Temporary range of validity for some mathematical models which involve the heat-diffusion equation
Author:
Lucio R. Berrone
Journal:
Quart. Appl. Math. 54 (1996), 1-19
MSC:
Primary 35K05; Secondary 80A20
DOI:
https://doi.org/10.1090/qam/1373835
MathSciNet review:
MR1373835
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Abstract: We study different initial and boundary value problems for the one-dimensional heat-diffusion equation, with the purpose of establishing conditions on initial and boundary data that ensure the solution is, during a certain interval of time, bounded by two predetermined constants. When the constants represent phase-change temperatures of a material medium, the desired conditions can be physically interpreted by saying that the medium does not undergo a phase-change during a certain lapse of time; i.e., the heat-conduction model will preserve its validity in the meantime. The tools employed to tackle this kind of problem consist in reducing the initial and boundary value problems to a Volterra integral equation. For these equations there exist simple methods to estimate their solutions. We improve some results that appeared in [8].
L. R. Berrone, Rango Temporal de Validéz de Modelos que Involucran a la Ecuación del Calor-Difusión, Cuadernos del Instituto de Matemática “Beppo Levi” 22, 1–32 (1993)
J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1984
E. Goursat, Cours d’Analyse Mathématique, Tome III, Gauthier-Villars, Paris, 1923
M. H. Protter and H. F, Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967
T. L. Saaty, Modern Nonlinear Equations, Dover, New York, 1981
A. D. Solomon, D. G. Wilson, and V. Alexiades, Explicit solutions to phase change problems, Quart. Appl. Math. 41, 237–243 (1983)
D. A. Tarzia, An inequality for the coefficient $\sigma$ of the free boundary $s\left ( t \right ) = 2\sigma \sqrt t$ of the Neumann solution for the two-phase Stefan problem, Quart. Appl. Math. 39, 491–497 (1981)
D. A. Tarzia and C. V. Turner, A note on the existence of a waiting time for a two-phase Stefan problem, Quart. Appl. Math. 50, 1–10 (1992)
W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970
K. Yosida, Lectures on Differential and Integral Equations, Interscience, New York, 1960
L. R. Berrone, Rango Temporal de Validéz de Modelos que Involucran a la Ecuación del Calor-Difusión, Cuadernos del Instituto de Matemática “Beppo Levi” 22, 1–32 (1993)
J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1984
E. Goursat, Cours d’Analyse Mathématique, Tome III, Gauthier-Villars, Paris, 1923
M. H. Protter and H. F, Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967
T. L. Saaty, Modern Nonlinear Equations, Dover, New York, 1981
A. D. Solomon, D. G. Wilson, and V. Alexiades, Explicit solutions to phase change problems, Quart. Appl. Math. 41, 237–243 (1983)
D. A. Tarzia, An inequality for the coefficient $\sigma$ of the free boundary $s\left ( t \right ) = 2\sigma \sqrt t$ of the Neumann solution for the two-phase Stefan problem, Quart. Appl. Math. 39, 491–497 (1981)
D. A. Tarzia and C. V. Turner, A note on the existence of a waiting time for a two-phase Stefan problem, Quart. Appl. Math. 50, 1–10 (1992)
W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970
K. Yosida, Lectures on Differential and Integral Equations, Interscience, New York, 1960
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Article copyright:
© Copyright 1996
American Mathematical Society
