IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2020i1p48-d469333.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/1/48/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/1/48/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Samad Noeiaghdam & Aliona Dreglea & Hüseyin Işık & Muhammad Suleman, 2021. "A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection," Mathematics, MDPI, vol. 9(12), pages 1-17, June.
    2. Samad Noeiaghdam & Sanda Micula & Juan J. Nieto, 2021. "A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library," Mathematics, MDPI, vol. 9(12), pages 1-26, June.
    3. Albertsen, Niels Christian & Chesneaux, Jean-Marie & Christiansen, Søren & Wirgin, Armand, 1999. "Comparison of four software packages applied to a scattering problem1Professor Ralph E. Kleinman, University of Delaware, USA, in memoriam.1," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 48(3), pages 307-317.
    4. Bansal, Komal & Mathur, Trilok & Agarwal, Shivi, 2023. "Fractional-order crime propagation model with non-linear transmission rate," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    5. Abidemi, Afeez & Owolabi, Kolade M. & Pindza, Edson, 2022. "Modelling the transmission dynamics of Lassa fever with nonlinear incidence rate and vertical transmission," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 597(C).
    6. Jézéquel, F. & Rico, F. & Chesneaux, J.-M. & Charikhi, M., 2006. "Reliable computation of a multiple integral involved in the neutron star theory," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 44-61.
    7. Joshi, Divya D. & Bhalekar, Sachin & Gade, Prashant M., 2023. "Controlling fractional difference equations using feedback," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    8. Attaullah, & Jan, Rashid & Yüzbaşı, Şuayip, 2021. "Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    9. Alt, R. & Lamotte, J.-L., 2001. "Experiments on the evaluation of functional ranges using a random interval arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 56(1), pages 17-34.
    10. Naik, Parvaiz Ahmad & Owolabi, Kolade M. & Yavuz, Mehmet & Zu, Jian, 2020. "Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    11. Mangal, Shiv & Misra, O.P. & Dhar, Joydip, 2023. "Fractional-order deterministic epidemic model for the spread and control of HIV/AIDS with special reference to Mexico and India," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 82-102.
    12. Alzahrani, Faris & Razzaq, Oyoon Abdul & Rehman, Daniyal Ur & Khan, Najeeb Alam & Alshomrani, Ali Saleh & Ullah, Malik Zaka, 2022. "Repercussions of unreported populace on disease dynamics and its optimal control through system of fractional order delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    13. Samad Noeiaghdam & Sanda Micula, 2021. "Dynamical Strategy to Control the Accuracy of the Nonlinear Bio-Mathematical Model of Malaria Infection," Mathematics, MDPI, vol. 9(9), pages 1-24, May.
    14. Samad Noeiaghdam & Denis Sidorov & Abdul-Majid Wazwaz & Nikolai Sidorov & Valery Sizikov, 2021. "The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    15. Gbenga O. Ojo & Nazim I. Mahmudov, 2021. "Aboodh Transform Iterative Method for Spatial Diffusion of a Biological Population with Fractional-Order," Mathematics, MDPI, vol. 9(2), pages 1-21, January.
    16. Guilain, S. & Vignes, J., 1994. "Validation of numerical software results — Application to the computation of apparent heat release in direct-injection diesel engines," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 37(1), pages 73-92.
    17. Samad Noeiaghdam & Sanda Micula, 2021. "A Novel Method for Solving Second Kind Volterra Integral Equations with Discontinuous Kernel," Mathematics, MDPI, vol. 9(17), pages 1-12, September.
    18. Arshad, Sadia & Siddique, Imran & Nawaz, Fariha & Shaheen, Aqila & Khurshid, Hina, 2023. "Dynamics of a fractional order mathematical model for COVID-19 epidemic transmission," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    19. Salkuyeh, Davod Khojasteh & Toutounian, Faezeh & Yazdi, Hamed Shariat, 2008. "A procedure with stepsize control for solving n one-dimensional IVPs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(2), pages 167-176.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:48-:d:469333. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.