Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Shape-preserving solutions of the time-dependent diffusion equation

Author: Frank S. Ham
Journal: Quart. Appl. Math. 17 (1959), 137-145
MSC: Primary 76.00
DOI: https://doi.org/10.1090/qam/108196
MathSciNet review: 108196
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Abstract: Exact solutions to the time-dependent diffusion equation are exhibited which correspond to the diffusion-limited growth of ellipsoidal precipitate particles with constant shape and dimensions proportional to the square root of the time. The asymmetry of the diffusion field in these solutions is consistent with the preservation of the particle's shape during growth even if the diffusivity is anisotropic. Limiting cases for simpler geometries are derived and shown to be in agreement with previously known results for radially symmetric particles and isotropic diffusion. Similar solutions for hyperboloidal surfaces are exhibited and generalizations are considered analogous to those discussed by Danckwerts for one-dimensional diffusion.

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DOI: https://doi.org/10.1090/qam/108196
Article copyright: © Copyright 1959 American Mathematical Society

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