## Approximated Solutions of Michaelis-Menten Diffusion Reaction Equation by Combination of Methods

Published: 2023-09-25

Page: 132-146

Eli I. Cleopas

Department of Mathematics and Statistics, Federal University Otuoke, Bayelsa, Nigeria.

Avievie Igodo

Department of Mathematics and Statistics, Federal University Otuoke, Bayelsa, Nigeria.

Edikan E. Akpanibah *

Department of Mathematics and Statistics, Federal University Otuoke, Bayelsa, Nigeria.

*Author to whom correspondence should be addressed.

### Abstract

A lot of real life problems when modelled result to nonlinear differential equations. One example of such problems is the two point boundary nonlinear second order ordinary differential equation known as the Michaelis-Menten diffusion reaction equation. This equation cannot be solved easily by analytical methods; hence, there is need for numerical methods of solutions of this problem. In this paper, the fourth and sixth order approximated solutions of Michaeli’s-Menten diffusion reaction equation is obtained by a combination of Taylor series approximation with Ying Buzu Shu Algorithm and Taylor series approximation with Newton-Raphson method. The solutions by Ying Buzu Shu Algorithm and Newton-Raphson method were compared in terms of rate of convergence, error and relative error. Also, some numerical simulations were presented using matlab programming software. It was observed that a combination of Taylor series approximation with Ying Buzu Shu Algorithm has a faster rate of convergence, smaller error and relative error compared to the combination of Taylor series approximation with Newton-Raphson method.

Keywords: Newton-raphson method, michaelis-menten diffusion reaction equation, ying buzu shu algorithm, taylor series approximation

#### How to Cite

Cleopas , E. I., Igodo , A., & Akpanibah , E. E. (2023). Approximated Solutions of Michaelis-Menten Diffusion Reaction Equation by Combination of Methods. Asian Basic and Applied Research Journal, 5(1), 132–146. Retrieved from https://globalpresshub.com/index.php/ABAARJ/article/view/1867

### References

Sun He, Zhao Hong. Chap 12: Drug elimination and hepatic clearance. In: Chargel, L., Yu, A. (eds.) Edrs, Applied Biopharmaceutics and Pharmacokinetics, 7th Edn. McGraw Hill, New York. 2016;309–355.

Akpanibah EE, Osu BO. Optimal Portfolio Selection for a Defined Contribution Pension Fund with Return Clauses of Premium with Predetermined Interest Rate under Mean variance Utility. Asian Journal of Mathematical Sciences. 2018;2(2):19–29.

He J, Kou S, Sedighi H. An ancient Chinese algorithm for two point boundary problems and its application to the Michaelis- Menten Kinematics. Mathematical Modelling and Control. 2021;1(4):172-176.

Osu BO, Akpanibah EE, Njoku KNC. On the effect of stochastic extra contribution on optimal investment strategies for stochastic salary under the affine interest rate model in a DC Pension Fund, General Letters in Mathematics. 2017;2(3):138-149 .

Akpanibah EE, Osu BO, Oruh BI, Obi CN. Strategic optimal portfolio management for a DC pension scheme with return of premium clauses. Transaction of the Nigerian Association of Mathematical Physics. 2019;8(1):121-130.

Ghori QK, Ahmed, Siddiqui AM. Application of homotopy perturbation method to squeezing flow of a Newtonian fluid, International Journal of Nonlinear Sciences and Numerical Simulation. 2007;8:179-184.

DOI:10.1515/IJNSNS.2007.8.2.179

Ozis T, Yildirim A. A comparative study of he’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities, International Journal of Nonlinear Sciences and Numerical Simulation. 2007;8:243-248. DOI:10.1515/IJNSNS.2007.8.2.243

SJ Li, Liu XY. An improved approach to non-linear dynamical system identification using PID neural networks. International Journal of Nonlinear Sci- ences and Numerical Simulation. 2006;7:177-182. DOI:10.1515/IJNSNS.2006.7.2.177

Mousa MM, Ragab SF, Nturforsch Z. Application of the homotopy perturbation method to linear and non-linear Schrödinger equations. Zeitschrift Für Naturforschung. 2008;63:140-144.

