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MathGroup Archive 2006

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Re: change of variables

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71265] Re: change of variables
  • From: bghiggins at ucdavis.edu
  • Date: Sun, 12 Nov 2006 06:48:11 -0500 (EST)
  • References: <ej4352$r9c$1@smc.vnet.net>

You can try the following approach:

Let suppose you have the 1-D heat equation where the dimensional
temperature v is a function of time(t) and the spatial coordinate (x).
Here is the equation  where K is the thermal diffusivity

heatEqn := D[v[x, t], t] == K D[v[x, t], {x, 2}]

Now let us introduce the following dimensionless varaibles bases on a
temperature scale V, a length scale L , and a time scale T

η[x_] := x/L
Ï?[t_] := t/T
v[x, t] := Ï?[η[x], Ï?[t]]/Φ

Then evaluating

In[5]:=

heatEqn /. {x/L -> η, t/T -> Ï?}

Out[5]=

Derivative[0, 1][Ï?][η, Ï?]/(T*Φ) ==
  (K*Derivative[2, 0][Ï?][η, Ï?])/(L^2*Φ)

One can apply additional rules to simplify the resulting dimensionless
equation if needed.

Hope this helps,

Cheers

Brian

hussain.alqahtani at gmail.com wrote:
> Dear Members,
>
> I have a differential equation that i need to nondimensionalize.
> Therefore, i introduced a new nondimensional varibles and so that i get
> a nondimensional equation. I am wondering how can i do that in
> Mathematica?. For example if i have this term:
>
> K*D[v,{x,2}],
>
> my new nondimensional variables are:
>
> x*=x/L and v*=v/L,
>
> I can do that by hand, but since i have a huge number of equations, i
> rather prefer to use Mathematica.
>
> I appreciate any help in this regard, and many thanks in anticipation.
> 
> HMQ


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