Approximated Solutions of Two-Point Nonlinear Boundary Problem by a Combination of Taylor Series Expansion and Newton Raphson Method

Authors: Chinwendu. B. Eleje, Udechukwu P. Egbuhuzor

Abstract:

One of the difficulties encountered in solving nonlinear Boundary Value Problems (BVP) by many researchers is finding approximated solutions with minimum deviations from the exact solutions without so much rigor and complications. In this paper, we propose an approach to solve a two point BVP which involves a combination of Taylor series expansion method and Newton Raphson method. Furthermore, the fourth and sixth order approximated solutions are obtained and we compare their relative error and rate of convergence to the exact solution. Finally, some numerical simulations are presented to show the behavior of the solution and its derivatives.

Keywords: Newton Raphson method, non-linear boundary value problem, Taylor series approximation, Michaelis-Menten equation.

References:

[1] D. Omari, A. K. Alomari, A. Mansour A. Bawaneh, A, Mansour. Analytical Solution of the Non-linear Michaelis–Menten Pharmacokinetics Equation, Int. J. Appl. Comput. Math, (2020), 6-10. https://doi.org/10.1007/s40819-019-0761-5
[2] https://www.rosehulman.edu/~brandt/Chem330/Enzyme_kinetics.pdf
[3] B. Choi, G. Rempala, A., Kim, J. Kyoung. Beyond the Michaelis–Menten equation: accurate and efficient estimation of enzyme kinetic parameters. Sci. Rep. 7(2017), 17-26.
[4] Sun, He, Zhao, Hong: Chap 12: drug elimination and hepatic clearance. In: Chargel, L., Yu, A. (eds.) Edrs, Applied Biopharmaceutics and Pharmacokinetics, 7th edn, pp. 309–355. McGraw Hill, New York (2016)
[5] L. Rulí˘sek, S. Martin. Computer modeling (physical chemistry) of enzymecatalysis,metalloenzymes.https://www.uochb.cz/web/document/cms_library/2597
[6] T. G. Sudha, H. V. Geetha and S Harshini. solution of heat equation by method of separation of variable using the Foss tools maxima. International Journal of pure and applied mathematics, 117(12), (2017), 281-288.
[7] N. V. Vaidya, Deshpande, AA and Pidurkar, (2021). Solution of heat equation (Partial Differential Equation) by various methods S R Journal of Physics: Conference Series 1913 (2021) 012144 IOP Publishing doi:10.1088/1742-6596/1913/1/012144. 1-12.
[8] A. Abdulla – Al – Mamun, S Ali and M. MunnuMiah 2018 A study on an analytic solution 1D heat equation of a parabolic partial differential equation and implement in computer programming, international Journal of Scientific & Engineering Research 9, Issue 9, 913 ISSN 2229-5518
[9] N. A. Udoh, U. P. Egbuhuzor, On the analysis of numerical methods for solving first order non linear ordinary differential equations. Asian Journal of Pure and Applied Mathematics, 4(3) (2022), 279-289.
[10] R. B. Ogunrinde, K. I Oshinubi. A Computational Approach to Logistic Model using Adomian Decomposition Method, Computing, Information System & Development Informatics Journal. 8(4) (2017). www.cisdijournal.org
[11] R. Bronson, G. Costa. Differential equations, third edition, Schaum’s outline series. (2006). McGraw-Hill, New York.
[12] S. Momani, S. Abuasad, Z. Odibat. Variational iteration method for solving nonlinear boundary value problems. Applied Mathematics and Computation, 183(2006), 1351-1358.
[13] J. He. Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Eng. J., 11 (2020), 1411–1414.
[14] C. Han, Y. Wang, Z. Li. Numerical solutions of space fractional variable-coefficient KdV modified KdV equation by Fourier spectral method, Fractals, (2021). https://doi.org/10.1142/S0218348X21502467.
[15] K. Wang, G. Wang. Gamma function method for the nonlinear cubic-quintic Du_ng oscillators, J. Low Freq. Noise V. A., (2021). https://doi.org/10.1177/14613484211044613.
[16] C. He, D. Tian, G. Moatimid, H. F. Salman, M. H. Zekry, Hybrid Rayleigh-Van der Pol Du_ng Oscillator (HRVD): Stability Analysis and Controller,
