Approximate Solution of Non-Linear Reaction-Diffusion in A Thin Membrane: Taylor Series Method

the nonlinear reaction-diffusion cycle in the thin membrane that describes the chemical reactions involving three species is studied. The model consists of the system of on nonlinear reaction-diffusion equations. The closed type of analytical expression of concentrations for the enzyme was developed by solving equations using the Taylor series formula. This results in the mixed Dirichlet and Neumann boundary conditions. Taylor series method similar to exponential function method. This technique provides approximate and simple solutions that are quick, easy to compute, and efficiently correct. These estimated findings are compared to the nuxmerical results. There is a good agreement with the simulation results. Keywords—Mathematical modelling; Non-linear reaction-diffusion equation; Thin membrane; Taylor series method; Numerical Simulation.


I. INTRODUCTION
According to the reaction mechanism product, a diffusion-controlled chemical reaction between two species A and B is considered to be a product. The reaction path consists of a coupled pair of simple, irreversible and fast reaction mechanisms [4].
Ariel et al. [1] used the HPM to measure the steady laminar flow of the third-grade fluid through a circular tube. Seidman et al. [2,3] and Kalacheve et al. [ 4] presented a detailed singular perturbation analysis of the steady-state problem. The corresponding non-steady state system of this problem was perceived by Haario Seidman [5] to describe the reactions in the film for the gas/liquid interface under the complex boundary conditions. Recently, Butuzov et al. [6,7] have addressed many issues related to this problem using a variety of techniques. Rajendranet al. [8] and Ananthaswamy et al. [9] developed approximate analytical expressions for steady-state concentrations using the homotopy perturbation and homotopy analysis methods. The diffusion coefficients of three species are considered to have an equal diffusion coefficient which is equal to one. The non-linear diffusion reaction equation in the thin membrane is described by the following nondimensional format [4]:

) ( u
denote the concentrations of the chemical species A, B and C respectively. We assume that the specie A is supplied with a given fixed concentration The reaction rate q is given by [15,16] suggests Taylor's series method to solve the Lane-Emden equation. This method also extended to all non-linear differential equation in fractional calculus [17,18]. He recently proposed the exp-function method for solving the non-linear equations [18]. Visuvasam et al. [19] have derived the analytical expression for concentration profile and current using a hyperbolic function method and the Taylor series method. We can also obtain concentrations and  w using Taylor's series method (Appendix D) as follows: The analytical expression for reaction rate q becomes, A higher order in Taylors series provides a better approximation. Table-1

TSE
The analytical expression for reaction rate q becomes, IV. NUMERICAL SIMULATION The nonlinear differential equation is solved numerically to investigate the accuracy of this analytical method. The detailed Matlab program for numerical simulation is provided in Appendix B. Figure 1 compares our empirical findings with the simulation results. In Tables2 to 4, our results are also compared with previous analytical results obtained using HPM and HAM. There is no significant difference in error percentage between the numerical and our analytical methods when the parameter The error percentage between the numerical other previous analytical methods is 8%. Also, our method has the simplest form when compared to all other previous methods.
V. DISCUSSION Equations (9) and (11)   VI. CONCLUSION In that paper a simple and efficient approach is introduced to solve the system of nonlinear reactiondiffusion equation in a thin membrane. Compared to other approaches the solution process is very simple and straightforward. Also, it can be extended to other boundary value problems in a thin membrane without any difficulty. Analytical expressions of the concentrations of species are derived by using the Taylor series method. Analytical and numerical simulation results are compared. Concentration and reaction rate, when the parameter  is less than one, give satisfactory agreement with simulation results.

A. Appendix A: Approximate analytical solution of the equations (3-5) using Taylor series method
Approximate analytical solution of the equations (3-5) using Taylors series method Taylor series method is accessible to all students and engineers; it might be the simplest analytical method [20][21][22]. More recently He [15] solved the convection-diffusion equation for E reaction arising in RDE using Taylors series method. The system of steady-state non-linear differential equations (3) -(5) in ECE reactions can be written as follows: Approximate Solution Of Non-Linear Reaction-Diffusion In A Thin Membrane: Taylor Series Method The given boundary conditions for the above equations are The Taylor series solution of (A. 1