A RELIABLE TAYLOR SERIES SOLUTION TO THE NONLINEAR REACTION-DIFFUSION MODEL REPRESENTING THE STEADY-STATE BEHAVIOUR OF A CATIONIC GLUCOSE-SENSITIVE MEMBRANE

The nonlinear reaction-diffusion model, which represents the steady-state behaviour of a cationic glucosesensitive membrane with consideration of oxygen limitation and swelling-dependent diffusivities of involved species inside the membrane, is discussed. Analytical expressions of substrate concentration of oxygen, glucose, and gluconic acid in planar coordinates at steady-state conditions are derived for all kinetic parameters, and hence the effect of various factors on the responsiveness of the membrane is analysed. Efficient approaches based on the hyperbolic function and Taylor’s series methods are used to derive the approximate analytical solutions of the nonlinear boundary value problem. A numerical simulation was generated by highly accurate and widely used computer generated routines. 8355 SOLUTIONS OF THE SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION The derived analytical expressions are shown to be in strong agreements with the numerical results established in the literature. It is concluded that each method is a powerful tool for solving high-order boundary value problem in engineering and science.


INTRODUCTION
Insulin-dependent diabetes has been a leading health concern worldwide because of the serious complications that are associated with it, such as retinopathy, neuropathy, and vascular disease.
As the ultimate goal of having a self-regulated insulin delivery system requires substantial time and effort to be fulfilled, it has not been fully achieved. However, there have been some remarkable developed strategies to reproduce the usual pattern of insulin kinetics. Testing whether a good metabolic control can prevent the long-term complications of diabetes includes intensified conventional therapy with multiple daily injections and continuous subcutaneous insulin infusion with external or implanted pumps [1].
Glucose-sensitive membranes are made using immobilized glucose oxidase (GOD) in pH-sensitive polymers [2], where in the presence of glucose; they swell and become more permeable to insulin.
As it is unlikely to attain optimal design of Glucose-sensitive membranes without setting up proper mathematical models for which approximate analytical solutions are determined, there has been 8356 MARY, DEVI, MEENA, RAJENDRAN, ABUKHALED some remarkable advances in theoretical modeling. Abdekhodaie and Wu [7] presented a theory describing the steady state behavior of a cationic glucose-sensitive membrane while taking into account oxygen limitation and swelling-dependent diffusivities of species inside the membrane.
Albin et al. [5] developed a mathematical model describing the steady-state behavior of two types of glucose sensitive membranes that are both synthetic hydrogels containing immobilized glucose oxidase enzyme. Leypoldt et al. [17]developed a model of two-substrate enzyme electrode for glucose. Klumb et al. [18] proposed a theoretical model to evaluate possible designs for an insulin delivery system based upon a glucose sensitive hydrogel containing immobilized glucose oxidase and catalase. Other theoretical models discussions can be found in [19,20].
Analytical solutions for nonlinear boundary value problems are more desired than numerical solutions because they provide a more accurate sensitive analysis of kinetic parameters on the governing system and hence facilitate the development of optimized models. The nonlinear mathematical model discussed in this paper has been solved analytically using the homotopy analysis method, Genocchi Polynomials and Adomian decomposition method [21,22]. Other widely used methods that are prone to deliver accurate analytical results for solving this kind of system include Green's function iterative method [23,24], variational iteration method [25,26], and homotopy perturbation method [27][28][29].
In this communication, we present efficient and reliable approaches to analytically solve a system of nonlinear differential equation in the cationic glucose-sensitive membrane. The simplicity and efficiency of the proposed approaches stem from the fact that basic conceptual mathematics is being used. Therefore, these approaches are easily accessible to researchers for further investigation of the effect of kinetic parameters and possibly obtain an optimal glucose-sensitive membranes. The reliability of the proposed methods will be investigated by direct comparison with numerical simulations from the literature and software built-in functions.

