I have the partial differential equation
dQ/dt = (1/2) d^2 Q / dx^2         for   t>0, -infinity<x<f(t)
(i.e. Q_t = (1/2) Q_xx)
on a semi-infinite plane.  The initial condition is
Q(x, t=0) = delta_d(x), with delta_d the Dirac delta function. 
The hard part is the boundary condition: Q is 0 along
the curve x=f(t) rather than along a line of constant x, i.e.
Q(x=f(t),t) = 0.
f(t) is a well-behaved function (continuous, monotonic, etc) which
I can derive numerically but can't write in a simple form.  It
is greater than 0 at t=0 and decreases monotonically to -infinity.
Is the solution for Q(x,t) known?  If so, what is it?
I really only need dQ/dx evaluated
on the boundary x=f(t), and would settle for that instead of
a complete solution for Q(x,t) everywhere.

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Request for Question Clarification by pythagoras-ga on 26 Jun 2002 13:33 PDT

Dear Sir,

What do you mean by:

" i.e. Q_t = (1/2) Q_xx "

Kind Regards,

Pythagoras

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Clarification of Question by chinaski-ga on 26 Jun 2002 14:34 PDT

Q_t is shorthand for dQ/dt, the first (partial) derivative of
Q with respect to t.  Q_xx is shorthand for d^2 Q / dx^2,
the second (partial) derivative of Q with respect to x.
So the equation in parentheses ("i.e. Q_t = ...") is
equivalent to the equation above it.  It's just easier to
read.

Dear chinasky:

Does this page help?
http://mathworld.wolfram.com/HeatConductionEquation.html

Best Regards,
blader-ga

No, unfortunately.  The page concerns the same equation, but it
assumes
a boundary condition that is fixed, not moving, and that makes the
solution much easier.  Sadly separation of variables doesn't seem to
work with my boundary conditions.  But thanks for the link.

Best, chinaski-ga

Try using a pullback. 

Set R(x,t) = Q(x+f(t),t). Then R is defined on t>0, x<0. R(x,t) then
solves
dR/dt = df/dt*dR/dx+1/2d^2R/dx^2. Thus, if you can solve this equation
you
can reproduce Q.  

I just did a mental calculation, so you'd better check the details.
Hope this helps.

Sincerely, 
ake-ga

Yes, that straightens the boundary condition, but then
the new equation you get is not separable (unless df/dt=const, which
is not the case) and I don't know how to solve it either.
But it's possible that recasting the problem this way will help
someone else think of a solution, even if it doesn't help me.
Thanks for the suggestion.

regards, chinaski-ga

The region of validity of the equation seems ill-defined.  For
example, if Q(x,t=0) = delta_d(x) for ALL real x, then the solution is
unique.  On the other hand, if Q(x,t=0)=delta_d(x) only for x < f(0),
with the domain of the function being {(t,x): t>=0 and x <= f(t)}, it
then makes sense.

Never mind...I should read the entire question before trying to answer it.

As far as I can see, there's no reason to suppose that you could map 
this to a separable equation, for the tranformation would have to 
map a non-separable equation into a separable one. In fact, I think 
it would be hard to come up with even a series representation of the
solution, except possibly locally. So why not solve the equation 
numerically? This seems reasonable since you don't have an 
explicit representation of the function f anyway. 

Sincerely, 
ake-ga

Dear hedgie-ga,

As you yorself has pointed out, this is not a free or moving boundary
problem. It is a PDE on a time-varying domain. The difference is
monumental. There's no reason why the techniques in Crank would help
here.

Furthermore, it was clear that chinansky asked for a representation
formula for the solution in terms of f and its derivatives. I think it
is far from obvious that, say, an explicit integral representation
formula doesn't exist though there are good reasons to doubt it (the
quote from Crank not being one of them).

That said, I can understand why you would think a reference on free
and moving boundaries would be helpful, but it would be far better to
supply that as a comment instead.

I suggest that chinanski ask for a refund on this one, and the Google
answers editors should think both once or twice if their "stringent
process for screening" in any way guarantees high quality answers to
mathematical questions.
 
Sincerely,
ake-ga

Dear ake
               I appreciate your comment, but beg to disagree. This
is, as I said,
    "prescribed  moving  boundary problem" , which is classified under
the generalised
    Stefan problem. I have said it is not 'classical Stefan Problem'.

 What I have given 

     J. Comput. Phys  in years 1971 - to 1981, which Crank says has
general
     mathematical formulation of adaptive finite elements for
prescribed
     moving boundary - which your case of Stefan-like problems (1974,
1975).
  
   is the  most appropriate relevant reference I was able to find.
   It is not a closed formula, and  reasons are
   given to believe  then none such exists. 
  
