Re: One-dimensional heat equation

From: Ken Honda (Honda_Kiai_at_hotmail.com)
Date: 10/25/04 

Date: 25 Oct 2004 08:02:52 -0700

Oh my, how embarassing! This is what I get for posting late at night
on very little sleep. I transcribe the problem incorrectly, and
thankyou very much to all for pointing this out!
We start with the heat equation

dU/dt = d^2U/dx^2

on a finite interval (0,L) with the following conditions:

U(t,0) = U(t,L) = 0
U(0,x) = U(x)

and furthermore f(x) admits a representation

f(x) = sum{A_n*sin(n*pi*x/L)} for n>0

and we want to know if the integral of u(t,x) with respect to x from
-infinity to positive infinity changes over time. (I think that the
above is a typo and that we are only supposed to evaluate the integral
of u(t,x) with respect to x from 0 to L). The next part of the
question reads "What is the behavior of this integral as t approaches
infinity? Give a physical explanation; think of a finite heated wire,
both of whose ends are embedded in an ice cube at constant temperature
0."

Sorry for the confusion, and thanks very much for all of your replies.
 I don't understand how admitting this representation for f(x) allows
one to show that the heat of the system converges; is this really dumb
of me? Is it something obvious that I've missed?
Thanks very much!
KH

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