# The Reaction–Diffusion Models in Biomedicine: Highly Accurate Calculations via a Hybrid Matrix Collocation Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. A Review of a New Class of Polynomials

#### 2.1. The Chatterjea Polynomials

**Lemma**

**1.**

**Proof.**

#### 2.2. Convergent of ChPs in the Sense of ${L}_{2}$

**Theorem**

**1.**

**Proof.**

## 3. Description of QLM–ChPs Matrix Approach

#### 3.1. The Essence of QLM

#### 3.2. The Main Algorithm

**Lemma**

**2.**

**Proof.**

#### 3.3. Theoretical Upper Bound for QLM–ChPs Approach

**Theorem**

**2.**

**Proof.**

#### 3.4. Error Measurement via REF Method

#### 3.5. The RC Methodology

## 4. Numerical Calculations

#### 4.1. Case Study I: Planar Particle ($n=0$)

#### 4.2. Case Study II: Spherical Particle ($n=2$)

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Scheller, F.; Schubeert, F. Biosensor; Elsevier: Amsterdam, The Netherlands, 1988; Volume 7. [Google Scholar]
- Wollenberger, U.; Lisdat, F.; Scheller, F.W. Enzymatic Substrate Recycling Electrodes. Frontiers in Biosensorics. B and II, Practical Applications; Birkhauser Verlag: Basel, Switzerland, 1997; pp. 45–70. [Google Scholar]
- Aris, R. Mathematical Modeling: A Chemical Engineer’s Perspective; Elsevier: Amsterdam, The Netherlands, 1999. [Google Scholar]
- Michaelis, L.; Menten, M. Die kinetic der invertinwirkung. Biochem. Z.
**1913**, 79, 333–369. [Google Scholar] - Lin, S.H. Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J. Theoret. Biol.
**1976**, 60, 449–457. [Google Scholar] [CrossRef] [PubMed] - McElwain, D.L.S. A Re-examination of oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. J. Theoret. Biol.
**1978**, 7, 255–263. [Google Scholar] [CrossRef] [PubMed] - Manimozhi, P.; Subbiah, A.; Rajendran, L. Solution of steady-state substrate concentration in the action of biosensor response at mixed enzyme kinetics. Sens. Actuators B Chem.
**2010**, 14, 290–297. [Google Scholar] [CrossRef] - Indira, K.; Rajendran, L. Analytical expression of the concentration of substrates and product in phenol-polypheneol oxidase system immobilized in laponite hydrogels Michaelis-Menten formalism in homogeneous medium. Electrochim. Acta
**2011**, 56, 6411–6419. [Google Scholar] [CrossRef] - Merchant, T.R. Cubic autocatalysis with Michaelis-Menten kinetics: Semi-analytical solutions for the reaction-diffusion cell. J. Chem. Eng. Sci.
**2004**, 59, 3433–3440. [Google Scholar] [CrossRef] - Devi, M.R.; Sevukaperumal, S.; Rajendran, L. Non-linear reaction diffusion equation with Michaelis-Menten kinetics and Adomian decomposition method. Appl. Math.
**2015**, 5, 21–32. [Google Scholar] - Mahalakshmi, M.; Hariharan, G.; Brindha, G.R. An efficient wavelet-based optimization algorithm for the solutions of reaction-diffusion equations in biomedicine. Comput. Methods Progams Biomed.
**2021**, 186, 105218. [Google Scholar] [CrossRef] - Tosaka, N.; Miyale, S. Analysis of a nonlinear diffusion problem with Michaelis Menten kinetics by an integral equation method. Bull. Math. Biol.
**1982**, 44, 841–849. [Google Scholar] [CrossRef] - Simpson, M.J.; Ellery, A.J. An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell. Appl. Math. Model.
**2012**, 36, 3329–3334. [Google Scholar] [CrossRef] - Selvi, M.S.M.; Seethalakshmi, R.; Rajendran, L. An analytical solution for diffusion and nonlinear uptake of oxygen in a planar, cylindrical and spherical cell using wavelet method. J. Crit. Rev.
**2020**, 7, 9729–9744. [Google Scholar] - Singh, R.; Wazwaz, A.M. Optimal homotopy analysis method for oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. MATCH Commun. Math. Comput. Chem.
**2018**, 80, 369–382. [Google Scholar] - Hadhoud, A.R.; Ali, K.K.; Shaalan, M.A. A septic B-spline collocation method for solving nonlinear singular boundary value problems arising in physiological models. Sci. Iran.
**2020**, 27, 1674–1874. [Google Scholar] - Roul, P. A new mixed MADM-collocation approach for solving a class of Lane–Emden singular boundary value problems. J. Math. Chem.
**2019**, 57, 945–969. [Google Scholar] [CrossRef] - Tripathi, V.M.; Srivastava, H.M.; Singh, H.; Swarup, C.; Aggarwal, S. Mathematical analysis of non-isothermal reaction-diffusion models arising in spherical catalyst and spherical biocatalyst. Appl. Sci.
