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

Î·[x_] := x/L
Ï?[t_] := t/T
v[x, t] := Ï?[Î·[x], Ï?[t]]/Î¦

Then evaluating

In[5]:=

heatEqn /. {x/L -> Î·, t/T -> Ï?}

Out[5]=

Derivative[0, 1][Ï?][Î·, Ï?]/(T*Î¦) ==
(K*Derivative[2, 0][Ï?][Î·, Ï?])/(L^2*Î¦)

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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