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Series Expansions of Generalized Temperature Functions in N Dimensions

Published online by Cambridge University Press:  20 November 2018

Deborah Tepper Haimo
Deborah Tepper Haimo*
Affiliation:
Southern Illinois University and Harvard University
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The generalized heat equation is given by

1.1

where Δxf(x) = f″(x) + (2v/x)f′(x), v a fixed positive number. In a recent paper (5), the author established criteria for representing solutions of (1.1) in either the form

1.2

or

1.3

where Pn,v(x, t) is t he polynomial solution of (1.1) given explicitly by

1.4

and Wn,v(x, t) is its Appell transform; cf. (1). Our object is to generalize these results by extending them to higher dimensions. D. V. Widder (8) studied the problem for the ordinary heat equation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 794 - 802
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bragg, Louis R., The radial heat polynomials and related functions, Trans. Amer. Math. Soc. (to appear).Google Scholar
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5. Haimo, D. T., Expansions in terms of generalized heat polynomials and of their Appell transforms J. Math. Mech. (to appear).Google Scholar
6. Haimo, D. T., L2-expansions in terms of generalized heat polynomials and of their Appell transforms, Pacific J. Math., 15 (1965), 865–875.Google Scholar
7. Rosenbloom, P. C. and Widder, D. V., Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc, 92 (1959), 220–266.Google Scholar
8. Widder, D. V., Series expansions of solutions of the heat equation in n dimensions, Ann. Mat. Pura Appl., 55 (1961), 389–409.Google Scholar