This paper discusses a mathematical model of the hysteresis behavior of the enzyme-substrate system in enzyme flow calorimetry. The hysteresis of the enzyme-substrate system due to enzyme flow calorimetry is based on the mathematical modeling of an immobilized enzyme system. The models are represented by the nonlinear differential equations of substrate inhibition kinetics in the enzymatic reaction. Two reliable wavelet-based computational algorithms for the approximate solutions for substrate concentration in planar, cylindrical, and spherical particles under steady-state conditions are discussed. To the best of our knowledge, no rigorous wavelet-based solution has been reported for the above models. The main idea of the proposed Taylor Wavelet Method (TWM) is that the nonlinear differential equations are converted into a system of algebraic equations. The validity and applicability of the proposed method are confirmed by comparing the results with those obtained by the Homotopy Perturbation Method (HPM) and the fourth-order Runge-Kutta (RK) method. Satisfactory agreement with the available results is observed. Moreover, the Chebyshev and Taylor wavelet methods are found to be efficient, simple, and straightforward tools for solving nonlinear reaction-diffusion models arising in bioprocess engineering.
- Open Access
- Article
A Robust Computational Algorithm for a Class of Nonlinear Enzyme Flow Calorimetric Models Using Taylor Wavelets: An Operational Matrix Method
- K. Harvinth,
- S. Kavipriya,
- R. Seethalakshmi,
- G. Hariharan *
Author Information
Received: 11 Apr 2026 | Revised: 01 Jun 2026 | Accepted: 22 Jun 2026 | Published: 17 Aug 2026
Abstract
Keywords
immobilized enzyme | planar particle | cylindrical particle | spherical particle | wavelets | substrate concentration
References
- 1.
Jesionowski, T.; Zdarta, J.; Krajewska, B. Enzyme immobilization by adsorption: A review. Adsorption 2014, 20, 801–821.
- 2.
Oliveira, S.C. Evaluation of effectiveness factor of immobilized enzymes using Runge-Kutta-Gill method: How to solve mathematical undetermination at particle center point? Bioprocess Eng. 1999, 20, 185–187.
- 3.
Fink, D.J.; Na, T.; Schultz, J.S. Effectiveness factor calculations for immobilized enzyme catalysts. Biotechnol. Bioeng. 1973, 15, 879–888.
- 4.
Manjón, A.; Gómez, J.L.; Gómez, E.; et al. Evaluation of the effectiveness factor along immobilized enzyme fixed-bed reactors: Design of a reactor with naringinase covalently immobilized into the glycophase-coated porous glass. Biotechnol. Bioeng. 1987, 30, 491–497.
- 5.
Vos, H.J.; Heederik, P.J.; Potters, J.; et al. Effectiveness factor for spherical biofilm catalysts. Bioprocess Eng. 1990, 5, 63–72.
- 6.
Bódalo, A.; Gómez, J.; Gómez, E.; et al. Analysis of diffusion effects on immobilized enzymes on porous supports with reversible Michaelis-Menten kinetics. Enzym. Microb. Technol. 1986, 8, 433–438.
- 7.
Gómez, J.L.; Bódalo, A.; Gómez, E.; et al. Two-parameter model for evaluating effectiveness factor for immobilized enzymes with reversible Michaelis–Menten kinetics. Chem. Eng. Sci. 2003, 58, 4287–4290.
- 8.
Rani, J.; Sevukaperumal, S.; Rajendran, L. Analytical expression of effectiveness factor for immobilized enzymes system with reversible Michaelis-Menten kinetics. Asian J. Sci. Appl. Technol. 2015, 4, 10–16.
- 9.
Avudainayagam, A.; Vani, C. Wavelet-Galerkin method for integro-differential equation. Appl. Numer. Math. 2000, 32, 247–254.
- 10.
Goyal, K.; Mehra, M. An adaptive meshfree spectral graph wavelet method for partial differential equations. Appl. Numer. Math. 2017, 113, 168–185.
- 11.
Guf, J.S.; Jiang, W.S. The Haar wavelets operational matrix of integration. Int. J. Syst. Sci. 1996, 27, 623–628.
- 12.
Razzaghi, M.; Yousefi, S. Legendre wavelets operational matrix of integration. Int. J. Syst. Sci. 2001, 32, 495–502.
- 13.
Keshavarz, E.; Ordokhani, Y. A fast numerical algorithm based on the Taylor wavelets for solving the fractional Integro-differential equations with weakly singular kernels. Math. Methods Appl. Sci. 2019, 42, 4427–4443.
- 14.
Keshavarz, E.; Ordokhani, Y.; Razzaghi, M. The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations. Appl. Numer. Math. 2018, 128, 205–216.
- 15.
Toan, P.T.; Vo, T.N.; Razzaghi, M. Taylor wavelet method for fractional delay differential equations. Eng. Comput. 2021, 37, 231–240.
- 16.
Pirabaharan, P.; Chandrakumar, R.D.; Hariharan, G. An efficient wavelet based approximation method for estimating the concentration of species and effectiveness factor in porous catalysts. MATCH Commun. Math. Comput. Chem. 2015, 73, 705–727.
- 17.
Sivasankari, M.K.; Rajendran, L. Analytical expression of the concentration of species and effectiveness factors in porous catalysts using the Adomian decomposition method. Kinet. Catal. 2013, 54, 95–105.
- 18.
Rajaraman, R.; Hariharan, G. An efficient wavelet based spectral method to singular boundary value problems. J. Math. Chem. 2015, 53, 2095–2113.
- 19.
Hariharan, G.; Rajaraman, R. A new coupled wavelet based method applied to the nonlinear reactions diffusion equation arising in mathematical chemistry. J. Math. Chem. 2013, 51, 2386–2400.
- 20.
Mahalakshmi, M.; Hariharan, G.; Kannan, K. The wavelet methods to linear and nonlinear reaction diffusion model arising in mathematical chemistry. J. Math. Chem. 2013, 51, 2361–2385.
- 21.
Mahalakshmi, M.; Hariharan, G. An efficient Chebyshev wavelet based analytical algorithm to steady state reaction-diffusion models arising in mathematical chemistry. J. Math. Chem. 2016, 54, 269–285.
- 22.
Mahalakshmi, M.; Hariharan, G. An efficient Wavelet Based Approximation Method onto steady state Reaction-Diffusion Model arising in Mathematical Chemistry. J. Membr. Biol. 2014, 247, 263–271.
- 23.
Yousefi, S.A. Legendre wavelets methods for solving differential equations of Lane–Emden type. Appl. Math. Comput. 2006, 181, 1417–1442.
- 24.
Yang, Y. Solving a nonlinear multi-order fractional differential equation using Legendre Pseudo-spectral methods. Appl. Math. 2013, 4, 113–118.
- 25.
Mohammadi, F. An extended Legendre Wavelet method for solving differential equation with non-analytic solution. J. Math. Ext. 2014, 8, 55–74.
- 26.
Hariharan, G.; Kannan, K. Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering. Appl. Math. Model. 2014, 38, 799–813.
- 27.
Hariharan, G.; Kannan, K.; Sharma, K.R. Haar wavelet in estimating the depth profile of soil temperature. Appl. Math. Comput. 2009, 210, 119–125.
- 28.
Devi, M.C.; Pirabaharan, P.; Rajendran, L.; et al. An efficient method for finding analytical expressions of substrate concentrations for different particles in an Immobilized enzyme system. React. Kinet. Mech. Catal. 2020, 130, 35–53.
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