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Functions of three variables which satisfy both the heat equation and Laplace's equation in two variables

Published online by Cambridge University Press:  09 April 2009

D. V. Widder
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In a recent paper on statistical fluid mechanics Professor J. Kampé de Fériet [1] employed several integrals of which the following is a typical example The function u(x, y, t), which it defines, formally satisfies the following three classical differential equations

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 3 , Issue 4 , November 1963 , pp. 396 - 407
Copyright
Copyright © Australian Mathematical Society 1963

References

[1]de Fériet, J. Kampé, Partial differential equations and continuum mechanics, edited by Langer, R. E., Madison, Wisconsin, 1961, pp. 107–136.Google Scholar
[2]Lamb, H., The dynamical theory of sound, London, 1931, p. 126.Google Scholar
[3]Widder, D. V., The rôle of the Appell transformation in the theory of heat conduction. To appear in the Transactions of the American Mathematical Society.CrossRefGoogle Scholar
[4]Bochner, S., Lectures on Fourier intetrals, Princeton, 1959, pp. 92–96.CrossRefGoogle Scholar
[5]Loomis, L. H. and Widder, D. V., The Poisson integral representation of functions which are positive and harmonic in a half-plane, Duke Mathematical Journal, vol. 9, (1942) pp. 643–645.CrossRefGoogle Scholar
[6]Widder, D. V., Positive temperatures on an infinite rod, Transactions of the American Mathematical Society, vol. 55 (1944) pp. 85–95.CrossRefGoogle Scholar