BOUNDARY VALE PROBLEM TO GLUCOSE-SENSITIVE MEMBRANE
The chemical reaction scheme inside a glucose-sensitive membrane is described by [7] The incorporated catalase then implies the conversion ( When the excess of catalase is immobilized with glucose oxidase, the overall mechanism of the reaction is described by It is evident from the reaction, that in the presence of catalase, only one-half of an oxygen molecule is consumed per molecule of glucose [7]. For the completion and self-consistency of the research, the derivation of the governing nonlinear differential equations in planar coordinates inside the cationic glucose-sensitive membrane is given in Appendix D of the supplementary material.   The main objective now is to obtain an analytical expression for the concentration profile ( )  u of oxygen, which will immediately lead to analytical expressions for the concentration profiles

DERIVATION OF ANALYTICAL EXPRESSIONS OF CONCENTRATIONS
In this section, we introduce simple, efficient and reliable techniques to derive analytical expressions of concentration for oxygen, which will immediately lead to the determination of analytical expressions for glucose and gluconic acid.

A modified hyperbolic function method
Special functions, in general, have always been used as an effective tool to solve nonlinear differential systems. For example, J. He utilized the exponential function to solve a nonlinear wave equation [30]. Furthermore, a recent research article used the gamma function to derive a semianalytic solution to a small amplitude oscillator equation [31]. The modified hyperbolic function method, considered a special case of the exponential function method [30] , is reliable and highly accurate in obtaining semi-analytic solutions of nonlinear models [32,33].
where the unknown parameter satisfies the equation Numerical values of for various values of the fundamental parameters can be obtained easily by using any computer algebra software. In Table S

Taylor series method
Taylor series method (TSM) is one of the simplest and most effective methods to solve nonlinear equations. Moreover, TSM is accessible to the broader research community because it requires no robust mathematical analysis background. Although some obstacles might emerge when using TSM, like in the case of strong nonlinear differential equations, the way around these obstacles is usually easy such as using more derivatives, and getting higher degreed polynomials or using Padé approximant. In recent years, TSM has been intensively employed to solve nonlinear ordinary and fractional differential equations such as Lane-Emden, third-order boundary value problems, fractional Bratu-type equations, and nonlinear oscillator problems [34][35][36][37][38][39][40][41][42][43].
Using Taylor

Previous analytical expression of concentrations
Sevukaperumal et al. [21] employed the homotopy analysis method (HAM) to derive the following analytical expressions for the concentration of oxygen inside the cationic glucose-sensitive membrane ( ) = cosh( ) + cosh( ) + ℎ cosh( ) + / sinh( ), ] (17) in which and ℎ is the convergence control parameter. Analytical expressions for the concentration of glucose and gluconic acid are obtained by substituting Eq. (16) into Eqs. (6) and (7), respectively.

Determination of pH profile inside the membrane
The pH in gluconic acid is determined by the concentration of buffer ions and gluconic acid in the microsphere. Gluconic acid production with a concentration of a inside the membrane changes pH to pH 2 via [8] From Eq. (D.5), we have = * [buffer] and hence from Eq. (11) pH is determined by

Estimation of kinetics parameters
From Eq. (D2), the following is easily obtained where , g and ox are known while the remaining parameters are unknown. Using multiple recursion analysis, we can obtain the kinetic parameter max and the Michaelis-Menten and ox .

NUMERICAL SIMULATION
The numerical results are noted.

DISCUSSION
By employing a modified hyperbolic function method, we derived Eqs.

APPENDIX A. BASIC CONCEPT OF THE MODIFIED HYPERBOLIC FUNCTION METHOD
We beging by expressing a general second order differential equation k p in the following form [28]: ) ) 5 .

( . 7)
The unknown constant is easily computed from the boundary condition (1) = 1. For example, using Eq.

APPENDIX-D. MATHEMATICAL FORMULATION OF THE PROBLEM.
The one-dimension steady-state nonlinear reaction-diffusion equations in cationic glucosesensitive membrane have been analyzed by Abdekhodaie and Wu [9]. For the self-consistency, the steady-state nonlinear equations for the concentrations of glucose, oxygen and gluconic acid are given below.
The terms a' ' and ox' ' , g ' ' provided to indicate glucose, oxygen and gluconic acid respectively, the stoichiometric coefficients, i v , are -1, -1/2, 1 in turn for a and ox g, i = , i C is the concentration, i D the corresponding diffusion coefficient in the membrane, x is the spatial coordinate and R is the overall reaction rate of the form where max v is the maximum reaction rate, Eq. (D1) reduced to dimensionless forms for various concentrations as follows: where u, v and w represent the dimensionless concentration of oxygen, glucose and gluconic acid. Eq. Eq. Eq. Eq. Eq.