6533b7d4fe1ef96bd126309c

RESEARCH PRODUCT

Two-Sided Estimates of the Solution Set for the Reaction–Diffusion Problem with Uncertain Data

Olli MaliSergey Repin

subject

Set (abstract data type)Range (mathematics)Uncertain dataBounded functionMathematical analysisReaction–diffusion systemSolution setFunction (mathematics)Robin boundary conditionMathematics

description

We consider linear reaction–diffusion problems with mixed Dirichlet–Neumann–Robin conditions. The diffusion matrix, reaction coefficient, and the coefficient in the Robin boundary condition are defined with an uncertainty which allow bounded variations around some given mean values. A solution to such a problem cannot be exactly determined (it is a function in the set of “possible solutions” formed by generalized solutions related to possible data). The problem is to find parameters of this set. In this paper, we show that computable lower and upper bounds of the diameter (or radius) of the set can be expressed throughout problem data and parameters that regulate the indeterminacy range. Our method is based on using a posteriori error majorants and minorants of the functional type (see [5, 6]), which explicitly depend on the coefficients and allow to obtain the corresponding lower and upper bounds by solving the respective extremal problems generated by indeterminacy of coefficients.

yearjournalcountryeditionlanguage
2009-10-19
https://doi.org/10.1007/978-90-481-3239-3_14