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[SOLVED] invariant PDE

dwsmith

Well-known member
Feb 1, 2012
1,673
Show the diffusion equation is invariant to a linear transformation in the temperature field
$$
\overline{T} = \alpha T + \beta
$$
Since $\overline{T} = \alpha T + \beta$, the partial derivatives are
\begin{alignat*}{3}
\overline{T}_t & = & \alpha T_t\\
\overline{T}_{xx} & = & \alpha T_{xx}
\end{alignat*}
So $T_t = \frac{1}{\alpha}\overline{T}_t$ and $T_{xx} = \frac{1}{\alpha}\overline{T}_{xx}$.
The diffusion equation is
$$
\frac{1}{\alpha}T_t = T_{xx}.
$$
By substitution, we obtain
$$
\frac{1}{\alpha}\overline{T}_t = \overline{T}_{xx}.
$$
Correct?
 

girdav

Member
Feb 1, 2012
96
Yes, as the set of solutions of such an equation is a vector space which contains constant functions.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Yes, as the set of solutions of such an equation is a vector space which contains constant functions.
So that is all that it was? It seems to simple.
 

girdav

Member
Feb 1, 2012
96
It may be for example the first question of a homework or a test, so it's not necessarily difficult. (maybe maybe the other question can be harder)
 

Jester

Well-known member
MHB Math Helper
Jan 26, 2012
183
So that is all that it was? It seems to simple.
Yes, it may be simple (in this case) but there's a deeper meaning. It means, given one solution $T_0$, you can construct a second solution $T = \alpha T_0 + \beta$.

You might also want to check that this same PDE is invariant under the change of variables

$\bar{t} = k^2 t,\;\;\; \bar{x} = k x$

i.e.

$ T_{\bar{t}}=\alpha T_{\bar{x} \bar{x}} \;\; \implies \;\; T_t = \alpha T_{xx}$.

The next question $-$ how is this useful?