   However, the customer is the ultimate arbitrer. In this case we
have a sophisticated client
   who will review this and other sources and will evaluate usefullnes
of what I have provided.
   I do hope s/he will provide rating abd feedback, as it help us to
decide wen and how to post
  a contribution.  I you, ake, can find a closed formula or more
relevant reference, I, and  I
 believe chinasky too, would be happy to see it.

I have answered this question. No analytical soultion is known, except
for few very special
cases. I have provided reference  to collection of the special cases.

Chinasky rejected the answer. It looks like under new google policy,
answer is removed
 this history of question is not shown.
 Since there is no analytical solution, and collecting evidence and
explanation why it is so,
is not acceptable answer, it is my opinion that no matter what you do,
you will not get paid.

Proceed at your own risk.

This problem can be mapped onto a random walk problem where a random
walker starts at the origin at time t=0 and diffuses in the presence
of a moving "trap" whose position is f(t).

Discretize the x axis into steps of delta x and time into steps of
delta t. The random walker moves randomly left/right once per time
step. If the walker lands on the trap, the walker is annihilated.

If you collect statistics on enough random walks you will achieve an
*approximate* answer to your problem. The density Q(x,t) is then equal
to the number of random walks at position x at time t, normalized by
the total number of random walkers released.

You say you really only need dQ/dx evaluated on the boundary x=f(t),
and this can be easily evaluated as it is something like the number of
walkers that get annihilated by the moving trap at exactly time t,
normalized by the total number of random walkers released.

Again these are only an approximate solutions but if you release
enough random walks you can achieve an answer of arbitrary accuracy.

The moral of the story is this: if you can't think like a
mathematician, try thinking like a physicist!

Alternatively, you could use the computer a different way to solve the
problem, e.g. by integrating the diffusion equation forward in time on
a discrete 1-d lattice with a finite time step. But I imagine you
already thought of that.

-askrobin

ps-- excuse me if this is not terribly coherent; it's late and i'm
worn out...!

Thanks for the comment, askrobin!  The random walk with annihilation
is exactly the problem I want to solve; ie, that is basically
the underlying physical situation.  What I really want to know
is how many walkers get annihilated as a function of time, as you guessed.
I just cast the question as a PDE because
I thought it would be easier to explain.  Unfortunately I'm not really
interested in solving this problem numerically, though I know it can be
done and have code to do it.  I'm more curious in learning what analytic
solutions are known.  I'm aware of exact solutions for the following
special cases of f(t):  f(t)=const, f(t) = A*t, and f(t) = B * t^(C/2).
But surely there are many others.

It's very difficult to solve moving-bounday problems analytically. 
However there is hope...
I can try to develop the problem by transforming it to an infinite
boundary problem.

Let's try mapping the x interval (-inf,f(t)]  to (-inf,+inf) by making
the
transformation:

R = Q/(x-f(t)) or Q = ( x-f(t) ) * R

In this case, we can transform the moving-boundary problem to a
partial
differential equation with infinite b.c.'s and then maybe use some
kind of
separation of variables. 

First, let 's do the substitution:

Q_t = - R * f_t + R_t * ( x - f(t) )

Q_x = R + R_x * (x - f(t) )

Q_xx = R_xx + R_xx * (x-f(t)) + R_x

Which transforms your original equation to:

-R*f_t + R_t * (x - f(t)) = 2* R_x + ( x - f(t) ) *R_xx

As you can see, the function f(t) is now integrally bound to the 
other variables by the product terms  f_t*R, f(t)*R_t,  etc... 

I have a strong feeling that this implies that there is no possible
analytical solution
to this problem without knowing f(t) a priori.

If you would like, I could try to develop the separation of variables
on known
functions, like f(t) = polynomial function of t.  

Regards,

Andy22

This is an erratum and addendum to my previous post.  

First of all,  apologies for some rather sloppy and incorrect math. 
I'll try to keep it clean this time.
I had inverted the transformation between Q and R.  The following is
what I meant to say:

If we make the transformation:
R = (x-f) * Q  or Q=R/(x-f),

the 1-d heat equation becomes:
R_t + f_t/(x-f) * R = 1/2*R_xx - R_x/(x-f) + R/(x-f)**2

subject to:  R=0 at x = -inf and x=+inf
and R = delta(x) * (x-f)  at t=0

I believe it is possible to solve for this by separation of variables
for the time
functions that chinaski needs ( const, A*t,, t^(c/2), etc...)

Dear chinaski-ga,
By Q(x,t=0) = delta_d(x) do you mean Q(x,t=0) = delta_d(x_o), where
x_o is some number between -infinity and f(0), or do you mean that at
t=0 the heat distribution is infinite everywhere from -infinity to
f(0)?

Thank you,
SFBP