He JH. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering. 1999;178:257-262. DOI:10.1016/S0045-7825(99)00018-3

Li X, He C. Homotopy perturbation method coupled with the enhanced perturbation method, J Low Freq. Noise VA. 2019;38:1399–1403.

Filobello-Nino U, Vazquez-Leal H, Palma-Grayeb B . The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform, Thermal Science. 2020;24: 1105–1115.

He J. Taylor series solution for a third order boundary value problem arising in architectural engineering. Ain Shams Eng J. 2020;11:1411–1414.

Han C, Wang Y, Li Z. Numerical solutions of space fractional variable-coefficient KdV modified KdV equation by Fourier spectral method. Fractals; 2021. Available:https://doi.org/10.1142/S0218348X21502467.

Wang K, Wang G. Gamma function method for the nonlinear cubic-quintic Dung oscillators, J Low Freq. Noise VA; 2021. Available:https://doi.org/10.1177/14613484211044613.

Udoh NA, Egbuhuzor UP. On the analysis of numerical methods for solving first order non linear ordinary differential equations. Asian Journal of Pure and Applied Mathematics. 2022;4(3):279-289.

Ogunrinde RB, Oshinubi KI. A computational approach to logistic model using adomian decomposition method. Computing, Information System & Development Informatics Journal. 2017;8(4). Available:www.cisdijournal.org

Bronson R, Costa G. Differential equations, third edition, Schaum’s Outline Series. McGraw-Hill, New York; 2016.

Momani S, Abuasad S, Odibat Z. Variational iteration method for solving nonlinear boundary value problems. Applied Mathematics and Computation. 2006;183(2006):1351-1358.

Choi B, Rempala Kim A, KyoungBeyond. The michaelis-menten equation: Accurate and efficient estimation of enzyme kinetic parameters. Sci. Rep. 2017;7:17-26.

Omari D, Alomari AK, Mansour A, Bawaneh A, Mansour A. Analytical solution of the non-linear michaelis–Menten pharmacokinetics equation, Int. J. Appl. Comput. Math. 2020;6-10. Available:https://doi.org/10.1007/s40819-019-0761-5

Amadi UC, Udoh NA. Solution of two point boundary problem using taylor series approximation and the Ying Buzu Shu algorithm, International Journal of Mathematical and Computational Sciences. 2022;16(8):68–73.

Traup JF. Iterative methods for the solution of equations. Prentice, Englewood Cliffs. New Jer-Sey; 1964.

Ortega JM, Rheinboldt WG. Iterative solution of nonlinear equations in several variables. Academic Press, New York; 1970.

Dennis JE, Schnabel RB. Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia; 1993.

Kelley CT. Solving nonlinear equations with Newton’s method. SIAM, Philadelphia; 2003.

Petković MS, Petković NB, Džunić LD. Multipoint methods for solving nonlinear equations: A survy. Appl Mat Comput. 2013;226:635–660.

Sharma JR, Guha RK, Sharma R. An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer Algorithms. 2013;62: 307–323.

Eleje CB, Egbuhuzor UP. Approximated solution of two point nonlinear boundary problem by a combination of taylor series expansion and Newton Raphson method. International Journal of Physical and Mathematical Sciences. 2023;17(3): 52–56.

He C. A simple analytical approach to a non-linear equation arising in porous catalyst. International Journal of Numerical Methods for Heat and Fluid-Flow. 2017;27:861–866.

He C. An introduction an ancient chinese algorithm and its modification. International Journal of Numerical Methods for Heat and Fluid-flow. 2016;26:2486–2491.

Golic M. Exact and approximate solutions for the decades-old Michaelis–Menten equation: progress curve analysis through integrated rate equations. Biochem Mol Biol Educ. 2011;39(2):117–125. Available:https://doi.org/10.1002/bmb.20479

Shanthi D, Ananthaswamy V, Rajendran L. Analysis of non-linear reaction-diffusion processes with Michaelis-Menten kinetics by a new Homotopy perturbation method, Natural Science. 2013;5(9):1034-1046. Available:http://dx.doi.org/10.4236/ns.2013.59128.