Subject: Re: differential equation
Date: Sun, 2 Apr 1995 12:35:24 -0700
From: [Permission pending]
To: rusin@math.niu.edu

> From: rusin@math.niu.edu (Dave Rusin)
> Message-Id: <9504021531.AA27521@clinch.math.niu.edu>
> To: [Permission pending]
> Subject: Re: differential equation
> Status: R
> 
> >> x  = 0 to L
> >> y''(x) = a(x).y(x) + b(x)
> >>  y'(0) = 0
> >>   y(L) = 0
> >About a's and b's: The intervel x = 0 to L is in fact very large. If one
> >divides it into three parts, x = 0 to x1, x = x1 to x2 and x = x2 to L, then
> >x1 >>>>> (x2 - x1), x1 >>>>> (L - x2) and ( (x2-x1) approx. = (L-x2) ).
> >And a(x), b(x) are non-zero only in the middle interval, i.e. x = x1 to x2.
> 
> That's a very strong statement. It says  y   is linear on those outer
> two intervals. Your additional axioms now say that y is constant from
> 0 to x1, then linear from x2 to L. If you would just tell me the values

	That's right! And all I really need, I think, is y'(x) for (x2,L)
which is constant. I don't think I can restrict the functions a and b
any more, other than that these are well behaved. Is there a possibility of
getting a solution by expanding a and b in an orthogonal basis of well
known functions? I would also appreciate any pointers to relevant numerical
methods; since it seems that's where I'll end up.

	For your interest, this problem has arisen in propagating electro-
magnetic waves (rays for now, geometrical optics approx.) through planetary
atmospheres for remote sensing of temerature and pressure profiles. Most
often, the atmosphere is assumed spherically symmetric and in that case we
do have an analytic formulation to find the bending angle (due to refraction).
But now that our equipment is much more sensitive, we need to account for
variations from spherical symmetry. The equation under study is obtained as
a first order approx. to find the additional bending due to non-spherically
symmetric variations.

[sig deleted --djr]


> y(x1) and y(x2), we'd know the whole behaviour of  y outisde (x1, x2).
> 
> On the other hand, you've used up just about all your information. All
> you've got left is that  y  is a funciton defined on the interval  (x1, x2)
> and satifying  y''=ay+b  for some functions a and b. If you can't say
> _anything_ else about a and b, then you can't say _anything_ else about
> y. Indeed, take any function you like for  y, let  a(x) = 0 and b(x) = y''(x);
> then the  y  you started with is a  solution to y''=ay+b ! So you'll have
> to know something else about  a  and  b  to proceed.
> 
> dave
> 

==============================================================================

Date: Thu, 6 Apr 95 00:48:52 CDT
From: rusin (Dave Rusin)
To: [Permission pending]
Subject: Your differential equation

Summary: you've got an ODE
	y''(x) = a(x).y(x) + b(x)
to solve on the interval  (x1, x2),  in which  y(x1)   and the ratio
y'(x2)/y(x2)  are given. (They are, respectively, y(0) and  1/(L-x2),
in the language you used earlier.) The ultimate goal is to find
y'(x2) (or, eqivalently, y(x2) ).

Unfortunately, no general solution is possible. One way to see the problem
is that, akthough you've told me  (x2-x1) is small, you don't seem to want
to commit to a description of  a  and  b.  If, for example,  b=0 but
a  is a large constant, y  will increase as a rapid exponential and can
cause  y(x2)  to be much larger than  y(x1). You can even arrange this
much variation and still have y'(x2)/y(x2) take the presecribed value.
There's just too much flexibility in the problem to allow you to
make the deductions you want.

On the other hand, your description of the problem of origination 
suggests there is more to this than meets the eye. You are attempting
to account for irregularities in observed data by suggesting
deviations from  an ideal model. Two suggestions come to mind:

1) If you are toying with non-spherical distributions, assume a
_slightly_ non-spherical condition, such as an ellipsoid. This is
often a good second-order model. It should lead to differential
equations with parameters which are constants (not fucntions).

2) To some extent your problem sounds like the situation in CAT scans.
There one assumes a completely unknown distribution (of matter in a skull)
and attempts to reconstruct it with projections measured from the
scanner in many directions. In its broadest outline, this is rather
like what you were proposing, except that one has not just one or two
but a continuous sampling of data points. Certainly this is the lesson
you should draw before attempting expansions with orthonormal families
or whatever: when you have infinitely many free parameters, you'll
need a lot of data to pin them down!
	(In case the CAT-scan model is more similar to your situation
than I am aware, you might want to know that the mathematical technique
of importance there is known as the Radon transform (that's a person, not
an element.) )

I'm afraid I'm at a loss to suggest something more concrete. I enjoyed
your description of the setting of your problem, but I don't think I
have enough insight yet to see through to what you need to do.

Good luck,
dave