Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-11T03:39:29.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact variational representations for the solution of the simple parabolic equation

Published online by Cambridge University Press:  17 April 2009

R.S. Anderssen
R.S. Anderssen
Affiliation:
Computer Centre, Australian National University, Canberra, ACT.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 3 , December 1971 , pp. 305 - 314
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Anderssen, R. S., “A class of densely invertible parabolic operator equations”, Bull. Austral. Math. Soc. 1 (1969), 363–374.CrossRefGoogle Scholar
[2]Carslaw, H. S. and Jaeger, J. C., Conduction of heat in solids (Clarendon Press, Oxford, 1947).Google Scholar
[3]Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G. (edited by), Tables of integral transforms, Vol. 1. Based, in part, on notes left by Harry Bateman. (McGraw-Hill, New York, Toronto, London, 1954).Google Scholar
[4]Mikhlin, S. G., Variational methods in mathematical physics (translated by Boddington, T.; Pergamon Press, Oxford; The Macmillan Co..Google Scholar
[5]Petryshyn, W. V., “Direct and iterative methods for the solution of linear operator equations in Hilbert space”, Trans. Amer. Math. Soc. 105 (1962), 136–175.CrossRefGoogle Scholar
[6]Titchmarsh, E. C., The theory of functions, 2nd ed., (Clarendon Press, Oxford, 1939).Google Scholar
[7]Zygmund, A., Trigonometria series, Vol. 1, 2nd ed., (Cambridge University Press, Cambridge, 1959).Google Scholar