Approximate analytical solution of a Stefan's problem in a finite domain

Author:
Shunsuke Takagi

Journal:
Quart. Appl. Math. **46** (1988), 245-266

MSC:
Primary 35R35; Secondary 35K05, 73B30, 80A20

DOI:
https://doi.org/10.1090/qam/950600

MathSciNet review:
950600

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A Stefan's problem in a finite domain may be given an approximate analytical solution. An example is shown with constant boundary and initial conditions. The solution is initially that of a semi-infinite domain, transits through infinitely many intermediate stage solutions, and finally becomes stationary. The solution is exact in the initial stage and also at the steady final stage, but approximate at the intermediate stages.

**[1]**I. G. Portnov,*Exact solution of freezing problem with arbitrary temperature variation on fixed boundary*, Soviet Phys. Dokl.**7**, 186 (1962)**[2]**F. Jackson,*The solution of problems involving the melting and freezing of finite slabs by a method due to Portnov*, Proc. Edinburgh Math. Soc. (2)**14**(1964/1965), 109–128. MR**0177639**, https://doi.org/10.1017/S0013091500025852**[3]**K. O. Westphal,*Series solution of freezing problem with the fixed surface radiation in a medium of arbitrary varying temperature*, Int. J. Heat Mass Trans.**10**, 195-205 (1967)**[4]**L. N. Tao,*The Stefan problem with arbitrary initial and boundary conditions*, Quart. Appl. Math.**36**(1978/79), no. 3, 223–233. MR**508769**, https://doi.org/10.1090/S0033-569X-1978-0508769-2**[5]**L. N. Tao,*On solidification problems including the density jump at the moving boundary*, Quart. J. Mech. Appl. Math.**32**(1979), no. 2, 175–185. MR**537351**, https://doi.org/10.1093/qjmam/32.2.175**[6]**L. N. Tao,*The analyticity of solutions of the Stefan problem*, Arch. Rational Mech. Anal.**72**(1979/80), no. 3, 285–301. MR**549645**, https://doi.org/10.1007/BF00281593**[7]**H. G. Landau,*Heat conduction in a melting solid*, Quart. Appl. Math.**8**(1950), 81–94. MR**0033441**, https://doi.org/10.1090/S0033-569X-1950-33441-3**[8]**H. S. Carslaw and J. C. Jaeger,*Conduction of Heat in Solids*, Oxford, at the Clarendon Press, 1947. MR**0022294****[9]**D. V. Widder,*The heat equation*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 67. MR**0466967****[10]**I. M. Gel’fand and G. E. Shilov,*Generalized functions. Vol. I: Properties and operations*, Translated by Eugene Saletan, Academic Press, New York-London, 1964. MR**0166596****[11]**B. A. Boley,*The embedding technique in melting and solidification problems*, In J. R. Ockendon and W. R. Hodgkins (Eds.),*Moving boundary problems in heat flow and diffusion*, 150-172, Clarendon Press, Oxford, 1975**[12]**C. Jordan,*Calculus of finite difference*, Budapest, 1939**[13]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[14]**A. M. Ostrowski,*Solutions of equations and systems of equations*(Second Edition), Academic Press, 1966**[15]**J. R. Rice,*Numerical methods, softwares, and analysis*: IMSM reference edition, McGraw-Hill, New York, 1983**[16]**William B. Jones and Wolfgang J. Thron,*Continued fractions*, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR**595864****[17]**J. F. Hart, et al.,*Computer approximations*, John Wiley & Sons, Inc., New York, 1968

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35R35,
35K05,
73B30,
80A20

Retrieve articles in all journals with MSC: 35R35, 35K05, 73B30, 80A20

Additional Information

DOI:
https://doi.org/10.1090/qam/950600

Article copyright:
© Copyright 1988
American Mathematical Society