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

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 integrated with respect to $x$ to obtain

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

where $f(t)$ is a function that depends on boundary conditions. If $f(t)=0$ identically (e.g. if the problem is to be solved on a periodic domain), then we get the diffusion equation

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

The diffusion equation can be solved, and the Cole-Hopf transformation inverted, to obtain the solution to the Burgers' equation:

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

Heat equation

References

↑ 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
↑ 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
↑ W. Wang and A. J. Roberts. Diffusion approximation for self-similarity of stochastic advection in Burgers’ equation. Communications in Mathematical Physics, July 2014.
↑ Eberhard Hopf (September 1950). "The partial differential equationy y_t + yy_x = μy_xx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.
↑ Landau, Lifshits, 'Fluid Mechanics', par. 93, Problem 2

External links

Burgers' Equation at EqWorld: The World of Mathematical Equations.
Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.
Burgers shock-waves and sound in a 2D microfluidic droplets ensemble Phys. Rev. Lett. 103, 114502 (2009).
Burgers' Equation Wolfram|Alpha search results.

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