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| Author |
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Maury Barbato
science forum Guru Wannabe
Joined: 13 Jun 2005
Posts: 169
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 Posted: Mon Jun 26, 2006 6:17 pm Post subject:
Heat Conduction
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Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
My Best Regards,
Maury |
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Robert B. Israel
science forum Guru
Joined: 24 Mar 2005
Posts: 2151
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| Posted: Mon Jun 26, 2006 8:05 pm Post subject:
Re: Heat Conduction
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In article <3951636.1151345874676.JavaMail.jakarta@nitrogen.mathforum.org>,
Maury Barbato <mauriziobarbato@aruba.it> wrote:
| Quote: | Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
|
No boundary conditions as x -> +infinity?
With h = 0, try the solution u(x,t) = x^2 + 2 t.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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David Moran
science forum Guru Wannabe
Joined: 13 May 2005
Posts: 252
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| Posted: Tue Jun 27, 2006 12:54 am Post subject:
Re: Heat Conduction
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"Maury Barbato" <mauriziobarbato@aruba.it> wrote in message
news:3951636.1151345874676.JavaMail.jakarta@nitrogen.mathforum.org...
| Quote: | Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
My Best Regards,
Maury
|
I'd set up a problem with the heat equation ut = k*uxx given u(0,t)=A,
u(L,t)=B and u(x,0)=f(x). Then take the limit as x -> infinity. I don't know
for sure if this is what you're looking for.
Dave |
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Abstract Dissonance
science forum Guru Wannabe
Joined: 29 Dec 2005
Posts: 201
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| Posted: Tue Jun 27, 2006 1:58 am Post subject:
Re: Heat Conduction
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"Maury Barbato" <mauriziobarbato@aruba.it> wrote in message
news:3951636.1151345874676.JavaMail.jakarta@nitrogen.mathforum.org...
| Quote: | Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
|
are you sure this is correct?
If you are applying a constant heat source then H(t) = h.
But your heat source is changing linearly with time since H(t) = h*t
and since u(0,t) = H(t) then
limit u(0,t) = limit H(t) = oo as t->oo.
Now if you mean H(t) = h then your answer is u(0,oo) = h. |
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Stephen Montgomery-Smith
science forum Guru
Joined: 01 May 2005
Posts: 487
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| Posted: Tue Jun 27, 2006 2:48 am Post subject:
Re: Heat Conduction
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Maury Barbato wrote:
| Quote: | Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
My Best Regards,
Maury
|
You can symmetrize the bar of metal, so it goes from -infinity to
+infinity. So it seems to me that you want to solve
u(x,0) = f(x)
u_t = k u_xx + h delta(x).
The solution is something like
F(x,t) + h int_0^t G(x,t-s) ds
where F is the solution when h=0, and G(x,t) is the usual heat kernel.
As best as I can see, it boils down whether int_0^infty 1/sqrt(t) dt
converges or not. |
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Maury Barbato
science forum Guru Wannabe
Joined: 13 Jun 2005
Posts: 169
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| Posted: Tue Jun 27, 2006 12:49 pm Post subject:
Re: Heat Conduction
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Robert Israel wrote:
| Quote: | In article
3951636.1151345874676.JavaMail.jakarta@nitrogen.mathf
orum.org>,
Maury Barbato <mauriziobarbato@aruba.it> wrote:
Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the
heat
transferred to the bar (at the point x=0) is
H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima
t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for
your
help.
No boundary conditions as x -> +infinity?
With h = 0, try the solution u(x,t) = x^2 + 2 t.
|
Surely right, but I have in mind intial conditions
that satisfy at least u(x,0)->0 when x tends to infinity.
Otherwise the problem is really trivial.
My Best Regards,
Maury |
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Maury Barbato
science forum Guru Wannabe
Joined: 13 Jun 2005
Posts: 169
|
| Posted: Tue Jun 27, 2006 1:22 pm Post subject:
Re: Heat Conduction
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David Moran wrote:
| Quote: |
"Maury Barbato" <mauriziobarbato@aruba.it> wrote in
message
news:3951636.1151345874676.JavaMail.jakarta@nitrogen.m
athforum.org...
Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the
heat
transferred to the bar (at the point x=0) is
H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima
t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much
for your
help.
My Best Regards,
Maury
I'd set up a problem with the heat equation ut =
k*uxx given u(0,t)=A,
u(L,t)=B and u(x,0)=f(x). Then take the limit as x -
infinity. I don't know
for sure if this is what you're looking for.
Dave
|
(I) I think it's not so difficult to find the explicit
solution to my problem (if we add some boundary
condition as x->+ infinity), but I don't know anything
about the Heat Equation, so I couldn't find it.
(II) The method you proposed is quite curious, if I have
understand it: you find the solution for a finite bar
and then you consider the limit as the length tends to
infinity!!! Are you sure you obtain a solution of the
Heat Equation? I have some (or more than some) doubt!
My Best Regards,
Maury |
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Robert B. Israel
science forum Guru
Joined: 24 Mar 2005
Posts: 2151
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| Posted: Tue Jun 27, 2006 5:58 pm Post subject:
Re: Heat Conduction
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Stephen Montgomery-Smith wrote:
| Quote: | Maury Barbato wrote:
Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
My Best Regards,
Maury
You can symmetrize the bar of metal, so it goes from -infinity to
+infinity. So it seems to me that you want to solve
u(x,0) = f(x)
u_t = k u_xx + h delta(x).
|
That doesn't fit the physics of the problem. The heat flux is
(proportional to)
-u_x, not u_t. So my interpretation of the problem is
u_t = k u_xx on [0, infty)
u(x,0) = f(x)
u_x(0,t) = -h
and some condition as x -> infty, perhaps u(x,t) bounded as x ->
infinity for each t.
