×

zbMATH — the first resource for mathematics

On the worst scenario method: a modified convergence theorem and its application to an uncertain differential equation. (English) Zbl 1199.47207
Summary: We propose a theoretical framework for solving a class of worst scenario problems. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. The main convergence theorem modifies and corrects the relevant results already published in the literature. The theoretical framework is applied to a particular problem with an uncertain boundary value problem for a nonlinear ordinary differential equation with an uncertain coefficient.

MSC:
47H05 Monotone operators and generalizations
47J05 Equations involving nonlinear operators (general)
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] J. Francu: Monotone operators. A survey directed to applications to differential equations. Appl. Math. 35 (1990), 257–301. · Zbl 0724.47025
[2] I. Hlaváček: Reliable solution of a quasilinear nonpotential elliptic problem of a non-monotone type with respect to uncertainty in coefficients. J. Math. Anal. Appl. 212 (1997), 452–466. · Zbl 0919.35047 · doi:10.1006/jmaa.1997.5518
[3] I. Hlaváček: Reliable solution of elliptic boundary value problems with respect to uncertain data. Nonlinear Anal., Theory Methods Appl. 30 (1997), 3879–3890. · Zbl 0896.35034 · doi:10.1016/S0362-546X(96)00236-2
[4] I. Hlaváček, J. Chleboun, I. Babuška: Uncertain Input Data Problems and the Worst Scenario Method. Elsevier, Amsterdam, 2004.
[5] I. Hlaváček, M. Křížek, J. Malý: On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 184 (1994), 168–189. · Zbl 0802.65113 · doi:10.1006/jmaa.1994.1192
[6] J. Chleboun: Reliable solution for a 1D quasilinear elliptic equation with uncertain coefficients. J. Math. Anal. Appl. 234 (1999), 514–528. · Zbl 0944.35027 · doi:10.1006/jmaa.1998.6364
[7] J. Chleboun: On a reliable solution of a quasilinear elliptic equation with uncertain coefficients: Sensitivity analysis and numerical examples. Nonlinear Anal., Theory Methods Appl. 44 (2001), 375–388. · Zbl 1002.35041 · doi:10.1016/S0362-546X(99)00274-6
[8] M. Křížek, P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Longman Scientific & Technical, New York, 1990.
[9] E. Zeidler: Applied Functional Analysis. Applications to Mathematical Physics. Springer, Berlin, 1995. · Zbl 0834.46002
[10] E. Zeidler: Applied Functional Analysis. Main Principles and Their Applications. Springer, New York, 1995. · Zbl 0834.46003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.