**2021**, 11, 10423. [Google Scholar] [CrossRef] - Jamal, B.; Khuri, S.A. Non-isothermal reaction-diffusion model equations in a spherical biocatalyst: Green’s function and fixed point iteration approach. Int. J. Appl. Comput. Math.
**2019**, 5, 120. [Google Scholar] [CrossRef] - Abuasbeh, K.; Qureshi, S.; Soomro, A.; Awadalla, M. An optimal family of block techniques to solve models of infectious diseases: Fixed and adaptive stepsize strategies. Mathematics
**2023**, 11, 1135. [Google Scholar] [CrossRef] - Qureshi, S.; Ramos, H. L-stable explicit nonlinear method with constant and variable step-size formulation for solving initial value problems. Int. J. Nonlinear Sci. Numer. Simul.
**2018**, 19, 741–751. [Google Scholar] [CrossRef] - Aydinlik, S. An efficient method for oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. Int. J. Biomath.
**2022**, 15, 2250019. [Google Scholar] [CrossRef] - Izadi, M.; Yüzbası, S.; Cattani, C. Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains. Ricerche Mat.
**2023**, 72, 235–261. [Google Scholar] [CrossRef] - Yüzbası, S.; Yildirim, G. A Laguerre approach for solving of the systems of linear differential equations and residual improvement. Comput. Methods Differ. Equ.
**2021**, 9, 553–576. [Google Scholar] - Abd-Elkawy, M.A.; Alqahtani, R.T. Shifted Jacobi spectral collocation method for solving two-sided fractional water wave models. Europ. Phys J. Plus
**2017**, 132, 50. [Google Scholar] [CrossRef] - Youssri, Y.H.; Hafez, R.M. Exponential Jacobi spectral method for hyperbolic partial differential equations. Math. Sci.
**2019**, 13, 347–354. [Google Scholar] [CrossRef] - Abd-Elhameed, W.M.; Al-Harbi, M.S.; Amin, A.K.; Ahmed, H.M. Spectral treatment of high-order Emden-Fowler equations based on modified Chebyshev polynomials. Axioms
**2023**, 12, 99. [Google Scholar] [CrossRef] - Srivastava, H.M.; Izadi, M. The Rothe-Newton approach to simulate the variable coefficient convection-diffusion equations. J. Mahani Math. Res.
**2022**, 11, 141–157. [Google Scholar] - Yadav, P.; Jahan, S.; Nisar, K.S. Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics. Ain Shams Eng. J.
**2023**, 14, 102299. [Google Scholar] [CrossRef] - Izadi, M.; Zeidan, D. A convergent hybrid numerical scheme for a class of nonlinear diffusion equations. Comp. Appl. Math.
**2022**, 41, 318. [Google Scholar] [CrossRef] - Srivastava, H.M.; Adel, W.; Izadi, M.; El-Sayed, A.A. Solving some physics problems involving fractional-order differential equations with the Morgan-Voyce polynomials. Fractal Fract.
**2023**, 7, 331. [Google Scholar] [CrossRef] - Krall, H.L.; Frink, O. A new class of orthogonal polynomials: The Bessel polynomials. Trans. Amer. Math. Soc.
**1949**, 65, 100–115. [Google Scholar] [CrossRef] - Chatterjea, S.K. New class of polynomials. Ann. Mat. Pura Appl.
**1964**, 65, 35–48. [Google Scholar] [CrossRef] - Izadi, M.; Srivastava, H.M.; Adel, W. An effective approximation algorithm for second-order singular functional differential equations. Axioms
**2022**, 11, 133. [Google Scholar] [CrossRef] - Izadi, M.; Roul, P. Spectral semi-discretization algorithm for a class of nonlinear parabolic PDEs with applications. Appl. Math. Comput.
**2022**, 429, 127226. [Google Scholar] [CrossRef] - Izadi, M.; Srivastava, H.M. Applications of modified Bessel polynomials to solve a nonlinear chaotic fractional-order system in the financial market: Domain-splitting collocation techniques. Computation
**2023**, 11, 130. [Google Scholar] [CrossRef] - Srivastava, H.M. An introductory overview of Bessel polynomials, the generalized Bessel polynomials and the q-Bessel polynomials. Symmetry
**2023**, 15, 822. [Google Scholar] [CrossRef] - Izadi, M. A combined approximation method for nonlinear foam drainage equation. Sci. Iran.
**2022**, 29, 70–78. [Google Scholar] - Aznam, S.M.; Ghani, N.A.; Chowdhury, M.S. A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method. Results Phys.
**2019**, 14, 102393. [Google Scholar] [CrossRef] - Izadi, M.; Srivastava, H.M. Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity. Discrete Contin. Dyn. Syst. Ser. S
**2023**, 2023, 1–23. [Google Scholar] [CrossRef]

