Upper bounds for the means of eigenvalues of random boundary value problems with weakly correlated coefficients
Authors:
William E. Boyce and Ning Mao Xia
Journal:
Quart. Appl. Math. 42 (1985), 439-454
MSC:
Primary 34F05; Secondary 34B05
DOI:
https://doi.org/10.1090/qam/766881
MathSciNet review:
766881
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Abstract: This paper concerns eigenvalue problems for second-order random differential equations with weakly correlated coefficients. The random problem and the mean (deterministic) problem are embedded in a parametrized problem whose eigenvalues are expanded in a power series in the parameter. This expansion leads, via the variational characterization of the eigenvalues, to computationally accessible upper bounds for the mean values of the eigenvalues of the original problem.
- William E. Boyce, Random eigenvalue problems, Probabilistic Methods in Applied Mathematics, Vol. 1, Academic Press, New York, 1968, pp. 1–73. MR 0263171
- William E. Boyce, On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems, Quart. Appl. Math. 38 (1980/81), no. 2, 241–245. MR 580882, DOI https://doi.org/10.1090/S0033-569X-1980-0580882-5
- William E. Boyce and Ning Mao Xia, The approach to normality of the solutions of random boundary and eigenvalue problems with weakly correlated coefficients, Quart. Appl. Math. 40 (1982/83), no. 4, 419–445. MR 693876, DOI https://doi.org/10.1090/S0033-569X-1983-0693876-9
L. Collatz, Eigenwertprobleme und ihre numerische Behandlung, Chelsea Publishing Co., New York, 1948
- Lothar Collatz, Eigenwertaufgaben mit technischen Anwendungen, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1963. Zweite, durchgesehene Auflage. MR 0152101
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
- Walter Purkert and Jürgen vom Scheidt, Zur approximativen Lösung des Mittelungsproblems für die Eigenwerte stochastischer Differentialoperatoren, Z. Angew. Math. Mech. 57 (1977), no. 9, 515–525 (German, with English and Russian summaries). MR 480129, DOI https://doi.org/10.1002/zamm.19770570904
W. Purkert and J. vom Scheidt, Eine Störungsrechnung für die Eigenwerte und Eigenvektoren zufälliger Matrizen, Beiträge Zur Analysis, 11 (1978), 113–135
- W. Purkert and J. vom Scheidt, Ein Grenzverteilungssatz für stochastische Eigenwertprobleme, Z. Angew. Math. Mech. 59 (1979), no. 11, 611–623 (German, with English and Russian summaries). MR 563447, DOI https://doi.org/10.1002/zamm.19790591104
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W. E. Boyce, Random eigenvalue problems, in Probabilistic Methods in Applied Mathematics, vol. 1, A. T. Bharucha-Reid (editor), Academic Press, New York, 1968, 1–73
W. E. Boyce, On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems, Q. Appl. Math., 38, 241–245 (1980)
W. E. Boyce and Ning-Mao Xia, The approach to normality of the solutions of random boundary and eigenvalue problems with weakly correlated coefficients, Q. Appl. Math., 40, 419–445 (1983)
L. Collatz, Eigenwertprobleme und ihre numerische Behandlung, Chelsea Publishing Co., New York, 1948
L. Collatz, Eigenwertaufgaben mit technischen Anwendungen, Geest and Portig, Leipzig, 1963
R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience, New York, 1953
W. Purkert and J. vom Scheidt, Zur approximativen Lösung des Mittelungsproblems für die Eigenwerte stochastischer Differentialoperatoren, ZAMM, 57, 515–525 (1977)
W. Purkert and J. vom Scheidt, Eine Störungsrechnung für die Eigenwerte und Eigenvektoren zufälliger Matrizen, Beiträge Zur Analysis, 11 (1978), 113–135
W. Purkert and J. vom Scheidt, Ein Grenzverteilungssatz für stochastische Eigenwertprobleme, ZAMM, 59 (1979), 611–623
W. Purkert and J. vom Scheidt, Stochastic eigenvalue problems for differential equations, Rep. Math. Phys., 15 (1979), 205–227
J. vom Scheidt and W. Purkert, Limit theorems for solutions of stochastic differential equation problems, Int. J. Math. and Math. Sci., 3 (1980), 113-149
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Article copyright:
© Copyright 1985
American Mathematical Society