Diffusion equation, dimensionless form
We note that the dimension of \( D \) is time/length$^2$.
Introducing the dimensionless variables \( \alpha\hat{x}=x \)
we get
$$
\begin{equation*}
\frac{\partial^2 T(x,t)}{\alpha^2\partial \hat{x}^2}=
D\frac{\partial T(x,t)}{\partial t},
\end{equation*}
$$
and since \( \alpha \) is just a constant we could define
\( \alpha^2D= 1 \) or use the last expression to define a dimensionless time-variable
\( \hat{t} \). This yields a simplified diffusion equation
$$
\begin{equation*}
\frac{\partial^2 T(\hat{x},\hat{t})}{\partial \hat{x}^2}=
\frac{\partial T(\hat{x},\hat{t})}{\partial \hat{t}}.
\end{equation*}
$$
It is now a partial differential equation in terms of dimensionless
variables. In the discussion below, we will however, for the sake
of notational simplicity replace \( \hat{x}\rightarrow x \) and
\( \hat{t}\rightarrow t \). Moreover, the solution to the \( 1+1 \)-dimensional
partial differential equation is replaced by \( T(\hat{x},\hat{t})\rightarrow u(x,t) \).