On the Fokker-Planck equation in the stochastic theory of mortality: II |
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Authors: | Ernesto Trucco |
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Institution: | (1) Division of Biological and Medical Research, Argonne National Laboratory, Argonne, Illinois |
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Abstract: | This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier
at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞
and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically,
because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section
6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic
value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those
derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow.
Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck
equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential
equations. |
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Keywords: | |
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