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A class of densely invertible parabolic operator equations

Published online by Cambridge University Press:  17 April 2009

R.S. Anderssen
R.S. Anderssen
Affiliation:
Computer Centre, Australian National University, Canberra, ACT.
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Abstract

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Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 3 , December 1969 , pp. 363 - 374
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Adler, György, “Sulla caratterizzabilità dell'eq.uaz.ione del calore dal punto di vista del calcalo delle variazioni”, Magyar Tud. Akad. Mat. Kutató Int. Közl. 2 (1957), 153–157.Google Scholar
[2]Courant, R. and Hilbert, D., Methods of mathematical physics, Vol. 1, (Interscience, New York, 1953).Google Scholar
[3]Mikhlin, S.G., Variational methods in mathematical physics (Pergamon Press, Oxford, 1964). Translated from Variatsionnyye metody v matematiaheskoi fizike (Gostekhizdat, Moscow, 1957).Google Scholar
[4]Mikhlin, S.G., The numerical performance of variational methods (to be published by Noordhoff, Groningen). Translated from Chislennaya realizatsiya variatsionnykh metodov (Nauka, Moscow, 1966).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]Petryshyn, W.V., “On a class of K-p.d. and non K-p.d. operator equations”, J. Math. Anal. Appl. 10 (1965), 1–24.CrossRefGoogle Scholar
[7]Friedman, Avner, Partial differential equations of parabolic type (Prentice-Hall, Englewood Cliffs, New Jersey, 1964).Google Scholar
[8]Anderssen, R.S., “The numerical solution of parabolic differential equations using variational methods”, Numer. Math. 13 (1969), 129–145.CrossRefGoogle Scholar
[9]Riesz, Frigyes and Sz.-Nagy, Béla, Functional analysis (Blackie & and Son, London, 1956).Google Scholar
[10]Lions, J.-L. et Malgrange, B., “Sur l'unicité rétrograde dans les problèmes mixtes paraboliques”, Math. Saand. 8 (1960), 277–286.Google Scholar
[11]Lions, J.-L., Equations différentieltes operationnelles et problèmes aux limites (Grundlehren der mathematischen Wissenschaften, Band 111, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1961).CrossRefGoogle Scholar