Towards the diffusion equation analysis

To show how the lifetime distribution of eqn. (346) is directly related to the diffusion equation analysis, first substitute eqn. (343) into eqn. (346) and take the Laplace Transform 

From the definition of the survival probability discussed in Chap. 7, Sect. 2.3 

On taking Laplace transforms (variable fescs), eqn. (347) becomes 

It was pointed out in Chap. 7, Sect. 3.4 that the Laplace transform of the time-dependent density, p(r, s r0), and the steady-state density distribution in the presence of an ion scavenger, pss(r, fescs[r0), are identical functionally. Both densities satisfy the same differential equation 

Finally, on integrating the delta function and using the partially reflecting boundary condition, noting there is no flux over the outer boundary 

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