The Stefan problem with arbitrary initial and boundary conditions
Author:
L. N. Tao
Journal:
Quart. Appl. Math. 36 (1978), 223-233
MSC:
Primary 80A10; Secondary 35K20
DOI:
https://doi.org/10.1090/qam/508769
MathSciNet review:
508769
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Abstract: The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed boundary condition at its face. An analytically exact solution of the problem is established, which is expressed in terms of some functions and polynomials of the similarity variable $x/{t^{1/2}}$ and time $t$. Convergence of the series solution is considered and proved. Hence the solution also serves as an existence proof. Some special initial and boundary conditions are discussed, which include the Neumann problem and the one-phase problem as special cases.
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H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, 2nd ed., Oxford Univ. Press, 1959
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R. W. Ruddle, The solidification of castings, Inst. Metals, London, 1957
L. I. Rubinstein, The Stefan problem (English translation), Am. Math. Soc., 1971
J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Oxford Univ. Press, 1975
S. G. Bankoff, Heat conduction or diffusion with change of phase, Adv. Chem. Eng. 5, 75–150 (1964)
J. C. Muehbauer and J. E. Sunderland, Heat conduction with freezing or melting, Appl. Mech. Rev. 18, 951–959 (1965)
W. A. Tiller, Principles of solidification, in Arts and sciences of growing crystals, ed. J. J. Gilman, Wiley, 1963
B. A. Boley, Survey of recent developments in the fields of heat conduction in solids and thermo-elasticity, Nuclear Eng. & Des. 18, 377–399 (1972)
Bateman Manuscript Project, Higher transcendental functions, 3 vols., McGraw-Hill, 1955
J. Kaye, A table of first eleven repeated integrals of the error function, J. Math. & Phys. 34, 119–125 (1955)
E. Jahnke and F. Emde, Tables of functions, Dover, 1945
C. Jordan, Calculus of finite differences, Chelsea, 1965
L. N. Tao, On the material time derivative of arbitrary order, Quart., Appl. Math., this issue
D. V. Widder, The heat equation, Academic Press, 1975
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Article copyright:
© Copyright 1978
American Mathematical Society
