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An efficient wavelet-based method for the numerical solutions of nonlinear coupled reaction–diffusion equations in biochemical engineering

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Abstract

The theoretical model of Saranya et al. (Bioprocess Biosyst Eng 41:281–294, 2017) glucose oxidation and hydrogen peroxide production in the Chitosan-alginate microsphere membrane is discussed. In this paper, an accurate and efficient Ultra-Spherical Wavelet Method (USWM) is successfully employed to investigate the numerical solutions for a set of nonlinear reaction–diffusion equations. The sensitivity analysis of various parameters on the concentrations of gluconic acid and hydrogen peroxide is also discussed. The numerical solutions are compared with Homotopy Perturbation Method and numerical simulation. Moreover, the use of USWM is found to be accurate, efficient, simple, flexible, and requires less computation costs.

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Abbreviations

Cg:

Concentration of glucose (mol/cm3)

COX :

Concentration of oxygen (mol/cm3)

Ca :

Concentration of gluconic acid(mol/cm3)

Ch :

Concentration of hydrogen peroxide (mol/cm3)

Dg :

Diffusion coefficient of glucose (cm2/s)

DOX :

Diffusion coefficient of oxygen (cm2/s)

Da :

Diffusion coefficient of gluconic acid (cm2/s)

Dh :

Diffusion coefficient of hydrogen peroxide (cm2/s)

Kg:

Michaelis–Menten constant for glucose (mol/cm3)

KOX :

Michaelis–Menten constant for oxygen (mol/cm3)

Vmax :

Maximal reaction velocity (cm/s)

Kg :

Michaelis–Menten constant for glucose:mol/cm3

S:

Radius of the microsphere (μ/m)

U:

Dimensionless concentration of glucose (None)

V:

Dimensionless concentration of oxygen (None)

W:

Dimensionless concentration of gluconic acid (None)

H:

Dimensionless concentration of hydrogen peroxide (None)

ℜ:

Overall reaction rate (None)

T:

Dimensionless time (None)

\( \gamma_{g} ,\;\gamma_{OX} ,\;\gamma_{h} ,\;\gamma_{l - } \) :

Dimensionless reaction–diffusion parameter

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Acknowledgments

The authors are very grateful to the referees for their valuable comments. This work was supported by the Naval Research Board (NRB), Government of India. Our hearty thanks are due to Prof. R. Sethuraman, Chairman, SASTRA Deemed University, Dr. S. Vaidhyasubramaniam, Vice-Chancellor, SASTRA Deemed University, Dr. S. Swaminathan, Dean/Planning and Development, SASTRA Deemed University for their kind encouragement and for providing good research environment. The author gratefully acknowledges the continued financial support from the Department of Science and Technology (DST), Government of India (SR/FST/MSI-107/2015).

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Authors and Affiliations

  1. Department of Mathematics, School of Humanities and Sciences, SASTRA Deemed University, Thanjavur, 613401, India

    M. Mahalakshmi & G. Hariharan

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  1. M. Mahalakshmi
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  2. G. Hariharan
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Correspondence to G. Hariharan.

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Mahalakshmi, M., Hariharan, G. An efficient wavelet-based method for the numerical solutions of nonlinear coupled reaction–diffusion equations in biochemical engineering. J Math Chem 57, 1154–1168 (2019). https://doi.org/10.1007/s10910-019-01014-3

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