Now we can get a solution of this problem as u(x,t) = v(x,t) + w(x,t)
where v(x,t) is a solution of the initial-value problem with insulating
boundary
condition
v_t = k v_xx
v(x,0) = f(x)
v_x(0,t) = 0
and some condition as x -> infty
while w(x,t) is a solution with initial value 0 and non-homogeneous
boundary
condition
w_t = k w_xx
w(x,0) = 0
w_x(0,t) = -h
and some condition as x -> infty.
Certainly v(x,t) will be uniformly bounded if f(x) is bounded. So the
initial
condition doesn't matter.
On the other hand, for w(x,t) I get the explicit solution
w(x,t) = h x (erf(x/(2 sqrt(k t))) - 1) + 2 h sqrt(kt/pi) exp(-x^2/(4 k
t))
This has w(0,t) = 2 h sqrt(kt/pi)
so the answer is no, w(0,t) -> +infty as t -> infty.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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Stephen Montgomery-Smith
science forum Guru
Joined: 01 May 2005
Posts: 487
|
| Posted: Tue Jun 27, 2006 8:32 pm Post subject:
Re: Heat Conduction
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|
Robert Israel wrote:
| Quote: | Stephen Montgomery-Smith wrote:
Maury Barbato wrote:
Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the heat
transferred to the bar (at the point x=0) is H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much for your
help.
My Best Regards,
Maury
You can symmetrize the bar of metal, so it goes from -infinity to
+infinity. So it seems to me that you want to solve
u(x,0) = f(x)
u_t = k u_xx + h delta(x).
That doesn't fit the physics of the problem. The heat flux is
(proportional to)
-u_x, not u_t. So my interpretation of the problem is
u_t = k u_xx on [0, infty)
u(x,0) = f(x)
u_x(0,t) = -h
and some condition as x -> infty, perhaps u(x,t) bounded as x -
infinity for each t.
Now we can get a solution of this problem as u(x,t) = v(x,t) + w(x,t)
where v(x,t) is a solution of the initial-value problem with insulating
boundary
condition
v_t = k v_xx
v(x,0) = f(x)
v_x(0,t) = 0
and some condition as x -> infty
while w(x,t) is a solution with initial value 0 and non-homogeneous
boundary
condition
w_t = k w_xx
w(x,0) = 0
w_x(0,t) = -h
and some condition as x -> infty.
Certainly v(x,t) will be uniformly bounded if f(x) is bounded. So the
initial
condition doesn't matter.
On the other hand, for w(x,t) I get the explicit solution
w(x,t) = h x (erf(x/(2 sqrt(k t))) - 1) + 2 h sqrt(kt/pi) exp(-x^2/(4 k
t))
This has w(0,t) = 2 h sqrt(kt/pi)
so the answer is no, w(0,t) -> +infty as t -> infty.
|
But I have this feeling that your solution is (surprizingly) identical
to mine. |
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David Moran
science forum Guru Wannabe
Joined: 13 May 2005
Posts: 252
|
| Posted: Tue Jun 27, 2006 8:46 pm Post subject:
Re: Heat Conduction
|
|
|
"Maury Barbato" <mauriziobarbato@aruba.it> wrote in message
news:837624.1151414574806.JavaMail.jakarta@nitrogen.mathforum.org...
| Quote: | David Moran wrote:
"Maury Barbato" <mauriziobarbato@aruba.it> wrote in
message
news:3951636.1151345874676.JavaMail.jakarta@nitrogen.m
athforum.org...
Hello,
the physical problem I'm studying is the following.
Consider a linear bar which lengthens from x=0 to
x=+ inf. Let us suppose that at the time t=0 we
apply a constant heat source in x=0, that is the
heat
transferred to the bar (at the point x=0) is
H(t)=ht,
where h is a positive constant. Let u(x,t) be the
temperature of the bar at the point x, at the tima
t. Is
lim_(t->+ inf) u(0,t) finite (for every initial
condition u(x,0))?
I don't know thw answer. So, thank you very much
for your
help.
My Best Regards,
Maury
I'd set up a problem with the heat equation ut =
k*uxx given u(0,t)=A,
u(L,t)=B and u(x,0)=f(x). Then take the limit as x -
infinity. I don't know
for sure if this is what you're looking for.
Dave
(I) I think it's not so difficult to find the explicit
solution to my problem (if we add some boundary
condition as x->+ infinity), but I don't know anything
about the Heat Equation, so I couldn't find it.
(II) The method you proposed is quite curious, if I have
understand it: you find the solution for a finite bar
and then you consider the limit as the length tends to
infinity!!! Are you sure you obtain a solution of the
Heat Equation? I have some (or more than some) doubt!
My Best Regards,
Maury
|
You are solving the heat equation for a bar with length L so your solution
will be a solution to the heat equation. Then since you considered an
arbitrary length, you can take the limit as L goes to infinity. If you want
to see how I did it, send me an e-mail at dmoran21@cox.net.
Dave |
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