A New Solution for The Enzymatic Glucose Fuel Cell Model with Morrison Equation via Haar Wavelet Collocation Method

Integral equations are essential tools in various areas of applied mathematics. A computational approach to an integral important in scientiﬁc The an the with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numer i cal solutions of glucose and hydrogen ion.


Introduction
Nonlinear differential equations (NDEs) play significant roles in various areas of applied sciences, for example, mathematical biology, fluid mechanics, chemical kinetics, temperature distribution, reaction-diffusion system, industrial process, and plasma physics.
In 1997, Chen and Hsiao [3] first developed the Haar wavelet method via an operational matrix of integration related to the Haar function vector. In 2005, Lepik [24] solved the higher order with nonlinear ordinary differential and evolution equations using the Haar wavelet method. Recently, the Haar wavelet method is used as a mathematical tool for reaction-diffusion partial differential equations in many applications [8] such as chemical reactions, biological chemistry and fluid mechanics.
In biology and chemistry systems, PDEs play a role in various processes in the structure of human beings, such as the carbon dioxide (CO 2 ) and oxygen (O 2 ) diffusions in the bloodstream [4], oxygen diffusion in absorbing tissue [13], enzyme kinetics reaction mechanisms [1], enzymatic glucose fuel cell [23], etc. Various types of enzymes play important roles in cells to help speed up numerous vital reactions. Enzymes are specific biocatalysts, meaning that they are specific to one type of reaction and a small group of reactants called substrates. Kinetics is the study of reaction rates, where the rate of the reaction means the rate of product formation. Enzyme kinetics has a long history, as it is one of the earliest areas where mathematics was utilized to understand biological phenomena. In addition, various structures for the pH, the concentrations of reaction, the temperature values of reaction media and mass transport effects are applied in enzyme kinetics.
The chemical reaction of enzymatic glucose fuel cell between enzyme and glucose is shown in Figure 1. Here, E is the enzyme, C G is the glucose (substrate), EC G is the enzyme-glucose complex and P is the product. In Figure  1, the Michaelis Menten equation [16] is one of the best-known reaction rates when On the other hand, when K M < [E T ], the appropriated reaction rate can be provided by the Morrison equation (Quadratic Velocity equation) [11] with the reaction rate f ([C G ]) as . Many researchers are interested in the performance of enzymatic glucose fuel cells. Scott Barton [21] studied enzymatic glucose fuel cells in various types of models. Rajendran [15] solved the approximate analytical solution for a mono enzymatic biosensor with the Michaelis Menten equation via the Homotopy perturbation method. Malinidevi [17] studied the reaction and diffusion of enzymes immobilized in an artificial membrane with a mathematical model. The enzymatic glucose fuel cell uses glucose as a fuel to generate electrical energy and enzymes as a biocatalyst to change chemical energy into electrical energy [14]. The anodic reaction is given by and the cathodic reaction is given by The diagram in Figure 2 assumes one-dimensional diffusion transport equations of glucose (C 6 H 12 O 6 ) and hydrogen ion (H + ) across a fuel cell. The glucose moves in from the anode, which has oxidation, to the cathode which has reduction. In the enzyme layer, a reaction occurs in which the glucose and enzyme become gluconic acid (C 6 H 12 O 7 ), hydrogen ion and The glucose concentration ([C G ]) and the hydrogen ion concentration ([C H + ]) can be determined by the model of the transport equation with the reaction rates of the Morrison equation as where 0 < x < L, t > 0. The initial conditions are and the boundary conditions are [C H + ](0, t) = 0, In this paper, we introduce the Haar wavelet collocation method (HWCM) and apply this method to find an approximate analytical solution of the enzymatic glucose fuel cell with the Morrison equation model. Coding of HWCM in the Maple program is developed to show some numerical results of the model, which can describe levels of glucose (C G ) and hydrogen ion concentration (C H + ) Eq.(5).

