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Convergence and Validity of Time Expansion Solutions: A Comparison to Exact and Approximate Solutions¹

ABSTRACT

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.

¹ Contribution from the Connecticut Agr. Exp. Sta., New Haven, Conn. 06504.

² Mathematician, Department of Ecology and Climatology.

Received for publication September 3, 1974. Accepted for publication October 22, 1974.