**Figure 1.**Approximate solutions (

**left**) and the related absolute values of REFs (

**right**) via the QLM–ChPs approach with $H=2,4,8,16$, and $\alpha =1,h=0.1,{s}_{h}=1$.

**Figure 2.**Approximate solutions (

**left**) and the related absolute values of REFs (

**right**) via the QLM–ChPs approach with $H=2$, $\alpha =1,{s}_{h}=1,$ and various $h=0.1,0.2,0.3,0.4,0.5$.

**Figure 3.**Approximate solutions using various $\alpha =0.1,\dots ,0.5$, $h=0.5$, ${s}_{h}=1$ (

**left**) and different ${s}_{h}=5,10,15,30,50$, $h=0.6$, $\alpha =0.5$ (

**right**) via the QLM–ChPs approach with $H=2$.

**Figure 4.**The error solution (

**left**) and the related absolute values of REFs (

**right**) via the RC technique with $(H,{H}^{\prime})=(4,6)$ and $\alpha ,h,{s}_{h}=1$.

**Figure 5.**A comparison of three-term approximate solutions using the ADM/LWM/QLM–ChPs approaches with $h=0.1$ (

**left**) and $h=0.5$ (

**right**). In the QLM–ChPs approach, we used $H=2$ and $\alpha ,{s}_{h}=1$ in the spherical case $n=2$.

**Figure 6.**The absolute values of REFs via the QLM–ChPs approach with $H=2,4,8,16$, and $\alpha =1,h=0.5,{s}_{h}=1$ in the spherical case $n=2$.

**Figure 7.**Approximate solutions using various $\alpha =0.1,\dots ,0.5$, $h=0.5$, ${s}_{h}=2$, (

**left**) and different ${s}_{h}=3,4,6,8,15$, $h=1$, $\alpha =1$ (

**right**) via the QLM–ChPs approach with $H=10$ in the spherical case $n=2$.

Parameter | Description | Unit |
---|---|---|

$\mathbb{S}:$ | Substrate concentration | mol/cm^{3} |

$M:$ | Dimensionless substrate concentration ($=\mathbb{S}/{\mathbb{S}}_{0}$) | - |

