scholarly journals Analysis of Mathematical Modelling on Potentiometric Biosensors

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
N. Mehala ◽  
L. Rajendran
Keyword(s):  
Homotopy Perturbation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Full Description ◽  
Response Curves ◽  
First Order ◽  
First Order Kinetics ◽  
Matlab Program ◽  
Flux Response ◽  
Analytical Expressions

A mathematical model of potentiometric enzyme electrodes for a nonsteady condition has been developed. The model is based on the system of two coupled nonlinear time-dependent reaction diffusion equations for Michaelis-Menten formalism that describes the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate and product and the corresponding flux response have been derived for all values of parameters using the new homotopy perturbation method. Furthermore, the complex inversion formula is employed in this work to solve the boundary value problem. The analytical solutions obtained allow a full description of the response curves for only two kinetic parameters (unsaturation/saturation parameter and reaction/diffusion parameter). Theoretical descriptions are given for the two limiting cases (zero and first order kinetics) and relatively simple approaches for general cases are presented. All the analytical results are compared with simulation results using Scilab/Matlab program. The numerical results agree with the appropriate theories.

scholarly journals Theoretical Analysis of an Amperometric Biosensor Based on Parallel Substrates Conversion

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
T. Praveen ◽  
L. Rajendran
Keyword(s):  
Homotopy Perturbation Method ◽  
Dimensionless Parameter ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Amperometric Biosensor ◽  
Steady State Condition ◽  
New Approach ◽  
Nonlinear Reaction ◽  
Analytical Result ◽  
Analytical Expressions

The behaviour of an amperometric biosensor based on parallel substrates conversion for steady-state condition has been discussed. This analysis contains a nonlinear term related to enzyme kinetics. Simple and closed form of analytical expressions of concentrations and of biosensor current is derived. This model was originally reported by Vytautas Aseris and his team (2012). Concentrations of substrate and product are expressed in terms of single dimensionless parameter. A new approach to Homotopy perturbation method (HPM) is employed to solve the system of nonlinear reaction diffusion equations. Furthermore, in this work, the numerical solution of the problem is also reported using Matlab program. The analytical results are compared with the numerical results. The analytical result provided is reliable and efficient to understand the behavior of the system.

Sparse Optimal Control of the Schlögl and FitzHugh–Nagumo Systems

2013 ◽  
Vol 13 (4) ◽  
pp. 415-442 ◽  
Author(s):  
Eduardo Casas ◽  
Christopher Ryll ◽  
Fredi Tröltzsch
Keyword(s):  
Optimal Control ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Optimal Controls ◽  
Wave Fronts ◽  
First Order ◽  
Schlögl Model ◽  
Sparse Optimal Control ◽  
Objective Functional

Abstract. We investigate the problem of sparse optimal controls for the so-called Schlögl model and the FitzHugh–Nagumo system. In these reaction–diffusion equations, traveling wave fronts occur that can be controlled in different ways. The L1-norm of the distributed control is included in the objective functional so that optimal controls exhibit effects of sparsity. We prove the differentiability of the control-to-state mapping for both dynamical systems, show the well-posedness of the optimal control problems and derive first-order necessary optimality conditions. Based on them, the sparsity of optimal controls is shown. The theory is illustrated by various numerical examples, where wave fronts or spiral waves are controlled in a desired way.

Kinetics of the Catalytic Combustion of Ethanol and Ethyl Acetate with Estimation of Activation Energy and Rate Constants: An Analytical Study

2021 ◽  
Vol 10 ◽  
Author(s):  
R. Joy Salomi ◽  
S. Vinolyn Sylvia ◽  
Marwan Abukhaled ◽  
L. Rajendran
Keyword(s):  
Activation Energy ◽  
Ethyl Acetate ◽  
Weight Ratio ◽  
Spherical Geometry ◽  
Reaction Diffusion ◽  
Kinetic Mechanism ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Reaction ◽  
Analytical Expressions

