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Convergence and Validity of Time Expansion Solutions: A Comparison to Exact and Approximate Solutions
Author(s) -
Parlange JeanYves
Publication year - 1975
Publication title -
soil science society of america journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.836
H-Index - 168
eISSN - 1435-0661
pISSN - 0361-5995
DOI - 10.2136/sssaj1975.03615995003900010006x
Subject(s) - convergence (economics) , series (stratigraphy) , mathematics , exact solutions in general relativity , diffusion , mathematical analysis , function (biology) , series expansion , basis (linear algebra) , physics , geometry , thermodynamics , paleontology , evolutionary biology , economics , biology , economic growth
The convergence of series solutions for the diffusion equation by time expansion is discussed quantitatively, on the basis of the linear and delta function solutions for a spherical cavity. It is shown that convergence alone is a poor criterion to justify the validity of the series solutions. A counter example, diffusion in the presence of an impervious wall, shows that the series may converge for all times but be entirely erroneous. By comparison an approximate integral technique yields a solution which agrees very well with the exact result.