An effective homotopy analysis method to solve the cubic isothermal auto-catalytic chemical system

K. M. Saad1,2, O. S. Iyiola3 and P. Agarwal4,5,∗ 1 Department of Mathematics, Faculty of Arts and Sciences, Najran, Najran University, Saudi Arabia 2 Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen 3 Department of Mathematical Sciences, Minnesota State University Moorhead, MN USA 4 Department of Mathematical, Anand International College of Engineering, Jaipur-303012, India 5 Center for Basic and Applied Sciences, Jaipur-302029, India


Introduction
Recently, Merkin et al. in [7] considered the following reaction-diffusion traveling waves system in region I as follows: for quadratic autocatalytic reaction A + B → 2B(rate k 1 ab), (1.1) together with a linear decay step B → C(rate k 2 b), (1.2) for cubic autocatalytic reaction A + 2B → 3B(rate k 3 ab 2 ) (1. 3) together with a linear decay step B → C(rate k 4 b), (1.4) where a and b are concentrations of reactant A and auto-catalyst B, k i (i = 1, 4) are the rate constants and C is some inert product of reaction. On the region II we assume that only the (1.1) and (1.3) are taking place for quadratic autocatalytic reaction and cubic autocatalytic reaction respectively. Here, we consider the following system for the dimensionless concentrations (α 1 , β 1 ) and (α 2 , β 2 ) in region I and II of species A and B, respectively with x > 0 and t > 0: ∂β 2 ∂t = ∂ 2 β 2 ∂x 2 + α 2 β 2 2 + γ(β 1 − β 2 ), (1.8) with the boundary conditions where k and γ are the strength of the auto-catalyst decay and the coupling between the two regions respectively. The present paper is organized as follows: In section-2, we described the idea of the standard HAM. Section-3, is devoted to the application of HAM to CIACS and Section-4, devoted to the numerical results. In the last section, we summarized the result in the conclusion.

Basic idea of HAM
In recent years, many authors presented homotopy analysis method and its application for differential equations in many ways (see, for example, [6,13,14,26,27,28,29] and also see for recent results [2,12,20,21,22,23,24,25]). After motivation with above mentioned works here we consider the following nonlinear differential equation: where N is nonlinear differential operator and y(t) is an unknown function. Liao [14] constructed the so-called zeroth-order deformation equation : where in the following, q ∈ [0, 1], h 0, H(t) 0, L, φ(t; q) be the embedding parameter, auxiliary parameter, auxiliary function, auxiliary linear operator and, respectively, and y 0 (t) be an initial guess for y(t) which satisfies the initial conditions. Clearly, when q = 0 and q = 1, the following relations hold respectively φ(t; 0) = y 0 (t), φ(t; 1) = y(t).
Expanding φ(t; q) in Taylor series with respect to q, one has Let us assume that the h, H(t), y 0 (t) and L are selected such that the series (2.12) converges at q = 1, and one has We can deduce the governing equation from the zero order deformation equation by defining the vector − → y n = {y 0 (t), y 1 (t), y 2 (t), . . . , y n (t)}.
Differentiating (2.11), m-times with respect to q, then by choosing q = 0 and dividing by m!, we get the so-called mth-order deformation equation where and More detailed analysis of HAM and the modified version of it together with various applications could be found in [4,8,9,10,11,17,18,19].

HAM solution of CIACS
Here, we apply the HAM on CIACS. We take the initial conditions to satisfy the boundary conditions, namely where λ n = nπ L . The HAM is based on a kind of continuous mapping such that, as the embedding parameter q increases from 0 to 1, φ i (x, t; q), ψ i (x, t; q) and i = 1, 2 varies from the initial approximation to the exact solution.
We define the nonlinear operators Now, we construct a set of equations, using the embedding parameter q with the initial conditions Where h 0 and H(x, t) 0 are the auxiliary parameter and function, respectively. We expand φ i (x, t; q) and ψ i (x, t; q) in a Taylor series with respect to q, and get Now, we construct the mth-order deformation equation from (2.14)-(2.15) as follows: If we take L i = d dt , (i = 1, 2) then the right inverse of L i = d dt will be t 0 (.)dτ (3.21) Let the initial approximation For m = 1, we obtain the first approximation as following:

Numerical results
Here, we compute the average residual error and the residual error and investigate the intervals of convergence by the h-curves. Finally, we checked the accuracy of the HAM solutions by comparing with another numerical method. The first approximation of α i1 (x, t) and β i1 (x, t) are ×b ri cos(δ r ) sin(λ r L/2)ht, And so on, in the same manner the rest of approximations can be obtained using the Mathematica package.

Average residual errors
We notice that h-curve does not give the best value for the h. Therefore, we evaluate its optimal values by the min of the averaged residual errors [1,3,5,12,15,16,26].

Comparison analysis
Now, we compare 5-terms of HAM solutions obtained with a numerical method using the commands with Mathematica 9 for solving the system of partial differential equations numerically. We plot the 5-terms of HAM solutions in Figure 3. Figure 3 shows the comparison of HAM solutions HAM solutions with numerical method for k = 0.1, γ = 0.2, L = 100, x = 3, a n 1 = 0.001, a n 2 = 0.002, b n 1 = 0.001, b n 2 = 0.002. We noted from this figure that the HAM solutions have a good agreement with the numerical method. Figure 4 shows the 3-terms HAM solutions obtained.

Conclusion
In the present research, the HAM was employed to analytically compute approximate solutions of CIACS. By comparing these approximate solutions with numerical solutions and the averaged residual error were found. We show the convergence region by h-curves. The agreement with the numerical solutions are very good. The results show that HAM accurate for solving CIACS. By increasing the number of iterations one can reach any desired accuracy. In this paper, we used Mathematica 9 in all calculations.