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Dual extremum principles for the heat equation†

Published online by Cambridge University Press:  14 February 2012

W. D. Collins
W. D. Collins
Affiliation:
Department of Applied Mathematics and Computing Science, University of Sheffield
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Synopsis

Dual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 3-4 , 1977 , pp. 273 - 292
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Noble, B. and Sewell, M. J.. On dual extremum principles in applied mathematics. J. Inst. Math. Appl. 9 (1972), 123–193.CrossRefGoogle Scholar
2Morse, P. M. and Feshbach, H.. Methods of theoretical physics I, 298–299, 313–314 (Toronto: McGraw-Hill, 1953).Google Scholar
3Gurtin, M. E.. Variational principles for linear initial-value problems. Quart. Appl. Math. 22 (1964), 252–256.CrossRefGoogle Scholar
4Noble, B.. Variational finite element methods for initial-value problems. Proc. Finite Elem. Conf. Brunei Univ. 143–151, Ed. Whiteman, J. (London: Academic Press, 1973).Google Scholar
5Herrera, I.. A general formulation of variational principles. Rep. Inst. Ing. Univ. Nac. Autónoma México E10 (1974).Google Scholar
6Herrera, I. and Bielak, J.. Dual variational principles for diffusion equations. Quart. Appl. Math., 34 (1976), 85–102.CrossRefGoogle Scholar
7Finlayson, B. A. and Scriven, L. E.. On the search for variational principles. Int. J. Heat Mass Transfer 10 (1967), 799–821.CrossRefGoogle Scholar
8Finlayson, B. A. and Scriven, L. E.. The method of weighted residuals and its relation to certain variational principles for the analysis of transport processes. Chem. Engrg Sci. 20 (1965), 395–404.CrossRefGoogle Scholar
9Finlayson, B. A. and Scriven, L. E.. The method of weighted residuals—a review. Appl. Mech. Rev. 19 (1966), 735–748.Google Scholar
10Collins, W. D.. Mathematics Research Center Report No. 1624 (Madison: Univ. Wisconsin, 1976).Google Scholar
11Kato, T.. Perturbation theory for linear operators, 275–276 (Berlin: Springer, 1966).Google Scholar
12Barnsley, M. F. and Robinson, P. D.. Bivariational bounds. Proc. Roy. Soc. Ser. A 338 (1974), 527–533.Google Scholar
13Sloan, D. M.. Extremum principles for magnetohydrodynamic channel flow. Z. Angew. Math. Phys. 24 (1973), 689–698.CrossRefGoogle Scholar
14Courant, R. and Hilbert, D.. Methods of mathematical physics I, 252–257 (New York: Interscience, 1953).Google Scholar
15Mikhlin, S. G.. Variational methods in mathematical physics 40–73 (Oxford: Pergamon, 1964)Google Scholar