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 and diffusion coefficient (or viscosity, as in the original fluid mechanical context) , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:
Added space-time noise forms a stochastic Burgers' equation[3]
This stochastic PDE is equivalent to the Kardar–Parisi–Zhang equation in a field upon substituting . 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:
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
Solution
Inviscid Burgers' equation
![](../I/m/Inviscid_Burgers_Equation_in_Two_Dimensions.gif)
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:
it can be put into the current density homogeneous form:
The solution of conservation equations can be constructed by the method of characteristics. This method yields that if is a solution of the ordinary differential equation
then is constant as a function of . For Burgers equation in particular is a solution of the system of ordinary equations:
The solutions of this system are given in terms of the initial values by
Substitute , then . Now the system becomes
Conclusion:
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
![](../I/m/Solution_of_the_viscous_two_dimensional_Burgers_equation.gif)
The viscous Burgers' equation can be linearized by the Cole–Hopf transformation [4]
which turns it into the equation
which can be integrated with respect to to obtain
where is a function that depends on boundary conditions. If identically (e.g. if the problem is to be solved on a periodic domain), then we get the diffusion equation
The diffusion equation can be solved, and the Cole-Hopf transformation inverted, to obtain the solution to the Burgers' equation:
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:
in Burgers' equation brings to:
that brings to:
where f(t) is an arbitrary function of time. With the transformation we can finally convert the latter to:
This is the searched heat equation, α being the diffusivity parameter. The initial condition is analogously transformed as:
where the fixed point of integration here is 0, but in general it can be set arbitrarily.
See also
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 yt + yyx = μyxx". 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.