The first difference we notice from the canonical equation is that
the coefficients depend on *x*. However, the equation is
homogeneous or invariant under the scaling of . The standard way to
simplify this and eliminate the explicit coordinate dependence is to
define a new variable
, where we have scaled *x* by *c* to
make it dimensionless. Then under ,
.
Since the equation is invariant under this, it cannot have any explicit
dependence on *u* in the coefficients. Changing variables to
*u* using
,
and defining
, the derivatives become

= | (7) | ||

= | (8) |

The 1/

Now we observe that even if
is independent of
*u*, it still grows as *e*^{rt} from the
term.
Factoring this out
at the start will remove the
term. We normalize this
behavior where the boundary condition is at *t*=*t*^{*} by writing the
solution as

Substituting this into Eq. 9 eliminates the term giving

We next scale towards a canonical form. First we scale *u* to
get a common coefficient for the *u* derivatives, and then absorb
that coefficient into a rescaling for *t*. The new variables are

(12) |

and

(13) |

With the equation has become

Now, even with a constant gradient in

(15) |

we finally get the canonical form of the diffusion equation with unit diffusion coefficient

The boundary conditions at *t*=*t*^{*} are now
at *t*'=0 where *z*=*u*'. translates into .
We now use the case where
so the condition on
*u* translates into
.
The boundary conditions are then