Modified Taylor solution of equation of oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics

Abstract

This work presents the application of a modified Taylor method to obtain a handy and easily computable approximate solution of the nonlinear differential equation to model the oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. The obtained solution is fully symbolic in terms of the coefficients of the equation, allowing to use the same solution for different values of the maximum reaction rate, the Michaelis constant, and the permeability of the cell membrane. Additionally, the numerical experiments show the high accuracy of the proposed solution, resulting 1.658509453 × 10 −15 as the lowest mean square error for a set of coefficients. The straightforward process to obtain the solution shows that the modified Taylor method is a handy alternative to a more sophisticated method because does not involve the solving of differential equations or calculate complicated integrals.

Optimized Direct Padé and HPM for Solving Equation of Oxygen Diffusion in a Spherical Cell

“…The employed integrating method was the Rule of Simpson 1/3. In all of the presented cases, the A.R.E. obtained with ODP is much lower compared to the Adomian and MTSM methods reported in [12,13] and matches with all the graphics presented in this work.…”

“…Thus, for Optimized Direct Padé and HPM, the proposed results in this work coincide with [12,13,39,48,49]; that is, the oxygen concentration is higher at the substrate and diminishes as it approaches the center of spherical cell, which is due to the oxygen diffusing into the cell; it reacts to give rise to the boundary condition at the center. Likewise, results are congruent with the spherical geometry proposed by Lane-Emden (2).…”

“…In [12], the equation that models oxygen diffusion in a spherical cell, including nonlinear absorption kinetics, is solved by transforming the Lane-Emden equation into its equivalent Volterra integral form, then Adomian decomposition was employed to solve the nonlinear Fredholm-Volterra integral. Furthermore, in [13] a fully symbolic solution was proposed to solve the aforementioned equation by means of a modified Taylor series obtaining an accuracy of 1.66×10 −15 as the lowest mean square error without the use of complicated integrals.…”

A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

“…In accordance with PSEM flexibility, series ( 21) can be obtained by some approximate methods from the literature: the Taylor series method [37], power series method [39], homotopy perturbation method [1,[17][18][19][20][21][22][23], perturbation method [34][35][36], homotopy analysis method [43], variational iteration method [44], differential transform method [45], and Adomian decomposition method [10][11][12][13][14][15], among others.…”

“…In effect, an exact solution to the proposed nonlinear problem can rarely be obtained [3], and for the same reason, several methods have been added to the best known classical methods. Next, we provide a list with some of most employed analytical methods in accordance with the literature: variational approaches [4][5][6], the tanh method [7], exp-function [8,9], Adomian's decomposition method [10][11][12][13][14][15], parameter expansion [16], the homotopy perturbation method (HPM) [1,6,[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], the perturbation method [34][35][36], the modified Taylor series method [37], the Picard method [38], the PSEM [39][40][41][42], the homotopy analysis method [25,43], the variational iteration method [44], and the differential transform method …”