Two half-spaces with uniform initial concentrations

2 Two half-spaces with uniform initial concentrations 

The infinite medium with one-dimensional diffusion and constant diffusion coefficient can be treated easily with the point source theory. Let us first assume that two half-spaces with uniform initial concentrations C0 for x 0 and 0 for x 0 are brought into contact with each other. The amount of substance distributed per unit surface between x and x + dx is just C0dx. From the previous result, at time t the effect of the point source C0 dx located at x on the concentration at x will be 

Summing over all the point sources at x from — oo to -t-oo and noting that the contribution from the half-space x 0 is zero will give the concentration distribution 

Introducing the error function erfx defined as (Appendix 8A) 

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