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) .