${\mathbb{S}}_{0}:$ | Bulk-substrate concentration | mol/cm^{3} |

$D:$ | Effective diffusivity inside the particle | cm^{3}/s |

${\mathbb{V}}_{M}:$ | Maximum reaction rate | mol/s cm^{3} |

${\mathbb{K}}_{M}:$ | Michaelis constant | mol/cm^{3} |

${k}_{m}:$ | External mass-transfer coefficient | mol/cm^{3} |

$x:$ | Spatial variable | cm |

$h:$ | Thiele modulus | - |

$\alpha :$ | Dimensionless Michaelis constant | - |

$L:$ | Half length of the particle | cm |

${s}_{h}:$ | Modified Sherwood number | - |

**Table 2.**Numerical evaluation of approximate solutions and REFs via the RC technique with $(H,{H}^{\prime})=(4,6)$ and $\alpha ,h,{s}_{h}=1$ at various $x\in [0,1]$.

x | ${\mathit{M}}_{4}^{\left(5\right)}\left(\mathit{x}\right)$ | $\left|{\mathfrak{R}}_{4}^{\left(5\right)}\left(\mathit{x}\right)\right|$ | ${\mathit{M}}_{4,6}^{\left(5\right)}\left(\mathit{x}\right)$ | $\left|{\mathfrak{R}}_{4,6}^{\left(5\right)}\left(\mathit{x}\right)\right|$ |
---|---|---|---|---|

$0.0$ | $0.26557303$ | $3.1936\times {10}^{-3}$ | $0.26550702$ | $4.7010\times {10}^{-4}$ |

$0.1$ | $0.26766356$ | $1.4356\times {10}^{-3}$ | $0.26760887$ | $2.7671\times {10}^{-4}$ |

$0.2$ | $0.27396962$ | $3.2785\times {10}^{-4}$ | $0.27393950$ | $6.9216\times {10}^{-5}$ |

$0.3$ | $0.28457805$ | $1.9296\times {10}^{-4}$ | $0.28457571$ | $5.3198\times {10}^{-5}$ |

$0.4$ | $0.29962257$ | $2.4287\times {10}^{-4}$ | $0.29964645$ | $5.1880\times {10}^{-5}$ |

$0.5$ | $0.31928378$ | $1.4663\times {10}^{-24}$ | $0.31933216$ | $5.4292\times {10}^{-5}$ |

$0.6$ | $0.34378916$ | $2.8270\times {10}^{-4}$ | $0.34386303$ | $1.9064\times {10}^{-4}$ |

$0.7$ | $0.37341307$ | $2.6150\times {10}^{-4}$ | $0.37351652$ | $2.2601\times {10}^{-4}$ |

$0.8$ | $0.40847670$ | $5.1746\times {10}^{-4}$ | $0.40861377$ | $3.1905\times {10}^{-5}$ |

$0.9$ | $0.44934817$ | $2.6407\times {10}^{-3}$ | $0.44951528$ | $8.4719\times {10}^{-4}$ |

$1.0$ | $0.49644248$ | $6.8532\times {10}^{-3}$ | $0.49661555$ | $2.5745\times {10}^{-3}$ |

**Table 3.**The numerical solutions via the QLM–ChPs approach with $\alpha ,{s}_{h}=1$ and various $h=0.1,0.2,\dots ,1$ in the spherical case $n=2$.

x | $\mathit{h}=0.1$ | $\mathit{h}=0.2$ | $\mathit{h}=0.3$ | $\mathit{h}=0.4$ | $\mathit{h}=0.5$ | $\mathit{h}=1$ |
---|---|---|---|---|---|---|

$0.0$ | $0.9950104174$ | $0.9801667075$ | $0.9558440569$ | $0.9226671500$ | $0.8815061913$ | $0.6008875096$ |

$0.1$ | $0.9950270425$ | $0.9802327071$ | $0.9559906738$ | $0.9229231020$ | $0.8818966446$ | $0.6021391527$ |

$0.2$ | $0.9950769180$ | $0.9804307140$ | $0.9564305662$ | $0.9236910908$ | $0.8830683354$ | $0.6058999347$ |

$0.3$ | $0.9951600457$ | $0.9807607524$ | $0.9571638580$ | $0.9249715150$ | $0.8850222552$ | $0.6121873814$ |

$0.4$ | $0.9952764279$ | $0.9812228627$ | $0.9581907562$ | $0.9267650382$ | $0.8877600545$ | $0.6210305927$ |

