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quat
science forum Guru Wannabe
Joined: 17 Jan 2006
Posts: 105
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 Posted: Thu Jun 08, 2006 5:01 am Post subject:
understanding PDE Problem statement,
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Let q_i denote the cartesian components of the heat flux vector, let u be
the temperature and ket L be the heat supply per unit volume. Assume the
heat flux vector is defined in terms of the temperature gradient by the
generalized Fourier law: q_i = -K_ij * u,_j
where u,_j means partial derivative of u wrt jth coordinate. The K_ij's are
given functions of x.
Given L: D-->R, f : T1-->R, h : T2-->R, find u : D-->R s.t.
q_i,_i = L in D (Heat Equation)
u = f on T1
-q_i *n_i = h on T2
I don't really understand how this is a heat equation. In my intro PDE
book, heat equation looks like:
u_t - k(u_xx + u_yy + u_zz) = 0 plus boundary/initial conditions.
Can someone explain the difference? |
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Abstract Dissonance
science forum Guru Wannabe
Joined: 29 Dec 2005
Posts: 201
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| Posted: Thu Jun 08, 2006 8:36 am Post subject:
Re: understanding PDE Problem statement,
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"quat" <spam@void.com> wrote in message
news:a8Ohg.13572$Jk2.2827@fed1read03...
| Quote: | Let q_i denote the cartesian components of the heat flux vector, let u be
the temperature and ket L be the heat supply per unit volume. Assume the
heat flux vector is defined in terms of the temperature gradient by the
generalized Fourier law: q_i = -K_ij * u,_j
where u,_j means partial derivative of u wrt jth coordinate. The K_ij's
are given functions of x.
Given L: D-->R, f : T1-->R, h : T2-->R, find u : D-->R s.t.
q_i,_i = L in D (Heat Equation)
u = f on T1
-q_i *n_i = h on T2
I don't really understand how this is a heat equation. In my intro PDE
book, heat equation looks like:
u_t - k(u_xx + u_yy + u_zz) = 0 plus boundary/initial conditions.
Can someone explain the difference?
|
I don't completely understand the notation. It seems that the index is
refering to the ith component but if this is true then there are probably
limits on it. If this is some type of einstien notation for tensors then I
can get close to the original heat equation but its off somewhat.
I'll show you what I can do and maybe you can fill in the rest... it could
be completely wrong.
in vector notation
if u = u(x,y,z)
q = -K*grad(u)
(i.e., in the original notation, u,_i means u_x = du/dx)
then
q_i,_i = L
would be
grad(q) = L
but q = -K*grad(u). if K = k is constant then
grad(q) = L ==> -k*laplacian(u) = L ==> -k*(u_xx + u_yy + u_zz) = L
where did u_t go? I don't know. speration of variables might have been
performeed on u to remove the time dependence. If thats the case then the
above make sense. If not then I don't know but a similar method might make
it work.
Jon |
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Abstract Dissonance
science forum Guru Wannabe
Joined: 29 Dec 2005
Posts: 201
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| Posted: Thu Jun 08, 2006 8:53 am Post subject:
Re: understanding PDE Problem statement,
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"Abstract Dissonance" <Abstract.Dissonance@hotmail.com> wrote in message
news:128foca905bd22f@corp.supernews.com...
| Quote: |
"quat" <spam@void.com> wrote in message
news:a8Ohg.13572$Jk2.2827@fed1read03...
Let q_i denote the cartesian components of the heat flux vector, let u be
the temperature and ket L be the heat supply per unit volume. Assume the
heat flux vector is defined in terms of the temperature gradient by the
generalized Fourier law: q_i = -K_ij * u,_j
where u,_j means partial derivative of u wrt jth coordinate. The K_ij's
are given functions of x.
Given L: D-->R, f : T1-->R, h : T2-->R, find u : D-->R s.t.
q_i,_i = L in D (Heat Equation)
u = f on T1
-q_i *n_i = h on T2
I don't really understand how this is a heat equation. In my intro PDE
book, heat equation looks like:
u_t - k(u_xx + u_yy + u_zz) = 0 plus boundary/initial conditions.
Can someone explain the difference?
I don't completely understand the notation. It seems that the index is
refering to the ith component but if this is true then there are probably
limits on it. If this is some type of einstien notation for tensors then I
can get close to the original heat equation but its off somewhat.
I'll show you what I can do and maybe you can fill in the rest... it could
be completely wrong.
in vector notation
if u = u(x,y,z)
q = -K*grad(u)
(i.e., in the original notation, u,_i means u_x = du/dx)
then
q_i,_i = L
would be
grad(q) = L
but q = -K*grad(u). if K = k is constant then
grad(q) = L ==> -k*laplacian(u) = L ==> -k*(u_xx + u_yy + u_zz) = L
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I guess I should say that q_i,_i = (grad(grad(-k*u)))_(i,i) but since one
has repeated indicies one sums over them(some type of tensor notation or
something).
i.e. grad(grad(u)) =
[u_xx u_xy u_xz]
[u_yx u_yy u_yz]
[u_zx u_zy u_zz]_(i,i)
=
u_xx if i = 1
u_yy if i = 2
u_zz if i = 3
but sense the index is repeat in the formula it is convention to sum over
it.
in essense one has
trace(grad(grad(-Ku))) = L
grad(u) is extendent to work on vectors and there is a certain convention to
treat it it as a tensor when done so.
i.e.,
given u:R^n->R then
trace(grad(grad(u))) = laplacian(u).
the reason for the notation with subscripts above is to get away from
talking about specific variables such as x, y, z. (which is what tensors
are all about).
I'm pretty sure thats whats going on but who knows.
Jon |
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