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On a Class of Partial Differential Equations with Nonlocal Dirichlet Boundary Conditions

  • Alain BensoussanEmail author
  • Janos Turi
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)

Abstract

We establish existence and uniqueness of solutions of a class of partial differential equations with nonlocal Dirchlet conditions in weighted function spaces. The problem is motivated by the study of the probability distribution of the response of an elasto-plastic oscillator when subjected to white noise excitation (see [1,2] on the derivation of the boundary value problem). Note that the developments in [1,2] are based on an extension of Khasminskii’s method (see, e.g. [5]) and in this paper we use a direct approach to achieve our objectives.

We refer the reader to [3, 4, 6, 7] for general background on modeling, theoretical, and computational issues related to elasto-plastic oscillators.

Keywords

Variational Inequality Bilinear Form Stochastic Stability Random Vibration Computational Issue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Bensoussan and J. Turi. Stochastic variational inequalities for elasto-plastic oscillators. C. R. Math. Acad. Sci. Paris, 343(6):399–406, 2006.zbMATHMathSciNetGoogle Scholar
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    A. Bensoussan and J. Turi. Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Optim., 58(1):1–27, 2007. DOI 10.1007/s00245-007-9027-4 (available online).CrossRefMathSciNetGoogle Scholar
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    R. Z. Khasminskii. Stochastic stability of differential equations. Sijthoff and Noordhoff, Alphen aan den Rijn, 1980.Google Scholar
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    J. B. Roberts and P.-T. D. Spanos. Random vibration and statistical linearization. John Wiley & Sons, Chichester, 1990.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.International Center for Decision and Risk Analysis, ICDRiA, School of ManagementUniversity of Texas at DallasRichardsonUSA
  2. 2.Programs in Mathematical SciencesUniversity of Texas at DallasRichardsonUSA

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