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Journal of Mathematical Physics / Volume 25 / Issue 2

J. Math. Phys. 25, 228 (1984); http://dx.doi.org/10.1063/1.526144 (9 pages)

Variational bounds on the temperature distribution

Kalman Kalikstein1, Larry Spruch2, and Alberto Baider3

1Physics Department, Hunter College, City University of New York, New York, New York 10021 and Physics Department, New York University, New York, New York 10003
2Physics Department, New York University, New York, New York 10003
3Mathematics Department, Hunter College, City University of New York, New York, New York 10021

(Received 1 March 1983; accepted 26 August 1983)

Upper and lower stationary or variational bounds are obtained for functions which satisfy parabolic linear differential equations. (The error in the bound, that is, the difference between the bound on the function and the function itself, is of second order in the error in the input function, and the error is of known sign.) The method is applicable to a range of functions associated with equalization processes, including heat conduction, mass diffusion, electric conduction, fluid friction, the slowing down of neutrons, and certain limiting forms of the random walk problem, under conditions which are not unduly restrictive: in heat conduction, for example, we do not allow the thermal coefficients or the boundary conditions to depend upon the temperature, but the thermal coefficients can be functions of space and time and the geometry is unrestricted. The variational bounds follow from a maximum principle obeyed by the solutions of these equations.

KEYWORDS and PACS

Keywords

functions, differential equations, errors, thermal conduction, mass transfer, electric conductivity, fluid flow, slowing−down, neutrons, random walk, variational methods

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.526144

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    E. Gerjuoy, A. R. P. Rau, and L. Spruch, Phys. Rev. A 8, 662 (1973)
    E. Gerjuoy, A. R. P. Rau, L. Rosenberg, and L. Spruch, J. Math. Phys. 16, 1104 (1975JMAPAQ000016000005001104000001).

    E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55, 725 (1983).

    L. Spruch and L. Rosenberg, Phys. Rev. 116, 1034 (1959).

    L. Rosenberg, L. Spruch, and T. F. O'Malley, Phys. Rev. 118, 184 (1960).

    L. Rosenberg, Phys. Rev. D 1, 1019 (1970).

    R. Shakeshaft and L. Spruch, Phys. Rev. A 8, 206 (1973).

    R. Shakeshaft, Phys. Rev. A 12, 2230 (1975).


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