Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The random eigenvalue problem for a differential equation containing small white noise

Author: Ning Mao Xia
Journal: Quart. Appl. Math. 46 (1988), 611-630
MSC: Primary 60H10; Secondary 34B25, 34F05
DOI: https://doi.org/10.1090/qam/973379
MathSciNet review: 973379
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Abstract | References | Similar Articles | Additional Information

Abstract: By means of the extended Ito integral and the shooting method, this paper concerns the random problem containing white noise $ B\left( {t, \omega } \right)$:

$\displaystyle {\left[ { - p\left( t \right)u'\left( t \right)} \right]'} + \lef... ...ght] u = \lambda u, \qquad u\left( 0 \right) = 0, \qquad u\left( 1 \right) = 0.$

When $ \varepsilon $ is small, the existence and asymptotic expansions for the solutions can be obtained, and the normal properties for the first correction terms can be proved. Formulas for evaluation are derived and one example of the Schrödinger equation is given to illustrate the whole procedure.

References [Enhancements On Off] (What's this?)

  • [1] 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 MR 0263171
  • [2] J. vom Scheidt and W. Purkert, Random Eigenvalue Problems, Akademie-Verlag, Berlin, 1983 MR 790850
  • [3] W. 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, 419-445 (1983) MR 693876
  • [4] W. E. Boyce and Ning-Mao Xia, Upper bounds for the means of eigenvalues of random boundary value problems with weakly correlated coefficients, Quart. Appl. Math. 42, 439-454 (1985) MR 766881
  • [5] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 3rd ed., Wiley, New York, 1978 MR 507190
  • [6] Ning-Mao Xia, The distribution function of the solution of the random eigenvalue problem for differential equations, J. Math. Anal. Appl. 130, 577-589 (1988) MR 929964
  • [7] Ning-Mao Xia, The solutions for the two-point boundary value problems of stochastic differential equations containing small white noises, Acta Mathematicae Applicate 8, 340-350 (1985) (in Chinese) MR 843408
  • [8] Ning-Mao Xia, The density function of the solution of a two-point boundary value problem containing small stochastic processes, Quart. Appl. Math. 46, 29-47 (1988) MR 934679
  • [9] Ning-Mao Xia, Two-point boundary value problem of linear random differential equations containing small parameter and applications to one-dimensional Helmholtz equation, Journal of Systems Science and Mathematical Sciences 7, 129-137 (1987) (in Chinese) MR 886620
  • [10] Ning-Mao Xia, Upper bounds for the means of the higher eigenvalues of random eigenvalue problems, Journal of East China Institute of Chemical Technology 12, 115-119 (1986) (in Chinese)
  • [11] I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1972
  • [12] L. Arnold, Stochastic Differential Equations: Theory and Applications, John-Wiley, New York, 1974 MR 0443083
  • [13] B. S. White and J. N. Franklin, A Limit Theorem for Stochastic Two-point Boundary Value Problems of Ordinary Differential Equations, Commun. Pure Appl. Math. 32, 253-275 (1979) MR 512421
  • [14] L. A. Pastur, Spectra of random self-adjoint operators, Russian Math. Surveys 28, 1-67 (1973) MR 0406251

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DOI: https://doi.org/10.1090/qam/973379
Article copyright: © Copyright 1988 American Mathematical Society

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