Spatial decay estimates in transient heat conduction
Authors:
C. O. Horgan, L. E. Payne and L. T. Wheeler
Journal:
Quart. Appl. Math. 42 (1984), 119-127
MSC:
Primary 80A20
DOI:
https://doi.org/10.1090/qam/736512
MathSciNet review:
736512
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Abstract: The spatial decay of solutions to initial-boundary value problems for the heat equation in a three-dimensional cylinder, subject to non-zero boundary conditions only on the ends, is investigated. It is shown that the spatial decay of end effects in the transient problem is faster than that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. The explicit spatial decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation. The results give rise to versions of Saint-Venant’s principle in transient heat conduction.
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V. G. Sigillito, On the spatial decay of solutions of parabolic equations, J. Appl. Math. Phys. (ZAMP), 21, 1078–1081 (1970)
J. K. Knowles, On the spatial decay of solutions of the heat equation, J. Appl. Math. Phys. (ZAMP), 22, 1050–1056 (1971)
C. O. Horgan and L. T. Wheeler, Spatial decay estimates for the heal equation via the maximum principle, J. Appl. Math. Phys. (ZAMP), 27, 371–376 (1976)
O. A. Oleinik and G. A. Yosifian, An analogue of Saint-Venant’s principle and the uniqueness of solutions of boundary value problems for parabolic equations in unbounded domains, Russian Math. Surveys, 31, 153–178 (1976)
C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant’s principle, Advances in Applied Mechanics (T. Y. Wu and J. W. Hutchinson, eds.), Vol. 23, Academic Press, New York (1983), 179–269
B. A. Boley, The determination of temperature, stresses and deflections in two-dimensional thermoelastic problems, J. Aero. Sci., 23, 67–75 (1956)
---, Some observations on Saint-Venant’s principle, Proc. 3rd U.S. Nat. Cong. Appl. Mech., ASME, New York (1958), 259–264
B. A. Boley and J. H. Weiner, Theory of thermal stresses, Wiley, New York, 1960
B. A. Boley, Upper bounds and Saint-Venant’s principle in transient heat conduction, Quart. Appl. Math. 18, 205–207 (1960)
M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, New Jersey, 1967
A. N. Tikhonov and A. A. Samarski, Partial differential equations of mathematical physics, Holden-Day, New York, 1964
M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, Dover, New York, 1965
H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, (2nd ed.), Oxford Univ. Press, Oxford, 1959
C. O. Horgan and L. T. Wheeler, A spatial decay estimate for pseudoparabolic equations, Lett. Appl. Eng. Sci., 3, 237–243 (1975)
G. I. Khil’kevich, An analogue of Saint-Venant’s principle, the Cauchy problem and the first boundary-value problem in an unbounded domain for pseudoparabolic equations, Russian Math. Surveys, 36, 252–253 (1981)
R. W. Carroll and R. E. Showalter, Singular and degenerate Cauchy problems, Academic Press, New York, 1976
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Article copyright:
© Copyright 1984
American Mathematical Society
