I have the following example in my notes:
Suppose we want to know how long it takes to cool down a long metal cylinder of radius $a$, with thermal diffusivity $\alpha$, initial with a uniform temperature $T_i$ , placed in a large bath of water at temperature $T_w$.
Physics tells us that that the temperature, $T$, in the cylinder satisfies the heat equation in cylindrical coordinates. As the initial temperature is uniform we can ignore any angular variation, and as the rod is “long" we can ignore any variation along its length. The problem then becomes one of examining how the temperature changes radially, i.e. we want to find a function $T = T (r , t )$. With all of this the heat equation we want to solve is,
$\dfrac{ \partial{T} }{ \partial{t}} = \alpha \dfrac{1}{r} \dfrac{\partial}{\partial{r}} \left(r \dfrac{\partial{T}}{\partial{r}} \right)$
For simplicity, assume that the water is able to remove all the heat which leaves the rod. This leads to the boundary condition $T(a,t) = T_w$. As the initial temperature in the rod was uniformly $T_i$ we have,
$T(r, 0) = T_i, 0 \le r < a$
We have 4 parameters: $\alpha$, $a$, $T_i$ and $T_w$. We can reduce the number of parameters by introducing dimensionless variables. So let
$u = \dfrac{T - T_w}{T_i - T_w}$, $r^* = \dfrac{r}{a}$, $t^* = \dfrac{t}{t_0}$,
where $t_0$ is some parameter with dimension of time.
Performing the change of variables we find,
$\dfrac{\partial{T}}{\partial{t}} = \dfrac{(T_i - T_w)}{t_0} \dfrac{\partial{u}}{\partial{t^*}}$
$\dfrac{\partial{T}}{\partial{r}} = \dfrac{(T_i - T_w)}{a} \dfrac{\partial{u}}{\partial{r^*}}$
The original PDE can be rewritten as
$\dfrac{1}{t_0} \dfrac{\partial{u}}{\partial{t^*}} = \dfrac{\alpha}{a^2 r^*} \dfrac{\partial}{\partial{r^*}} \left( r^* \dfrac{\partial{u}}{\partial{r^*}} \right)$
What I don't understand is how we went from
$$u = \dfrac{T - T_w}{T_i - T_w}, r^* = \dfrac{r}{a}, t^* = \dfrac{t}{t_0}$$
to
$$\dfrac{\partial{T}}{\partial{t}} = \dfrac{(T_i - T_w)}{t_0} \dfrac{\partial{u}}{\partial{t^*}}$$
$$\dfrac{\partial{T}}{\partial{r}} = \dfrac{(T_i - T_w)}{a} \dfrac{\partial{u}}{\partial{r^*}}$$
$$\dfrac{1}{t_0} \dfrac{\partial{u}}{\partial{t^*}} = \dfrac{\alpha}{a^2 r^*} \dfrac{\partial}{\partial{r^*}} \left( r^* \dfrac{\partial{u}}{\partial{r^*}} \right)$$
It says that it uses change of variables, but I can't see how this was done, leading to the derivation of these 3 equations.
I would greatly appreciate it if people could please take the time to explain this to me.