Non-classical diffusion equations related to birth-death processes with two boundaries

Authors:
Z. G. Mansourati and L. L. Campbell

Journal:
Quart. Appl. Math. **54** (1996), 423-443

MSC:
Primary 60J80; Secondary 35B20, 35K55, 60J70

DOI:
https://doi.org/10.1090/qam/1402403

MathSciNet review:
MR1402403

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Abstract: A pair of forward and backward diffusion equations is considered. In the forward equation, boundary values appear in the differential equation, and in the backward equation, boundary values are related to average values of the solution in the interior of the domain. The forward equation can be regarded as a diffusion approximation to a type of birth-death process with returns to the interior, or as a heat equation in one dimension where heat flowing out from the boundaries is returned to the interior. Existence and uniqueness theorems are proved, and some properties of the associated eigenvalues and eigenfunctions are deduced. An expression for the steady-state solution is obtained. Some information on the goodness of the diffusion approximation is also obtained.

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Additional Information

DOI:
https://doi.org/10.1090/qam/1402403

Article copyright:
© Copyright 1996
American Mathematical Society