Jan Franců

Reliable Solutions of Problems with Uncertain Hysteresis Operators

abstract:
Problems in technology lead to initial boundary value problems for partial differential equations. Material properties which appear in constitutive relations are obtained by measurements. These data are uncertain and thus are known to some extent only. Using their mean values in numerical modelling cause several serious failures in technology.
The problem of finding a reliable solution by uncertain data is solved by the so-called worst scenario method introduced by Ivo Babuška and Ivan Hlaváček. The method consists in looking for the worst scenario that may appear in the case of any admissible data, the badness of situation is estimated by means of a~criterion-functional evaluating critical parts of the body.
In the contribution, the worst scenario method is applied to boundary value problems for nonlinear equation with a scalar hysteresis operator $\mathcal F$ or its inverse $\mathcal G$ of Prandtl-Ishlinskii type. The method demands special construction of admissible data and estimates the hysteresis operators. The existence of a reliable solution for the initial boundary value problem for the heat conduction or the diffusion equation $c\,u_t=(\mathcal F_\eta[u_x])_x+f$ with various types of criterion-functionals is proved.

Mathematics Subject Classification: 47J40, 47Hxx, 35K55

Key words and phrases: Prandtl-Ishlinskii hysteresis operator, reliable solution, uncertain data, worst scenario method, heat conduction equation, diffusion equation