display="block">\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}.</math>
When the diffusion term is absent (i.e. <math>\nu=0</math>), Burgers' equation becomes the inviscid Burgers' equation: <math display="block">\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,</math> 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 <math display="block">\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial(u^2)}{\partial x} = 0.</math>
Terms
There are 4 parameters in Burgers' equation: <math>u, x, t</math> and <math>\nu</math>. In a system consisting of a moving viscous fluid with one spatial (<math>x</math>) and one temporal (<math>t</math>) 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:[8]
- <math>x</math>: spatial coordinate
- <math>t</math>: temporal coordinate
- <math>u(x,t)</math>: speed of fluid at the indicated spatial and temporal coordinates
- <math>\nu</math>: 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, more generally a first order quasilinear hyperbolic equation. The solution to the equation and along with the initial condition <math display="block">\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0, \quad u(x,0) = f(x)</math> can be constructed by the method of characteristics. The characteristic equations are <math display="block">\frac{dx}{dt} = u, \quad \frac{du}{dt}=0.</math>
Integration of the second equation tells us that <math>u</math> is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e., <math display="block">u=c, \quad x = ut + \xi </math> where <math>\xi</math> is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn. Since <math>u</math> at <math>x</math>-axis is known from the initial condition and the fact that <math>u</math> is unchanged as we move along the characteristic emanating from each point <math>x=\xi</math>, we write <math>u=c=f(\xi)</math> on each characteristic. Therefore, the family of trajectories of characteristics parametrized by <math>\xi</math> is <math display="block">x=f(\xi) t+ \xi.</math>
Thus, the solution is given by <math display="block">u(x,t) = f(\xi) = f(x-ut), \quad \xi = x - f(\xi) t.</math>
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[9] <math display="block">t_b = \inf_x\left(\frac{-1}{ f'(x)}\right)</math>
Inviscid Burgers' equation for linear initial condition
Subrahmanyan Chandrasekhar provided the explicit solution in 1943 when the initial condition is linear, i.e., <math>f(x) = ax + b</math>, where a and b are constants.[10] The explicit solution is <math display="block">u(x,t) = \frac{ax+b}{at+1}.</math>
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.[11]Шаблон:Better source Using this complete integral, Chandrasekhar obtained the general solution described for arbitrary initial conditions from the envelope of the complete integral.
Viscous Burgers' equation
The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation,[12][13][14] <math display="block">u = -2\nu \frac{1}{\phi}\frac{\partial\phi}{\partial x},</math> which turns it into the equation <math display="block"> \frac{\partial}{\partial x} \left( \frac{1}{\phi}\frac{\partial\phi}{\partial t}\right) = \nu \frac{\partial}{\partial x} \left( \frac{1}{\phi}\frac{\partial^2\phi}{\partial x^2}\right) </math> which can be integrated with respect to <math>x</math> to obtain <math display="block"> \frac{\partial\phi}{\partial t} = \nu \frac{\partial^2\phi}{\partial x^2} + g(t) \phi </math> where <math>g(t)</math> is a function that depends on boundary conditions. If <math>g(t)=0</math> identically (e.g. if the problem is to be solved on a periodic domain), then we get the diffusion equation <math display="block">\frac{\partial\phi}{\partial t}=\nu\frac{\partial^2\phi}{\partial x^2}.</math>
The diffusion equation can be solved, and the Cole–Hopf transformation inverted, to obtain the solution to the Burgers' equation: <math display="block">u(x,t)=-2\nu\frac{\partial}{\partial x}\ln\left\{(4\pi \nu t)^{-1/2}\int_{-\infty}^\infty \exp\left[-\frac{(x-x')^2}{4\nu t} - \frac{1}{2\nu}\int_0^{x'}f(x)dx\right]dx'\right\}.</math>
Other forms
Generalized Burgers' equation
The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e., <math display="block">\frac{\partial u}{\partial t} + c(u) \frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}.</math> where <math>c(u)</math> is any arbitrary function of u. The inviscid <math>\nu=0</math> equation is still a quasilinear hyperbolic equation for <math>c(u)>0</math> and its solution can be constructed using method of characteristics as before.[15]
Stochastic Burgers' equation
Added space-time noise <math>\eta(x,t) = \dot W(x,t)</math>, where <math>W</math> is an <math>L^2(\mathbb R)</math> Wiener process, forms a stochastic Burgers' equation[16] <math display="block">\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}-\lambda\frac{\partial\eta}{\partial x}.</math>
This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field <math>h(x,t)</math> upon substituting <math>u(x,t)=-\lambda\partial h/\partial x</math>.
See also
References
External links
- Burgers' Equation at EqWorld: The World of Mathematical Equations.
- Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.
- ↑ Шаблон:Cite journal
- ↑ It relates to the Navier–Stokes momentum equation with the pressure term removed Шаблон:Citation: here the variable is the flow speed y=u
- ↑ It arises from Westervelt equation with an assumption of strictly forward propagating waves and the use of a coordinate transformation to a retarded time frame: here the variable is the pressure
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Whitham, G. B. (2011). Linear and nonlinear waves (Vol. 42). John Wiley & Sons.
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite report
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.
- ↑ Шаблон:Cite journal
