Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order

The aim of this study is to present a novel method to find the optimal solution of the reverse osmosis (RO) system. We apply the Sinc integration rule with single exponential (SE) and double exponential (DE) decays to find the approximate solution of the RO. Moreover, we introduce the stochastic arithmetic (SA), the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library instead of the mathematical methods based on the floating point arithmetic (FPA). Applying this technique, we would be able to find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorems are proved to support the method analytically. Based on these theorems, we can apply a new stopping condition in the numerical procedure instead of the traditional absolute error. These theorems show that the number of common significant digits (NCSDs) of exact and approximate solutions are almost equal to the NCSDs of two successive approximations. The numerical results are obtained for both SE and DE Sinc integration rules based on the FPA and the SA. Moreover, the number of iterations for various $\epsilon$ are computed in the FPA. Clearly, the DE case is more accurate and faster than the SE for finding the optimal approximation, the optimal error and the optimal iteration of the RO system.

The basic ideas of mathematical modeling of optimal dynamic measurements are presented. The key thing is to construct a mathematical model of the measuring transducer, which allows simulating also a complex measuring transducer. In this manuscript we propose such a mathematical model and we discuss its results obtained through numerical studies in a variety of cases: from the simplest, without considering disturbances, to the case when a deterministic disturbance, for example resonances, occur at the input and at the output of the measuring transducer. Lastly, the importance of the description of the set of admissible measurements in the modeling of dynamic measurements is discussed.