$0.5$ | $0.9954260681$ | $0.9818171015$ | $0.9595115499$ | $0.9290725885$ | $0.8912840398$ | $0.6324700779$ |

$0.6$ | $0.9956089709$ | $0.9825435413$ | $0.9611266108$ | $0.9318953568$ | $0.8955971683$ | $0.6465575239$ |

$0.7$ | $0.9958251419$ | $0.9834022709$ | $0.9630363924$ | $0.9352347957$ | $0.9007030429$ | $0.6633554950$ |

$0.8$ | $0.9960745874$ | $0.9843933951$ | $0.9652414301$ | $0.9390926176$ | $0.9066059048$ | $0.6829370650$ |

$0.9$ | $0.9963573150$ | $0.9855170346$ | $0.9677423404$ | $0.9434707921$ | $0.9133106255$ | $0.7053853799$ |

$1.0$ | $0.9966733333$ | $0.9867733263$ | $0.9705398207$ | $0.9483715442$ | $0.9208226968$ | $0.7307931523$ |

**Table 4.**The maximum absolute value of REFs and the associated ${\mathrm{Ord}}_{H}$ achieved via the QLM–ChPs procedure using $\alpha ,h,{s}_{h}=1$ and various H.

$\mathit{n}=0$ | $\mathit{n}=2$ | |||||||
---|---|---|---|---|---|---|---|---|

$\mathit{h}=\mathbf{0}.\mathbf{5}$ | $\mathit{h}=\mathbf{1}$ | $\mathit{h}=\mathbf{0}.\mathbf{5}$ | $\mathit{h}=\mathbf{1}$ | |||||

$\mathit{H}$ | ${\mathit{e}}_{\infty}$ | ${\mathbf{Ord}}_{\mathit{H}}$ | ${\mathit{e}}_{\infty}$ | ${\mathbf{Ord}}_{\mathit{H}}$ | ${\mathit{e}}_{\infty}$ | ${\mathbf{Ord}}_{\mathit{H}}$ | ${\mathit{e}}_{\infty}$ | ${\mathbf{Ord}}_{\mathit{H}}$ |

2 | $3.7498\times {10}^{-5}$ | − | $2.7776\times {10}^{-1}$ | − | $1.2500\times {10}^{-5}$ | − | $1.1905\times {10}^{-1}$ | − |

4 | $2.4301\times {10}^{-8}$ | $10.592$ | $6.8532\times {10}^{-3}$ | $5.3409$ | $2.2738\times {10}^{-9}$ | $12.425$ | $1.5816\times {10}^{-3}$ | $6.2339$ |

8 | $3.2518\times {10}^{-15}$ | $22.833$ | $6.0441\times {10}^{-6}$ | $10.147$ | $3.9753\times {10}^{-18}$ | $29.091$ | $2.7289\times {10}^{-7}$ | $12.501$ |

16 | $1.2902\times {10}^{-29}$ | $47.841$ | $1.1835\times {10}^{-12}$ | $22.284$ | $2.3086\times {10}^{-33}$ | $50.613$ | $3.1188\times {10}^{-15}$ | $26.383$ |

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**MDPI and ACS Style**

Izadi, M.; Srivastava, H.M.
The Reaction–Diffusion Models in Biomedicine: Highly Accurate Calculations via a Hybrid Matrix Collocation Algorithm. *Appl. Sci.* **2023**, *13*, 11672.
https://doi.org/10.3390/app132111672

**AMA Style**

Izadi M, Srivastava HM.
The Reaction–Diffusion Models in Biomedicine: Highly Accurate Calculations via a Hybrid Matrix Collocation Algorithm. *Applied Sciences*. 2023; 13(21):11672.
https://doi.org/10.3390/app132111672

**Chicago/Turabian Style**

Izadi, Mohammad, and Hari M. Srivastava.
2023. "The Reaction–Diffusion Models in Biomedicine: Highly Accurate Calculations via a Hybrid Matrix Collocation Algorithm" *Applied Sciences* 13, no. 21: 11672.
https://doi.org/10.3390/app132111672