Skip to main content
SpringerLink
Log in

Numerical Treatment by Using a Hybrid Efficient Technique for the Biochemical Reaction Model

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this article, we implement a spectral collocation method by using the properties of Legendre and Lagrange polynomials for solving the resulting nonlinear system of ODEs of the biochemical reaction model. This technique reduces the proposed model to a system of algebraic equations. We prove the uniqueness and present the local stability of the given model. A comparison with the numerical solution is obtained by using the RK4 method and the previously published results using the Picard-Padè method. The proposed method introduces a promising tool for solving many nonlinear systems of differential equations. Numerical illustrations are stated to demonstrate utility, validity and the great potential of the introduced method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bell, W.W.: Special Functions for Scientists and Engineers. Great Britain, Butler and Tanner Ltd, Frome (1968)

    MATH  Google Scholar 

  2. Borhanifara, A., Khader, M.M.: Jacobi operational matrix and its application for solving systems of ODEs. Differ. Equ. Dyn. Syst. 24(4), 459–473 (2015)

    Article  MathSciNet  Google Scholar 

  3. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2003)

    Book  Google Scholar 

  4. Goha, S.M., Noorani, M.S.M., Hashim, I.: Introducing variational iteration method to a biochemical reaction model. Nonlinear Anal. Real World Appl. 11, 2264–2272 (2010)

    Article  MathSciNet  Google Scholar 

  5. Hashim, I., Chowdhurly, M.S.H., Mawa, S.: On multistage homotopy perturbation method applied to nonlinear biochemical reaction model. Chaos Solitons Fract. 36, 823–827 (2008)

    Article  Google Scholar 

  6. Khader, M.M.: On the numerical solutions to nonlinear biochemical reaction model using Picard–Padé technique. World J. Model. Simul. 9(1), 38–46 (2013)

    MathSciNet  Google Scholar 

  7. Khader, M.M.: Numerical treatment for solving fractional logistic differential equation. Differ. Equ. Dyn. Syst. 24(1), 99–107 (2016)

    Article  MathSciNet  Google Scholar 

  8. Kumar, R., Kumar, S.: A new fractional modelling on susceptible-infected-recovered equations with constant vaccination rate. Nonlinear Eng. 3(1), 11–19 (2013)

    Google Scholar 

  9. Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)

    Article  MathSciNet  Google Scholar 

  10. Matignon, D.: Stability results for fractional differential equations with applications to control processing. Computational engineering in systems and application. In: Multiconference, IMACS, IEEE-SMC, Lille, France, vol. 2, pp. 963–968 (1996)

  11. Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186, 286–293 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Schnell, S., Mendoza, C.: Closed form solution for time-dependent enzyme kinetics. J. Theor. Biol. 187, 207–212 (1997)

    Article  Google Scholar 

  13. Sen, A.K.: An application of the Adomian decomposition method to the transient behavior of a model biochemical reaction. J. Math. Anal. Appl. 131, 232–245 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks Deanship of Academic Research, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, KSA, for the Financial support of the project number (371204).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

    M. M. Khader

  2. Department of Mathematics, College of Science, Benha University, Benha, Egypt

    M. M. Khader

Authors
  1. M. M. Khader
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. M. Khader.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khader, M.M. Numerical Treatment by Using a Hybrid Efficient Technique for the Biochemical Reaction Model. Differ Equ Dyn Syst 28, 707–713 (2020). https://doi.org/10.1007/s12591-019-00455-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-019-00455-1

Keywords

Mathematics Subject Classification

Search

Navigation