Scaling of the Diffusion Equation

Under certain conditions, boundary-value diffusion problems can be solved conveniently by scaling. First, introduce the dimensionless variable tj, 

suppose that for the particular boundary-value problem under consideration, the initial and boundary conditions are unchanged by scale change  

Then i) is invariant under the scaling corresponding to Eq. 4.19 and c becomes a function of the single variable, r). The diffusion equation becomes an ordinary differential equation (i.e., d — d). 

If the boundary-value diffusion problem can be scaled according to Eq. 4.19, it is considerably easier to solve.1 Consider the one-dimensional step-function diffusion problem shown in Fig. 4.2, where 

The initial and boundary conditions given by Eq. 4.23 are transformed by scaling into 

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