I want to know the "analytical" solution for the
1-d nonlinear Burger's equation, given by :
with a computational domain of (0,1)
and periodic boundary conditions : u(x=0)=u(x=1),
the initial condition is a sine wave given by:

u= 2 + (1/(4*pi))*sin(2*pi*x);

thanks, I am looking for the
mathematics of it, is for a CFD study and I am looking at the wavenumber of
3 numerical schemes,

best regards,

Javier

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Request for Question Clarification by mathtalk-ga on 15 Feb 2005 10:03 PST

Hi, javier5440-ga:

I suspect you accidentally omitted the partial differential equation itself.

Are you referring to this 1-D nonlinear 2nd order PDE:

  u_t + u*u_x - c*u_xx = 0

where subscripts x,t refer to derivatives of u(x,t)?

regards, mathtalk-ga

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Clarification of Question by javier5440-ga on 15 Feb 2005 10:30 PST

thanks for your quick reply:

the equation is : 
ut+ (0.5*(u^2u))x=0

where I have partial derivative of U w.r.t  time (ut)
and partial derivative of (0.5*(u^2)) w.r.t. space ((0.5*(u^2u))x)



thanks

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Clarification of Question by javier5440-ga on 15 Feb 2005 10:32 PST

In the last e-mail I accidenatlly miswrote
the equation it should be:

ut + (0.5*(u^2))x=0

where I have the pde of u , w.r.t. time
and pde of (0.5*(u^2)), w.r.t space(x)

thanks

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Request for Question Clarification by mathtalk-ga on 16 Feb 2005 05:57 PST

Hi, javier5440-ga:

The equation you've asked about is called the inviscid Burgers equation:

  u_t + u*u_x = 0

and is often given as a model for evolution of singularities from
smooth initial conditions in finite time.

It is possible to give for the case of periodic initial and boundary
conditions an analytical solution in terms of Fourier series in x
whose coefficients are functions of t, valid up to the time at which
the singularity appears.

Would a reference to the literature on this subject be satisfactory
for your purposes?

regards, mathtalk-ga

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Clarification of Question by javier5440-ga on 16 Feb 2005 06:24 PST

Thanks for your reply,

I am actually looking for an analyticl condition because
I have a olution using numerical schemes using Runge-Kutta method and
different spatial differencing methods,

If it is too much work is alright, i have to have a solution for tomorrow,

I thank you for you time and compliment you on your diligent work,

best regards,

Javier5440

Hi, Javier:

Your question certainly brought back memories of graduate classes in
numerical PDE's and fluid dynamics.

Here's a recent paper that details the approach for analytically
expressing solutions to the inviscid Burgers equation in 1-D, for
smooth periodic initial data defined on the entire x-axis:

[A Simple Illustration Of A Weak Spectral Cascade - David J. Muraki]
http://www.math.sfu.ca/~dmuraki/research/cascade.pdf

Given the inviscid Burgers equation:

  u_t + u*u_x = 0

and smooth initial data:

  u(x,0) = f(x)

that is periodic in x, one can give an implicit "solution" for Burgers equation by:

  u(x,t) = f(x - u*t)

It is easily verified that for t = 0, such a function u must satisfy
the initial condition.  Furthermore implicit differentiation with
respect to x,t will show that the PDE is satisfied as well.  For fixed
time t > 0, the periodicity of u follows from the periodicity of f.

For consistency with the notation in that paper, we will assume that
f(x) is of period 2pi, rather than of period 1 as assumed in your
problem.  It is easy to show that a common change of scale for both
x,t preserves the solutions of Burgers equation while adjusting the
period of initial values f.

As the above linked paper shows, Platzman's earlier analysis of the
Fourier series for u(x,t):

Platzman, G.W.
  An exact integral of complete spectral equations
  for unsteady onedimensional flow,
  Tellus, XVI (1964), pp. 422?431

can be generalized to give (up to the time of singularity):

            a_0     +oo
  u(x,t) =  ---  +  SUM [ a_n(t)*cos(nx) + b_n(t)*sin(nx) ]
             2      n=1

where for n > 0:
             
     a_0  =  (1/pi) INTEGRAL f(z) dz  OVER [-pi,+pi]

               1
   a_n(t) =  -----  INTEGRAL sin(nz + nt*f(z)) dz OVER [-pi,+pi]
             pi*nt

              -1
   b_n(t) =  -----  INTEGRAL cos(nz + nt*f(z)) dz OVER [-pi,+pi]
             pi*nt

Note that the "constant term" a_0 is truly a constant, independent of
time t, corresponding to the conservation of the mean exhibited by
Burgers equation.

This Fourier series expansion of u(x,t) determines a single-valued
function for all time t, but it only satisfies Burgers equation up to
the point in time when a singularity develops.  The author of the
paper above, David J. Muraki, explains the distinction between the
single-valued function defined by the Fourier series, the multi-valued
function defined by the method of characteristics, and the "entropy"
solution to which one ascribes "physical reality".

Please feel free to post a Request for Clarification if I can be of
further assistance.

regards, mathtalk-ga

Thanks for your time, I greatly appreciate your answer and compliment you
on the refernce you attached, good work,

best regards,

Javier5440

The PDE which Javier has asked about is a nonlinear evolution
equation.  Without loss of generality we may view the initial
conditions as extended periodically to the entire x-axis.

The particular initial data consists of a constant plus a superimposed
"fundamental" sine wave:

  u(x,0) = 2 + (1/(4pi))*sin(2pi*x)

so that the period in x is length 1.

As the solution evolves all the higher harmonics are "excited".  In
1964 G.W. Platzmann published a paper in an obscure meterology journal
Tellus which gives the exact solution (up to singularity) for the case
of initial data which is purely sinusoidal.  The "time" coefficients
of the various harmonics are then given explicitly in terms of Bessel
functions.

This approach has recently been generalized to all smooth periodic
initial conditions, but it would require some effort to work out the
details for the particular initial condition posed here.

Other approaches to obtaining exact solutions consider the extension
of the domain to complex or even hypercomplex numbers.

regards, mathtalk-ga

this information is enough for now,
i greatly appreciate it and you CAN go ahead and bill me
the amount I had agreed when I posted the response ($10)
your service is great and I am quite impressed,

I will try to find your reference,

best regards,

javier5440