Solutions Involving the Error Function

The instantaneous local-source solutions in Table 5.1 can be used to build up solutions for general initial distributions of diffusant by using the method of superposition (see Section 4.2.3). 

Section 4.2.2 shows how to use the scaling method to obtain the error function solution for the one-dimensional diffusion of a step function in an infinite medium given by Eq. 4.31. The same solution can be obtained by superposing the onedimensional diffusion from a distribution of instantaneous local sources arrayed to simulate the initial step function. The boundary and initial conditions are 

The initial distribution is simulated by a uniform distribution of point, line, or planar sources placed along x 0 as in Fig. 5.5. The strength, or the amount of diffusant contributed by each source, must be Co dx. The superposition can be achieved by replacing in Table 5.1 with c(x)dV [c(x)dx in one dimension] and integrating the sources from each point. 

Consider the contribution at a general position x from a source at some other position . The distance between the general point x and the source is — x, thus 

So the solution corresponding to the conditions given by Eq. 5.20 must be the integral over all sources, 

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