: A mathematical model for the combustion of ethanol and ethyl acetate mixtures using Mn9Cu1 (mixture of manganese and copper with a weight ratio of 9:1) catalyst is discussed. The model’s kinetic mechanism is expressed in terms of nonlinear reaction-diffusion equations with common initial and boundary conditions in a finite planar, cylindrical, and spherical geometry. A Taylor series approach is used to derive general approximate analytical expressions of ethanol, acetaldehyde, and ethyl acetate molar concentrations inside the particle and reactor phase for various values of rate constants, diffusion, and kinetic parameters. The effect of shape factor for the planar, cylindrical, and spherical geometry of dispersed particles was examined for the first time. Activation energy and rate constant at the reference temperature of ethanol, acetaldehyde, and ethyl acetate are also obtained from the rate equations. A direct comparison with numerical simulations confirms the accuracy of the derived analytical results. Background: A mathematical model for the combustion of ethanol and ethyl acetate mixtures using Mn9Cu1 (mixture of manganese and copper with a weight ratio of 9:1) catalyst is discussed. The model’s kinetic mechanism is expressed in terms of nonlinear reaction-diffusion equations with common initial and boundary conditions in a finite planar, cylindrical, and spherical geometry. Objective: Derive general approximate analytical expressions of ethanol, acetaldehyde, and ethyl acetate molar concentrations inside the particle and reactor phase for various parameter values. Method: We employ the simple and reliable Taylor series method. Results: semi-analytic expressions of the concentration and bulk concentration of ethanol, ethyl acetate, and acetaldehyde. Conclusion: Approximate analytical expressions of the concentrations of ethanol, acetaldehyde and ethyl acetate were derived for arbitrary catalyst particle (planar, cylindrical and spherical) by using a simple, reliable, and robust method. In addition, the concentration of the species in reactor phase was also reported. The effects of the kinetic parameters, which are influenced by adsorption equilibrium constant, effective diffusivity, activation energy, on concentration, were discussed.

scholarly journals Comments on “Homotopy Perturbation Method for Solving Reaction-Diffusion Equations”

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Afgan Aslanov
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Diffusion Type ◽  
Homotopy Perturbation ◽  
False Conclusion ◽  
Correct Form

The paper entitled“Homotopy perturbation method for solving reaction diffussion equation”contains some mistakes and misinterpretations along with a false conclusion. Applying the homotopy perturbation method (HPM) in an incorrect manner, the authors have drawn the false conclusion that this approach is efficient for reaction-diffusion type of equation. We show that HPM in the proposed form is not efficient in most cases, and hence, we will introduce the correct form of HPM.

scholarly journals Exponential Attractor for a First-Order Dissipative Lattice Dynamical System

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoming Fan
Keyword(s):  
Dynamical System ◽  
Fractal Dimension ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Exponential Attractor ◽  
Spatial Discretization ◽  
First Order ◽  
Lattice Dynamical System

We construct an exponential attractor for a first-order dissipative lattice dynamical system arising from spatial discretization of reaction-diffusion equations in . And we obtain fractal dimension of the exponential attractor.

scholarly journals Analytical Expressions Pertaining to the Concentration of Substrates and Product in Phenol-Polyphenol Oxidase System Immobilized in Laponite Hydrogels: A Reciprocal Competitive Inhibition Process

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
K. Indira ◽  
L. Rajendran
Keyword(s):  
Homotopy Perturbation Method ◽  
Competitive Inhibition ◽  
Coupled System ◽  
Current Response ◽  
Product Concentration ◽  
Matlab Program ◽  
Nonlinear Coupled System ◽  
Diffusion And Kinetics ◽  
Analytical Expressions ◽  
Kinetics Of

Theoretical analysis corresponding to the diffusion and kinetics of substrate and product in an amperometric biosensor is developed and reported in this paper. The nonlinear coupled system of diffusion equations was analytically solved by Homotopy perturbation method. Herein, we report the approximate analytical expressions pertaining to substrate concentration, product concentration, and current response for all possible values of diffusion and kinetic parameters. The numerical solution of this problem is also reported using Scilab/Matlab program. Also, we found excellent agreement between the analytical results and numerical results upon comparison.

scholarly journals Homotopy Perturbation Method for Solving Reaction-Diffusion Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Yu-Xi Wang ◽  
Hua-You Si ◽  
Lu-Feng Mo
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Reaction Diffusion ◽  
High Accuracy ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Unknown Parameters ◽  
Weighted Residuals ◽  
Homotopy Perturbation ◽  
Method Of Weighted Residuals

The homotopy perturbation method is applied to solve reaction-diffusion equations. In this method, the trial function (initial solution) is chosen with some unknown parameters, which are identified using the method of weighted residuals. Some examples are given. The obtained results are compared with the exact solutions, revealing that this method is very efficient and the obtained solutions are of high accuracy.

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