Article
MSC:
35D30,
35G30,
35J62,
47H05,
47J05,
49M25,
49N45,
65J15,
65N30 | MR 2852066 | Zbl 1249.35043 | DOI: 10.1007/s10492-011-0026-z
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Keywords:
worst scenario problem; nonlinear differential equation; uncertain input parameters; Galerkin approximation; Kachanov method
worst scenario problem; nonlinear differential equation; uncertain input parameters; Galerkin approximation; Kachanov method
Summary:
We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. Furthermore, it is shown that the Galerkin approximation of the state solution can be calculated by means of the Kachanov method as the limit of a sequence of solutions to linearized problems.
References:
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