| Topic: |
Science > Physics |
| User: |
"vsgdp" |
| Date: |
09 Oct 2005 04:40:51 PM |
| Object: |
heat pde proof |
I need to show Integral( u(x, t)^2 dx from a to b) <= Integral( u(x, 0)^2 dx
from a to b). In other words, energy is lost with time.
This is what I have so far, but I get stuck near the end.
Let F(t) = Integral( u(x, t)^2 dx from a to b)
F(t) - F(0) = Integral( dF/dt dt from 0 to t)
= Integral( Integral( 2u(x, t) * u_t(x, t) dx from a to b ) dt from 0 to t)
By heat equation: u_t = u_xx
= Integral( Integral( 2u(x, t) * u_xx(x, t) dx from a to b ) dt from 0 to t)
Now I integrate by parts and get:
= 2 Integral( u(b,t) u_x(b, t) - u(a, t) u_x(a, t) - Integral( u_x(x, t)^2
dx from a to b ) dt from 0 to t)
At this point I know Integral( u_x(x, t)^2 dx from a to b ) >= 0.
I would like to conclude that the entire RHS is <= 0 so that F(t) - F(0) <=
0, but for arbitrary boundary conditions I am stuck.
.
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| User: "" |
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| Title: Re: heat pde proof |
09 Oct 2005 08:24:04 PM |
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You'll find you have absolutely no use for all that once you get out of
college.
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| User: "Eric Gisse" |
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| Title: Re: heat pde proof |
09 Oct 2005 09:39:00 PM |
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wrote:
You'll find you have absolutely no use for all that once you get out of
college.
How would you know? You aren't an engineer, physicst, or anything
useful.
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| User: "Eric Gisse" |
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| Title: Re: heat pde proof |
09 Oct 2005 09:48:58 PM |
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vsgdp wrote:
I need to show Integral( u(x, t)^2 dx from a to b) <= Integral( u(x, 0)^2 dx
from a to b). In other words, energy is lost with time.
This is what I have so far, but I get stuck near the end.
Let F(t) = Integral( u(x, t)^2 dx from a to b)
F(t) - F(0) = Integral( dF/dt dt from 0 to t)
= Integral( Integral( 2u(x, t) * u_t(x, t) dx from a to b ) dt from 0 to t)
By heat equation: u_t = u_xx
In this heat queation, there is no loss of energy.
I am not sure how you intend to conclude that energy is lost with time.
= Integral( Integral( 2u(x, t) * u_xx(x, t) dx from a to b ) dt from 0 to t)
Now I integrate by parts and get:
= 2 Integral( u(b,t) u_x(b, t) - u(a, t) u_x(a, t) - Integral( u_x(x, t)^2
dx from a to b ) dt from 0 to t)
At this point I know Integral( u_x(x, t)^2 dx from a to b ) >= 0.
I would like to conclude that the entire RHS is <= 0 so that F(t) - F(0) <=
0, but for arbitrary boundary conditions I am stuck.
.
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