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A Valid Dynamical Control on the Reverse Osmosis System Using the CESTAC Method

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  • Samad Noeiaghdam

    (Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
    Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia)

  • Denis Sidorov

    (Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
    Energy Systems Institute of Russian Academy of Science, 664033 Irkutsk, Russia)

  • Alyona Zamyshlyaeva

    (Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia)

  • Aleksandr Tynda

    (Higher and Applied Mathematics Department, Penza State University, 440026 Penza, Russia)

  • Aliona Dreglea

    (Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia)

Abstract

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 ε 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.

Suggested Citation

  • Samad Noeiaghdam & Denis Sidorov & Alyona Zamyshlyaeva & Aleksandr Tynda & Aliona Dreglea, 2020. "A Valid Dynamical Control on the Reverse Osmosis System Using the CESTAC Method," Mathematics, MDPI, vol. 9(1), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:48-:d:469333
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    References listed on IDEAS

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    1. Naik, Parvaiz Ahmad & Zu, Jian & Owolabi, Kolade M., 2020. "Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Vignes, J., 1993. "A stochastic arithmetic for reliable scientific computation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 35(3), pages 233-261.
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