Normalizing boundary conditions

  • Thread starter garcijon
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Hi All,

This is my first post on these forums. I am not looking for a solution to this problem but more interested in seeing if someone can point me to a resource that can explain the following. Thanks in advance for any help.

I'm trying to solve a pde which gives a temperature profile.

upload_2014-11-21_15-54-40.png

We end up changing over to spherical coordinates where our boundary conditions are not completely clear although our initial condition is.

upload_2014-11-21_15-56-36.png


The professor writes something like this. basically it normalizes the boundary conditions to (0,1). I can't wrap my head around this and googling for a while hasn't given me any results that clarify this.

upload_2014-11-21_15-58-26.png
 
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pasmith

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The professor writes something like this. basically it normalizes the boundary conditions to (0,1). I can't wrap my head around this and googling for a while hasn't given me any results that clarify this.

View attachment 75728
This is a straightforward linear rescaling of the form [tex]
f : [a,b] \to [0,1] : x \mapsto \frac{x - a}{b - a}.[/tex] Observe that [itex]f[/itex] is strictly increasing with [itex]f(a) = 0[/itex] and [itex]f(b) = 1[/itex].

If what you are struggling with is "why does he bother to do this?" then the concept you are looking for is nondimensionalization. Also it's easier to work with [itex]G(t = 0) = G(r \to \infty) = 0[/itex] rather than [itex]G(t = 0) = rT_0[/itex] and [itex]G(r \to \infty) = \infty[/itex].
 

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