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Optimum One‐term Solutions for Heat Conduction Problems
Author(s) -
Kleinstein G.,
Ting L.
Publication year - 1971
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19710510101
Subject(s) - eigenvalues and eigenvectors , term (time) , mathematics , mathematical analysis , exact solutions in general relativity , series (stratigraphy) , thermal conduction , moment (physics) , zero (linguistics) , physics , thermodynamics , classical mechanics , quantum mechanics , paleontology , linguistics , philosophy , biology
A series solution in descending powers of time is developed for heatconduction problems with initial profiles prescribed at an instant t* > 0. The leading term of the series solution, the first eigensolution, represents a source with strength given by the integral of the initial profile. It is shown that the coefficients of the eigensolutions corresponding to the second eigenvalue can be equated to zero when the origin is shifted to the center of gravity of the initial profile and that the coefficient of the symmetric eigensolution of the third eigenvalue will vanish when the time shift t* is related to the second moment of the initial profile. The first eigensolution with these optimum shifts in space and time variables yields an optimum one term representation of the exact solution as time increases. The optimum solution fulfills two properties of the exact solution, the conservation of energy integral and the invariance of the center of gravity of the temperature profile. Proofs for these properties of the exact solution are presented.