Re: One-dimensional heat equation

From: Rouben Rostamian (rouben_at_pc18.math.umbc.edu)
Date: 10/25/04 

Date: Mon, 25 Oct 2004 15:30:37 +0000 (UTC)

In article <7156fcf8.0410250702.4537571d@posting.google.com>,
Ken Honda <Honda_Kiai@hotmail.com> wrote:
>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?

First let's clarify: what is f(x) and what does it have to do
with this problem? 

-- 
Rouben Rostamian

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