Re: One-dimensional heat equation
From: Robert Israel (israel_at_math.ubc.ca)
Date: 10/25/04
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Date: 25 Oct 2004 20:23:38 GMT
In article <7156fcf8.0410250702.4537571d@posting.google.com>,
Ken Honda <Honda_Kiai@hotmail.com> wrote:
>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)
I think you mean f(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."
> 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?
Hints:
The solution U(x,t) can be written in terms of a very similar series
with some exponential factors involving t ...
What do you get when you integrate sin(n*pi*x/L) for x from 0 to L?
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
- Next message: Tim Wescott: "Re: any deep thinking on why linear systems are commutative?"
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