From: "Kai Xiu" <kaixiu@students.uiuc.edu>
Subject: Minimize integral under conditions
Date: Thu, 16 Dec 1999 20:50:07 -0600
Newsgroups: sci.math
Keywords: typical calculus of variations problem

I have to solve this math problem in electron optics:

Given
u''+f(z)u=0 and boundary condition
u(z1)=u(z2)=0,

Minimize (or Maximize)
Integral of f(z)^2 dz, (z from z1 to z2)

It might belong to a branch of mathematics but I
don't know how to start with, thanks a lot.

Kai Xiu
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From: udaypatil@home.com (Uday Patil)
Subject: Minimize integral under conditions
Date: 17 Dec 1999 16:14:52 -0500
Newsgroups: sci.math

I am not sure I understand the problem. 

is it to optimize Int{z1->z2} dz (u''/u)^2  ?

If so, then consider 
u(z) = sin(n*PI*(z-z1)/(z2-z1)) . . . (n integer)
The integral is simply (n*PI/(z2-z1))^2 (unbounded)

Any way, the field of math you are looking for is 
'calculus of variations' where you solve integral 
optimization problems (usually by converting them
into differential equations).

Uday

[original article was quoted --djr]
==============================================================================

From: "Kai Xiu" <kaixiu@students.uiuc.edu>
Subject: Re: Minimize integral under conditions
Date: Fri, 17 Dec 1999 16:05:37 -0600
Newsgroups: sci.math

Thanks,

Yes, it's equivalent to optimize Int{z1->z2} dz (u''/u)^2,
but the form of f(z) is undetermined, so it's impossible to
solve the differential equation at first.

Kai


Uday Patil <udaypatil@home.com> wrote in message
news:lji9t9v7omdt@forum.swarthmore.edu...
[previous article quoted --djr]
==============================================================================

From: Lynn Killingbeck <killbeck@pointecom.net>
Subject: Re: Minimize integral under conditions
Date: Fri, 17 Dec 1999 17:47:25 -0600
Newsgroups: sci.math

Kai Xiu wrote:
[previous article quoted --djr]

The branch of mathematics is the Calculus of Variations. The integral
clearly has no maximum (e.g., any u that goes through 0 at some point
with u'' non-zero gives an infinite integral). Glancing back a couple of
decades into Gelfand and Fomin "Calculus of Variations", this is in
section 11, 'Functionals Depending on Higher-Order Derivatives'. If you
happen to have that text, you are after "Euler's equation", eq. (22).
The material is covered in just about any text I've seen; and, it is a
common branch, so there are many books available. The most complete book
I recall is by Forsythe - but it has a notation that makes it nearly
unreadable!

Lynn Killingbeck