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I am studying diffusion equations to model the diffusion of fluorescence particles into a cell monolayer. The easiest way is to model them starting from a point source (x = 0 at t = 0 -> $x_\mathrm{0}$), that can diffuse freely (for example with the diffusivity factor D = 0.25). At x = 5 the particles are absorbed. I have found the following formula describing the survival function (the number of particles that have not been absorbed):

$$ S(t) = erf(\frac{x_f}{\sqrt{4Dt}}) $$

erf is the error function; $x_\mathrm{f}$ is the location of the absorbing boundary (therefore x = 5). I am able to match my simulations (monte-carlo approach) with this function. However, if i simulate particles to be released at $x_\mathrm{0}$ = 0 or $x_\mathrm{0}$ = -1 or $x_\mathrm{0}$ = -2, the above function obviously fails. I can however match the function with an average of the survival functions with $x_\mathrm{f}$ = 5, $x_\mathrm{f}$ = 6 and $x_\mathrm{f}$ = 6:

$$S(t) = \frac{\sum_{i=5}^{6}erf(\frac{i}{\sqrt{4Dt}})}{3}$$

In the end, I would like to find a way to convert the function S(t) that it depends on the minimum and maximum of the starting positions. Can someone point me in the right direction (maybe a related mathematical technique), as math and physics are not my major fields (more a field of enthusiasm)? Thank you

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The answer can be derived via basic calculus. To get a survival function from the stated problem above, one can simply integrate over the location variable. Therefore:

$$ S(t) = \frac{\int_a^b{erf(\frac{x_f}{\sqrt{4Dt}})dx_f}}{b-a} $$

which gives: $$ S(t) = \left[\frac{\sqrt{4Dt}*exp(-\frac{b^2}{4Dt})}{\sqrt\pi} - \frac{\sqrt{4Dt}*exp(-\frac{a^2}{4Dt})}{\sqrt\pi} - a*erf(\frac{a}{\sqrt{4Dt}}) + b*erf(\frac{b}{\sqrt{4Dt}})\right] * \frac{1}{b-a}$$

Hope this can help others aswell.

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