IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i9p1436-d404651.html
   My bibliography  Save this article

A Comparative Study of the Fractional-Order Clock Chemical Model

Author

Listed:
  • Hari Mohan Srivastava

    (Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40204, Taiwan
    Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan)

  • Khaled M. Saad

    (Department of Mathematics, College of Arts and Sciences, Najran University, Najran P.O. Box 1988, Saudi Arabia
    Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz P.O. Box 6803, Yemen)

Abstract

In this paper, a comparative study has been made between different algorithms to find the numerical solutions of the fractional-order clock chemical model (FOCCM). The spectral collocation method (SCM) with the shifted Legendre polynomials, the two-stage fractional Runge–Kutta method (TSFRK) and the four-stage fractional Runge–Kutta method (FSFRK) are used to approximate the numerical solutions of FOCCM. Our results are compared with the results obtained for the numerical solutions that are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation (LPI). Firstly, the accuracy of the results is checked by computing the absolute error between the numerical solutions by using SCM, TSFRK, FSFRK, and LPI and the exact solution in the case of the fractional-order logistic equation (FOLE). The numerical results demonstrate the accuracy of the proposed method. It is observed that the FSFRK is better than those by SCM, TSFRK and LPI in the case of an integer order. However, the non-integer orders in the cases of the SCM and LPI are better than those obtained by using the TSFRK and FSFRK. Secondly, the absolute error between the numerical solutions of FOCCM based upon SCM, TSFFRK, FSFRK, and LPI for integer order and non-integer order has been computed. The absolute error in the case of the integer order by using the three methods of the third order is considered. For the non-integer order, the order of the absolute error in the case of SCM is found to be the best. Finally, these results are graphically illustrated by means of different figures.

Suggested Citation

  • Hari Mohan Srivastava & Khaled M. Saad, 2020. "A Comparative Study of the Fractional-Order Clock Chemical Model," Mathematics, MDPI, vol. 8(9), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1436-:d:404651
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/9/1436/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/9/1436/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ghanbari, Behzad & Günerhan, Hatıra & Srivastava, H.M., 2020. "An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. M. M. Khader & Khaled. M. Saad, 2020. "Numerical treatment for studying the blood ethanol concentration systems with different forms of fractional derivatives," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 31(03), pages 1-13, January.
    3. Harendra Singh & Rajesh K. Pandey & Hari Mohan Srivastava, 2019. "Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials," Mathematics, MDPI, vol. 7(3), pages 1-24, February.
    4. West, Bruce J., 2015. "Exact solution to fractional logistic equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 429(C), pages 103-108.
    5. Khader, M.M. & Saad, K.M., 2018. "A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 169-177.
    6. Baleanu, Dumitru & Jajarmi, Amin & Mohammadi, Hakimeh & Rezapour, Shahram, 2020. "A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    7. Saad, Khaled M. & Srivastava, H.M. & Gómez-Aguilar, J.F., 2020. "A Fractional Quadratic autocatalysis associated with chemical clock reactions involving linear inhibition," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hari M. Srivastava & Khaled Mohammed Saad & Walid M. Hamanah, 2022. "Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations," Mathematics, MDPI, vol. 10(7), pages 1-13, March.
    2. Mohammad Izadi & Şuayip Yüzbaşi & Samad Noeiaghdam, 2021. "Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach," Mathematics, MDPI, vol. 9(16), pages 1-16, August.

    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. Saad, Khaled M. & Gómez-Aguilar, J.F. & Almadiy, Abdulrhman A., 2020. "A fractional numerical study on a chronic hepatitis C virus infection model with immune response," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    2. Srivastava, H.M. & Saad, Khaled M. & Khader, M.M., 2020. "An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Ghanbari, Behzad & Günerhan, Hatıra & Srivastava, H.M., 2020. "An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    4. Kritika, & Agarwal, Ritu & Purohit, Sunil Dutt, 2020. "Mathematical model for anomalous subdiffusion using comformable operator," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    5. Moghaddam, B.P. & Machado, J.A.T. & Behforooz, H., 2017. "An integro quadratic spline approach for a class of variable-order fractional initial value problems," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 354-360.
    6. Fawaz E. Alsaadi & Amirreza Yasami & Christos Volos & Stelios Bekiros & Hadi Jahanshahi, 2023. "A New Fuzzy Reinforcement Learning Method for Effective Chemotherapy," Mathematics, MDPI, vol. 11(2), pages 1-25, January.
    7. Vasily E. Tarasov, 2020. "Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients," Mathematics, MDPI, vol. 8(12), pages 1-11, December.
    8. Singh, C.S. & Singh, Harendra & Singh, Somveer & Kumar, Devendra, 2019. "An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1440-1448.
    9. Khader, M.M. & Inc, Mustafa, 2021. "Numerical technique based on the interpolation with Lagrange polynomials to analyze the fractional variable-order mathematical model of the hepatitis C with different types of virus genome," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    10. Ullah, Ihsan & Ahmad, Saeed & Rahman, Mati ur & Arfan, Muhammad, 2021. "Investigation of fractional order tuberculosis (TB) model via Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    11. Chimmula, Vinay Kumar Reddy & Zhang, Lei, 2020. "Time series forecasting of COVID-19 transmission in Canada using LSTM networks," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    12. Khater, Mostafa M.A. & Mohamed, Mohamed S. & Attia, Raghda A.M., 2021. "On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    13. Mahmoud, Emad E. & Trikha, Pushali & Jahanzaib, Lone Seth & Almaghrabi, Omar A., 2020. "Dynamical analysis and chaos control of the fractional chaotic ecological model," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    14. Etemad, Sina & Avci, Ibrahim & Kumar, Pushpendra & Baleanu, Dumitru & Rezapour, Shahram, 2022. "Some novel mathematical analysis on the fractal–fractional model of the AH1N1/09 virus and its generalized Caputo-type version," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    15. Pandey, Divyansh & Pandey, Rajesh K. & Agarwal, R.P., 2023. "Numerical approximation of fractional variational problems with several dependent variables using Jacobi poly-fractonomials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 28-43.
    16. Khan, Hasib & Alam, Khurshaid & Gulzar, Haseena & Etemad, Sina & Rezapour, Shahram, 2022. "A case study of fractal-fractional tuberculosis model in China: Existence and stability theories along with numerical simulations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 198(C), pages 455-473.
    17. Hussain, Ghulam & Khan, Amir & Zahri, Mostafa & Zaman, Gul, 2022. "Ergodic stationary distribution of stochastic epidemic model for HBV with double saturated incidence rates and vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    18. H. Çerdik Yaslan, 2021. "Numerical solution of the nonlinear conformable space–time fractional partial differential equations," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(2), pages 407-419, June.
    19. Alam, Mehboob & Zada, Akbar, 2022. "Implementation of q-calculus on q-integro-differential equation involving anti-periodic boundary conditions with three criteria," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    20. Liaqat, Muhammad Imran & Akgül, Ali, 2022. "A novel approach for solving linear and nonlinear time-fractional Schrödinger equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).

    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:8:y:2020:i:9:p:1436-:d:404651. 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.