Burgers' equation

Burgers' equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics,[1] nonlinear acoustics,[2] gas dynamics, traffic flow. It is named for Johannes Martinus Burgers (1895–1981).

For a given field y(x,t) and diffusion coefficient (or viscosity, as in the original fluid mechanical context) d, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:

\frac{\partial y}{\partial t} + y \frac{\partial y}{\partial x} = d \frac{\partial^2 y}{\partial x^2}.

Added space-time noise \eta(x,t) forms a stochastic Burgers' equation[3]

\frac{\partial y}{\partial t} + y \frac{\partial y}{\partial x} = d \frac{\partial^2 y}{\partial x^2}-\lambda\frac{\partial\eta}{\partial x}

This stochastic PDE is equivalent to the Kardar-Parisi-Zhang equation in a field h(x,t) upon substituting y(x,t)=-\lambda\partial h/\partial x. But whereas Burgers' equation only applies in one spatial dimension, the Kardar-Parisi-Zhang equation generalises to multiple dimensions.

When the diffusion term is absent (i.e. d=0), Burgers' equation becomes the inviscid Burgers' equation:

\frac{\partial y}{\partial t} + y \frac{\partial y}{\partial x} = 0,

which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the 'advection form' of the Burgers' equation. The 'conservation form' is

\frac{\partial y}{\partial t} + \frac{1}{2}\frac{\partial}{\partial x}\big(y^2\big) = 0.

Solution

Inviscid Burgers' equation

The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. In fact by defining its current density as the kinetic energy density:

j(y)= \frac 1 2 y^2

it can be put into the current density homogeneous form:

y_t + j_x (y) = 0.

The solution of conservation equations can be constructed by the method of characteristics. This method yields that if X(t) is a solution of the ordinary differential equation

\frac{dX(t)}{dt} = y[X(t),t]

then Y(t) := y[X(t),t] is constant as a function of t. For Burgers equation in particular [X(t),Y(t)] is a solution of the system of ordinary equations:

\frac{dX}{dt}=Y,\quad\mbox{and}\quad \frac{dY}{dt}=0.

The solutions of this system are given in terms of the initial values by

X(t)=X(0)+tY(0),\quad\mbox{and}\quad Y(t)=Y(0).

Substitute X(0)= \eta, then Y(0)=y[X(0),0]=y(\eta,0). Now the system becomes

X(t)=\eta+ty(\eta,0)\quad\mbox{and}\quad Y(t)=Y(0).

Conclusion:

y(\eta,0)=Y(0)=Y(t)=y[X(t),t]=y[\eta+ty(\eta,0),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.

Viscous Burgers' equation

This is a numerical solution of the viscous two dimensional Burgers equation using an initial Gaussian profile. We see shock formation, and dissipation of the shock due to viscosity as it travels.

The viscous Burgers' equation can be linearized by the Cole–Hopf transformation [4]

y=-2d \frac{1}{\phi}\frac{\partial\phi}{\partial x},

which turns it into the equation

\frac{\partial}{\partial x} \Bigl( \frac{1}{\phi}\frac{\partial\phi}{\partial t}\Bigr) = d \frac{\partial}{\partial x} \Bigl( \frac{1}{\phi}\frac{\partial^2\phi}{\partial x^2}\Bigr)

which can be rewritten as

\frac{\partial\phi}{\partial t} = d \frac{\partial^2\phi}{\partial x^2} + f(t) \phi

with f(t) an arbitrary function. Assuming it vanishes, we get the diffusion equation

\frac{\partial\phi}{\partial t}=d\frac{\partial^2\phi}{\partial x^2}.

This allows one to solve an initial value problem:

y(x,t)=-2d\frac{\partial}{\partial x}\ln\Bigl\{(4\pi d t)^{-1/2}\int_{-\infty}^\infty\exp\Bigl[-\frac{(x-x')^2}{4d t} -\frac{1}{2d}\int_0^{x'}y(x'',0)dx''\Bigr]dx'\Bigr\}.

Heat equation

See also: Heat equation

Burgers' equation can notably be converted to a heat equation through a nonlinear substitution, as suggested by E. Hopf in 1950.[5] In fact substituting:

y(x,t) \rightarrow - 2 \alpha \frac \partial{\partial x} \ln y(x,t)

in Burgers' equation brings to:

2 \alpha \frac \partial{\partial x} \left[\frac 1 y \left(- \frac {\partial y}{\partial t} + \alpha \frac {\partial^2 y}{\partial^2 x} \right) \right] = 0

that brings to:

\frac {\partial y}{\partial t} - \alpha \frac {\partial^2 y}{\partial^2 x} = y \frac {d f(t)}{dt}

where f(t) is an arbitrary function of time. With the transformation y \rightarrow e^f y we can finally convert the latter to:

\frac {\partial y}{\partial t} - \alpha \frac {\partial^2 y}{\partial^2 x} = 0

This is the searched heat equation, α being the diffusivity parameter. The initial condition is analogously transformed as:

y_0 (x) \rightarrow - \frac 1 {2 \alpha} \int_0^x y_0(x') dx'

where the fixed point of integration here is 0, but in general it can be set arbitrarily.

See also

References

  1. ↑ It relates to the Navier-Stokes momentum equation with the pressure term removed Burgers Equation (PDF): here the variable is the flow speed y=u
  2. ↑ 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
  3. ↑ W. Wang and A. J. Roberts. Diffusion approximation for self-similarity of stochastic advection in Burgers’ equation. Communications in Mathematical Physics, July 2014.
  4. ↑ Eberhard Hopf (September 1950). "The partial differential equationy yt + yyx = μxx". Communications on Pure and Applied Mathematics 3 (3): 201–230. doi:10.1002/cpa.3160030302.
  5. ↑ Landau, Lifshits, 'Fluid Mechanics', par. 93, Problem 2

External links

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