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Burgers' equation

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Burgers' equation

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

Contents

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:

∂ y ∂ t + y ∂ y ∂ x = d ∂ 2 y ∂ x 2 .

Added space-time noise η ( x , t ) forms a stochastic Burgers' equation

∂ y ∂ t + y ∂ y ∂ x = d ∂ 2 y ∂ x 2 − λ ∂ η ∂ x

This stochastic PDE is equivalent to the Kardar–Parisi–Zhang equation in a field h ( x , t ) upon substituting y ( x , t ) = − λ ∂ h / ∂ 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:

∂ y ∂ t + y ∂ y ∂ 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

∂ y ∂ t + 1 2 ∂ ∂ x ( y 2 ) = 0.

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 ) = 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

d X ( t ) d t = 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:

d X d t = Y , and d Y d t = 0.

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

X ( t ) = X ( 0 ) + t Y ( 0 ) , and Y ( t ) = Y ( 0 ) .

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

X ( t ) = η + t y ( η , 0 ) and Y ( t ) = Y ( 0 ) .

Conclusion:

y ( η , 0 ) = Y ( 0 ) = Y ( t ) = y [ X ( t ) , t ] = y [ η + t y ( η , 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

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

y = − 2 d 1 ϕ ∂ ϕ ∂ x ,

which turns it into the equation

∂ ∂ x ( 1 ϕ ∂ ϕ ∂ t ) = d ∂ ∂ x ( 1 ϕ ∂ 2 ϕ ∂ x 2 )

which can be integrated with respect to x to obtain

∂ ϕ ∂ t = d ∂ 2 ϕ ∂ x 2 + f ( t ) ϕ

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

∂ ϕ ∂ t = d ∂ 2 ϕ ∂ 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 ) = − 2 d ∂ ∂ x ln ⁡ { ( 4 π d t ) − 1 / 2 ∫ − ∞ ∞ exp ⁡ [ − ( x − x ′ ) 2 4 d t − 1 2 d ∫ 0 x ′ y ( x ″ , 0 ) d x ″ ] d x ′ } .

Heat equation

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

y ( x , t ) → − 2 α ∂ ∂ x ln ⁡ y ( x , t )

in Burgers' equation brings to:

2 α ∂ ∂ x [ 1 y ( − ∂ y ∂ t + α ∂ 2 y ∂ 2 x ) ] = 0

that brings to:

∂ y ∂ t − α ∂ 2 y ∂ 2 x = y d f ( t ) d t

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

∂ y ∂ t − α ∂ 2 y ∂ 2 x = 0

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

y 0 ( x ) → − 1 2 α ∫ 0 x y 0 ( x ′ ) d x ′

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

References

Burgers' equation Wikipedia
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