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

## DOI:

https://doi.org/10.14419/ijamr.v4i2.4273## Published:

2015-03-09## Keywords:

Taylor method, Power series method, Boundary valued problems, Approximate solutions.## 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 1.658509453Ã…~10âˆ’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.

## References

[1] S. H. Lin, â€Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kineticsâ€, *J. Theor. Biol.*, Vol.60, (1976),pp.449â€“457.

[2] D.L.S. McElwain, â€A re-examination of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kineticsâ€, *J. Theor. Biol.*, Vol.71, (1978), pp.255â€“263.

[3] H. Vazquez-Leal, B. Benhammouda, U.A. Filobello-Nino, A. Sarmiento-Reyes, V.M. Jimenez-Fernandez, A. Marin-Hernandez, A.L. Herrera-May, A. Diaz-Sanchez, J. Huerta-Chua, â€Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervalsâ€, *Springer Plus*, Vol.3, No.160, (2014), pp.1â€“7.

[4] P. Hiltmann, P. Lory, â€On oxygen diffusion in a spherical cell with Michaelis Menten oxygen uptake kineticsâ€, *Bull. Math. Biol.*, Vol.45, No.5, (1983), pp.661â€“664.

[5] M. J. Simpson, A. J. Ellery, â€An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cellâ€, *Appl. Math. Model.*, Vol.36, (2012), pp.3329â€“3334.

[6] A.M.Wazwaz, â€The variational iteration method for solving nonlinear singular boundary value problems arising in various physical modelsâ€, *Commun. Nonlinear Sci. Numer. Simulat.*, Vol.16, (2011), pp.3881â€“3886.

[7] R. Rach, A.M.Wazwaz, J. S. Duan , â€A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kineticsâ€, *International Journal of Biomathematics*, Vol.7, No.2, (2014), pp.1â€“12.

[8] R. Rach, J. S. Duan, A.M.Wazwaz, â€Solving coupled LaneEmden boundary value problems in catalytic diffusion reactions by the Adomian decomposition methodâ€, *J. Math. Chem.*, Vol.52, (2014), pp.255â€“267.

[9] A. M. Wazwaz, R. Rach, J. S. Duan, â€Adomian decomposition method for solving the Volterra integral form of the LaneEmden equations with initial values and boundary conditionsâ€, *Appl. Math. Comput.*, Vol.219, (2013), pp.5004-5019.

[10] A. M. Wazwaz, R. Rach, J. S. Duan, â€A study on the systems of the Volterra integral forms of the LaneEmden equations by the Adomian decomposition methodâ€, *Math. Models Methods Appl. Sci.*, Vol.37, (2014), pp.10-19.

[11] G. Adomian, Nonlinear Stochastic Operator Equations, *Academic*, (2014).

[12] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, *Kluwer Academic*, (1994).

[13] G. Adomian, R. Rach, â€Inversion of nonlinear stochastic operatorsâ€, *J. Math. Anal. Appl.*, Vol.91, (1983), pp.39â€“46.

[14] G.A. Sod, â€A numerical study of oxygen diffusion in a spherical cell with the Michaelis-Menten oxygen uptake kineticsâ€, *J. Math. Biol.*, Vol.24, (1986), pp.279â€“289.