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Heat equation on a hemisphere

Published online by Cambridge University Press:  14 November 2011

M. van den Berg
M. van den Berg
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K.
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Synopsis

Consider the following heat conduction problem: let Hm−1 be a hemisphere of Sm−1 and suppose that Hm−1 has temperature 1 at time t = 0, while the boundary of Hm−1 is kept at temperature 0 for all time t>0. We obtain a combinatorial formula and a uniform estimate for the amount of heat in Hm−1 at time t.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 1-2 , 1991 , pp. 5 - 12
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Elworthy, K. D.. The method of images for the heat kernel of S 3. Proceedings of International Conference on Stochastic Processes, Geometry cand Physics, Ascona-Locarno 1988, ed. S. Albeverio et al.Google Scholar
2Dante De Blassie, R.. Exit times from cones in ∝n of brownian motion. Probab. Theory Related Fields 74 (1987), 1–29.Google Scholar
3Dante De Blassie, R.. Remark on exit times from cones in ∝n of brownian motion. Probab. Theory Related Fields 79 (1988), 95–97.Google Scholar
4Ndumu, M. N.. The heat kernel of the standard 3-sphere and some eigenvalue problems (preprint, 1988).Google Scholar
5van den Berg, M. and Davies, E. B.. Heat flow out of regions in ∝m. Math. Z. 202 (1989), 463–482.CrossRefGoogle Scholar
6van den Berg, M. and Gall, J.-F. Le. Mean curvature and the heat equation (preprint, 1991).Google Scholar
7Abramowitz, M. and Stegun, I. A.. Pocketbook of Mathematical Functions (Thun: Hani Deutsch, 1984).Google Scholar
8Rudin, W.. Principles of mathematical analysis (Kosaido: McGraw-Hill, 1982).Google Scholar
9Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.. Higher Transcendental Functions II (New York: McGraw-Hill, 1953).Google Scholar
10Robbins, H. E.. A remark on Stirling's formula. Amer. Math. Monthly 62 (1955), 26–29.Google Scholar
11Whittaker, E. T. and Watson, G. N.. A course of modern analysis (Cambridge: Cambridge University Press, 1973).Google Scholar
12Gradshteyn, I. S. and Ryzhik, I. M.. Table of Integrals, Series, and Products (New York: Academic Press, 1980).Google Scholar
13Titchmarsh, E. C.. The theory of functions (Oxford: Oxford University Press, 1979).Google Scholar