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Oct22-04, 03:42 AM
Sci Advisor
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Clausius2's Avatar
P: 1,479
Well, this is my question. It's something like a philosophical thinking.

Look at this problem of the Heat Equation:

[tex] \frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2}[/tex]

with: 0<x<L

[tex] T(x,0)=g(x)[/tex] and

[tex] T(0,t)=T_1[/tex]

[tex] T(L,t)=T_o[/tex]

This is a parabolical PDE equation. It has only one characteristic line:

[tex] \frac{dx}{dt}=\infty [/tex]

So that, only information happened along the entire domain and in past times affects to what will happen in the future. It has sense. But the difference with hyperbolic equations, like Wave Equation is there is not any physical velocity of propagation of the phenomena. I will explain that:

A Wave has a finite velocity in any medium (except in a perfect liquid). One perturbation produced in some place takes a time away for being propagated to another zone.

On the other hand, in a change of time of [tex] dt[/tex] in Heat Equation, the initial distribution g(x) is smoothed and every point of the domain feels the heat flux instantaneously.

This seems rather disappointing, isn't it?. I don't know neither instantaneous interaction or instantaneous effect. With Heat Equation, the effects appear to be transmitted with infinite velocity. In fact, nobody talks never about velocity propagation of the heat.

What do you think about this strange effect?
Or am I missing something?.
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