New analytic solutions of the one-dimensional heat equation for temperature and heat flow rate both prescribed at the same fixed boundary (with applications to the phase change problem)
Author:
David Langford
Journal:
Quart. Appl. Math. 24 (1967), 315-322
MSC:
Primary 35.78
DOI:
https://doi.org/10.1090/qam/211094
MathSciNet review:
211094
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Abstract: New solutions of the heat equation are exhibited for the case in which both the temperature and heat flow rate are prescribed at a single fixed boundary. The prescribed temperature and heat flow rate may be any arbitrary infinitely differentiable functions of time. The new solutions are applicable for one-dimensional (radial) heat flow in spheres, cylinders, and slabs.
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- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
David Langford, A closed form solution for the constant velocity solidification of spheres initially at the fusion temperature, Br. J. Applied Physics, 17(2), 286 (1966)
David Langford, Stefan’s melting problem, Doctoral Dissertation, Rensselaer Polytechnic Institute, Troy, N. Y., 1965
- David Langford, Pseudo-similarity solutions of the one-dimensional diffusion equation with applications to the phase change problem, Quart. Appl. Math. 25 (1967), 45–52. MR 209686, DOI https://doi.org/10.1090/S0033-569X-1967-0209686-2
C. L. Pekeris and L. B. Slichter, Problems of ice formation, J. Applied Physics, 10(2), 135–137 (1939)
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J. Stefan, On the theory of ice formation, especially on ice formation in polar seas (Über die Theorie der Eisbildung, insbesondere uber die Eisbildung im Polarmeere). Sitzungsberichte der Kaiserlichen Akademie Wiss. Wien., Math.-naturwiss. Kl. 98(2a), 965–973 (1890). (In German)
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- D. V. Widder, Series expansions of solutions of the heat equation in $n$ dimensions, Ann. Mat. Pura Appl. (4) 55 (1961), 389–409. MR 136870, DOI https://doi.org/10.1007/BF02412092
H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, 2nd Ed., Oxford Univ. Press, London, 1959
R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, Partial Differential Equations, pp. 39–54, 227, 237–9. Interscience, New York, 1962
David Langford, A closed form solution for the constant velocity solidification of spheres initially at the fusion temperature, Br. J. Applied Physics, 17(2), 286 (1966)
David Langford, Stefan’s melting problem, Doctoral Dissertation, Rensselaer Polytechnic Institute, Troy, N. Y., 1965
David Langford, Pseudo-similarity solutions of the one-dimensional diffusion equation with applications to the phase change problem, Quart. Appl. Math., 1967 (to appear)
C. L. Pekeris and L. B. Slichter, Problems of ice formation, J. Applied Physics, 10(2), 135–137 (1939)
P. C. Rosenbloom and D. V. Widder, Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc. 92, 220–266 (1959)
J. Stefan, On the theory of ice formation, especially on ice formation in polar seas (Über die Theorie der Eisbildung, insbesondere uber die Eisbildung im Polarmeere). Sitzungsberichte der Kaiserlichen Akademie Wiss. Wien., Math.-naturwiss. Kl. 98(2a), 965–973 (1890). (In German)
D. V. Widder, Analytic solutions of the heat equation, Duke Math. J. 29(4), 497–504 (1962)
D. V. Widder, Series expansions of solutions of the heat equation in n dimensions. Ann. Mat. Pura Appl. IV-55, 389–410 (1961) (In English)
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© Copyright 1967
American Mathematical Society
