Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-07-06T07:28:49.582Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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