[17] D. Tian, Q. Ain, N. Anjum. Fractal N/MEMS: from pull-in instability to pull-in stability, Fractals, 29 (2021), 2150030.
[18] D. Tian, C. He, A fractal micro-electromechanical system and its pull-in stability, J. Low Freq. Noise V. A., 40 (2021), 1380–1386.
[19] C. He, S. Liu, C. Liu, H. Mohammad-Sedighi. A novel bond stress-slip model for 3-D printed concretes, Discrete and Continuous Dynamical System, (2021). http://dx.doi.org/10.3934/dcdss.2021161.
[20] Q.K Ghori, M. Ahmed, A. M. Siddiqui. Application of Homotopy perturbation method to squeezing flow of a Newtonian fluid, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 179- 184. doi:10.1515/IJNSNS.2007.8.2.179
[21] T. Ozis, A. Yildirim. 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, 8 (2007), 243-248. doi:10.1515/IJNSNS.2007.8.2.243
[22] S.J. Li, X.Y. Liu. An Improved approach to non-linear dynamical system identification using PID neural networks, International Journal of Nonlinear Sci- ences and Numerical Simulation, 7 (2006), 177-182. doi:10.1515/IJNSNS.2006.7.2.177
[23] M. M. Mousa, S.F Ragab, Z. Nturforsch. Application of the Homotopy perturbation method to linear and non-linear Schrödinger equations. Zeitschrift für Naturforschung, 63 (2008), 140-144.
[24] J.H. He. Homotopy perturbation technique. Com- puter Methods in Applied Mechanics and Engineering, 178 (1999), 257-262. doi:10.1016/S0045-7825(99)00018-3
[25] X. Li, C. He. Homotopy perturbation method coupled with the enhanced perturbation method, J. Low Freq. Noise V. A., 38 (2019), 1399–1403.
[26] U. Filobello-Nino, H. Vazquez-Leal, B. Palma-Grayeb. The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform, Thermal Science, 24 (2020), 1105–1115.
[27] J. He, S. Kou, H, Sedighi. An Ancient Chinese Algorithm for two point boundary problems and its application to the Michaelis-Menten Kinematics, Mathematical Modelling and Control, 1(4) (2021), 172-176.
[28] M. Golic. Exact and approximate solutions for the decades-old Michaelis–Menten equation: progress curve analysis through integrated rate equations. Biochem. Mol. Biol. Educ. 39(2) (2011), 117–125. https://doi.org/10.1002/bmb.20479
[29] D. Shanthi, V. Ananthaswamy, L. Rajendran. Analysis of non-linear reaction-diffusion processes with Michaelis-Menten kinetics by a new Homotopy perturbation method, Natural Science, 5(9) (2013), 1034-1046.http://dx.doi.org/10.4236/ns.2013.59128.
[30] D. Omari, A. K. Alomari, A. Mansour A. Bawaneh, A, Mansour. Analytical Solution of the Non-linear Michaelis–Menten Pharmacokinetics Equation, Int. J. Appl. Comput. Math, (2020), 6-10. https://doi.org/10.1007/s40819-019-0761-5
[31] C. He, A Simple Analytical Approach to a Non-Linear Equation Arising in Porous Catalyst, International Journal of Numerical Methods for Heat and Fluid-Flow, 27 (2017), 861–866.
[32] C. He, An Introduction an Ancient Chinese Algorithm and Its Modification, International Journal of Numerical Methods for Heat and Fluid-flow, 26 (2016), 2486–2491.
[33] J. F. Traup. Iterative methods for the solution of equations. Prentice, Englewood Cliffs, New Jer-sey (1964).
[34] J. M. Ortega and W. G. Rheinboldt Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York. (1970)
[35] J. E Dennis and R. B. Schnabel Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia. (1993).
[36] C. T Kelley. Solving nonlinear equations with Newton’s method. SIAM, Philadelphia. (2003).
[37] M. S. Petković, N. B, Petković and LD, Džunić Multipoint methods for solving nonlinear equations: a survy. Appl Mat Comput 226 (2013) 635–660
[38] J. R. Sharma, R. K Guha, R. Sharma. An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer Algorithms 62, (2013), 307–323.
[39] U. C. Amadi and N. A. Udoh Solution of two point boundary problem using Taylor series approximation and the Ying Buzu Shu algorithm, International Journal of mathematical and computational sciences, 16(8) (2022), 68–73.