The Haar Wavelet Collocation Method
The Haar wavelet is known as the simplest example of an orthogonal wavelet, which is defined by a square wave function and the scaling functions, for i = 2, 3, . . ., where α 1 = k m , α 2 = k+0.5 m , α 3 = k+1 m and the dilations integer m = 2 j , j = 0, 1, . . . , J indicates the level of the wavelet with the maximal level of resolution J and k = 0, 1, . . . , m − 1 is the translation parameter as shown in the following Figure 3.
The index i in Eq. (9) is expressed by the integers m and k as i = m + k + 1. It is obvious that the minimal value of i is i = 2 when m = 1, k = 0 and the maximal value of i is The Haar wavelet functions satisfy the following properties: and the orthogonal property for i, l = 1, 2, . . .
The Haar wavelet function can construct a very good transform basis which is represented by any square integrable func- any function u(x) can be expressed in terms of an infinite sum of the Haar wavelets as follows: when i = 2 j + k, j ≥ 0, 0 ≤ k ≤ 2 j and the Haar coefficients In general, the series expansion of u(x) involves infinite terms. Practically, the continuous function u(x) can be approximated by the finite sum of the Haar wavelet, that is where M = 2 J and the integral square error is defined as The approximation u 2M (x) can be written in the matrix form T is the Haar function vector and A T = a 1 , a 2 , . . . , a 2M is called the coefficient vector. Defining the wavelet collocation points x l as We construct the Haar wavelet matrix H 2M of order 2M in which its columns are the Haar function vectors evaluated at For example, if J = 1 ⇒ 2M = 2 J+1 = 4, then the Haar wavelet matrix of order 4 is and if J = 2 ⇒ 2M = 2 J+1 = 8, then the Haar wavelet matrix of order 8 is In consequence, we have We also concisely provide the basic idea of the integrals of the Haar functions h i (x) of order n denoted by p i,n (x) which can be calculated analytically as follows: For i = 1: the integral of the Haar wavelet, h 1 (x) of order n is x n n! , where n = 1, 2, . . .
and the integrals of the Haar wavelet, h i (x), order (n) as [20] p i,n (x) Then, we define the 2M operational matric of integrations P n and its element order n, index i is computed using the relation P n (i, l) = p i,n (x l ), where x l is defined in (17) For example, if J = 1 ⇒ 2M = 4, and n = 1, then from (22) the Haar integral matrix of first order and when n = 2, the Haar wavelet integral matrix of second order is computed by using (23) as For example, if J = 2 ⇒ 2M = 8, and n = 1, then from (22) we have the Haar integral matrix of first order as with the graphs as Figure 4. Considering at n = 2, the Haar wavelet integral matrix of second order is computed by (23) as with the graphs as shown in Figure 5.

Kronecker product
Let a matrix A = (a ij ) be dimension m × n and let a marix B = (b ij ) be dimension p × q (not necessarily square) as where A, B and C are any matrices, 0 is a zero matrix and k is a scalar. The Kronecker product of matrics A, B, C and D can write the matrix products AC and BD as which is called the mixed-product property. The accuracy and efficiency of this method had been studied by comparing some HWCM solutions with exact solutions of second order differential equations with several types of boundary conditions [20].

HWCM for a reaction-diffusion equation
The diffusion equation is a special partial differential equation [2] occur in numerous engineering problems. The simple 193 problem requires one initial and two boundary conditions as [10] where k is constant. The initial condition is u(x, 0) = f (x), and boundary conditions are u(0, t) = g 0 (t), ∂u ∂x (1, t) = g 1 (t). Applying HWCM, assuming u (2,1) (x, t) = ∂ 3 u ∂x 2 ∂t that expands in terms of two-variable truncated Haar wavelet series can be expressed in the matrix form as where
Putting x = 1 into (30), it provides Substituting (32) into Eqs. (30) and (31), we get Next, we integrate (29) twice from 0 to x which obtains Putting x = 1 into (35), then we have Substituting (37) into Eqs. (35), and (36), we obtain So the HWCM solution is Substituting Eqs. (29) and (34) into (26), the system becomes Considering the Haar wavelet collocation points of space and time are defined by We can generate a system of algebraic equations as We also rewrite the (42) into the matrix from as where the matrices are given by We solve (43) with the developing coding HWCM via the Maple program to find the coefficient matrix of the Haar wavelet A. After that, we substitute A into (39), and obtain the approximate analytical solution of the diffusion equation (26) with the initial and boundary conditions.

HWCM solutions of the enzymatic glucose fuel cell model
Consider (5), we have the transport equations for glucose concentration ([C G ] = u) and hydrogen ion concentration From Eq. (44), we get In this example, the collocation points of space (x i ) and time (t j ) when M = 2 are defined by Assuming can be expressed by the Haar wavelet (M = 2) as the matrix form where A, B are coefficient matrices, N = 2M , that is Considering the collocation points (48), we apply (29) with initial condition (6), we get a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 From (34) and the boundary conditions (7), obtains a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 From (38) and the boundary conditions (7), obtains and and the HWCM solution of the hydrogen ion concentration is given  Table 1. Figure 6 shows that the glucose concentration decreases along with a distance (x) and increasing time (t). Initially, glucose enters the cell with full constant concentration, 1 mol · m −3 at t = 0. After a while, the glucose concentration decreases exponentially. The hydrogen ion concentration increases across the cell. At the beginning of time (t) hydrogen ion concentration rapidly increases.

Conclusions
This paper proposes the enzymatic glucose fuel cell model by applying enzyme kinetics, the Morrison equation. The approximate analytical solutions for glucose and hydrogen ion concentrations in enzymatic glucose fuel cell can be solved by the Haar wavelet collocation method (HWCM). It is obvious that glucose concentration decreases across the cell. Moreover, glucose concentration rapidly decreases when diffusion coefficients of glucose decrease, but the hydrogen ion concentration increases when the diffusion coefficients of hydrogen ion decrease.  KMUTNB-64-NEW-14, and partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thai-land.