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

  • M. M. KhaderEmail author
Original Research


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.


Biochemical reaction model Spectral collocation method Legendre–Lagrange Polynomials RK4 method Stability analysis 

Mathematics Subject Classification

65N12 41A30 



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).


  1. 1.
    Bell, W.W.: Special Functions for Scientists and Engineers. Great Britain, Butler and Tanner Ltd, Frome (1968)zbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2003)CrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)CrossRefzbMATHGoogle Scholar
  6. 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)Google Scholar
  7. 7.
    Khader, M.M.: Numerical treatment for solving fractional logistic differential equation. Differ. Equ. Dyn. Syst. 24(1), 99–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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. 9.
    Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)Google Scholar
  11. 11.
    Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186, 286–293 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Schnell, S., Mendoza, C.: Closed form solution for time-dependent enzyme kinetics. J. Theor. Biol. 187, 207–212 (1997)CrossRefGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of ScienceAl Imam Mohammad Ibn Saud Islamic University (IMSIU)RiyadhSaudi Arabia
  2. 2.Department of Mathematics, College of ScienceBenha UniversityBenhaEgypt

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