Comparative study of a cubic autocatalytic reaction via different analysis methods

In this paper we discuss an approximate solutions of the space-time fractional cubic autocatalytic chemical system (STFCACS) equations. The main objective is to find and compare approximate solutions of these equations found using Optimal q-Homotopy Analysis Method (Oq-HAM), Homotopy Analysis Transform Method (HATM), Varitional Iteration Method (VIM) and Adomian Decomposition Method (ADM).


1.
Introduction. If two chemicals, which we label A and B, react through a mechanism known as cubic autocatalysis, we have the chemical reaction equation [28] A + 2B → 3B, rate k uv 2 . (1) Here k is the reaction rate constant and u and v are the concentrations of the two chemicals which are measured in moles. The chemical B is known as the auto catalyst, since it catalyses its own production. The greater the concentration of B, the faster it is produced by the reaction (1). If these two chemicals then react in a long thin tube, so that their concentrations only vary in the x-direction along the tube, the main physical processes that act, in the absence of any underlying fluid flow, are chemical reaction and one dimensional diffusion. The equation governing the chemical reaction (1) is the reaction-diffusion system Here t is time and D the constant diffusivity of the chemicals, with the diffusivity of both species assumed equal. The reaction-diffusion system (2)-(3) can be non-dimensionalised, so that D = 1 and k = 1. Furthermore, this system can be replaced by its equivalent space-time fractional system by replacing u t , v t by u α t , v α t and u xx , v xx by u 2β x , v 2β x , respectively, where 0 < α, β ≤ 1. We then obtain the space-time fractional derivative STFCACS We take the initial conditions a sin(0.5(nπ) cos (0.5(L − 2x)µ n ) , v(x, 0) = ∞ n=1 b sin(0.5(nπ) cos (0.5(L − 2x)µ n ) , where µ n = nπ L , 0 ≤ L ≤ L 0 , L 0 > 0 and the boundary conditions u(0, t) = u(L, t) = 1, v(0, t) = v(L, t) = 0.
The present paper is organized as follows: The second section is devoted to the basic idea of the fractional calculus. The third, fourth, fifth and sixth are devoted to applying the Oq-HAM, HATM, VIM and ADM on STFCACS respectively. The seventh section is devoted to the comparison analysis. In the last section, conclusion is presented.
, the set of all integrable functions, and α > 0, then the Rimann-Liouville fractional integral of order α, denoted by J α a+ is defined by Definition 2.2. For α > 0, the Caputo fractional derivative of order α, denoted by C D α a+ , is defined by where n is such that n − 1 < α < n and D = d dτ If α is an integer, then this derivative takes the ordinary derivative Finally the Caputo fractional derivative on the whole space R is defined by Definition 2.3. For α > 0 the Caputo fractional derivative of order α on the whole space, denoted by C D α a+ , is defined by 2.1. Basic idea of Oq-HAM. The principles of the HAM and its applicability for various kinds of differential equations are given in [4,19,33,34,37,60,64,66].
Also new results obtained into [2,25,47,48,49,50,53,54,55,59,62,63] using the homotopy analysis method. For convenience, we will present a review of the HAM [34]. To describe the basic idea of the standard Oq-HAM [15,51], we consider the nonlinear differential equation where N is nonlinear differential operator and u(x, t) is an unknown function. Liao [33] constructed the so-called zeroth-order deformation equation : where q ∈ [0, 1 n ] is an embedding parameter , h = 0 is an auxiliary parameter, H(x, t) = 0 is an auxiliary function, L ia an auxiliary linear operator, φ(x, t; q) is an unknown function , and u 0 (x, t) is an initial guess for u(x, t) which satisfies the initial conditions. It should be emphasized that one has great freedom in choosing the initial guess u 0 (x, t), L, h and H(x, t). Obviously, when q = 0 and q = 1 n , the following relations hold respectively Expanding φ(x, t; q) in Taylor series with respect to q, one has where We assume that the auxiliary parameter h, the auxiliary function H(x, t), the initial approximation u 0 (x, t) and the auxiliary linear operator L are selected such that the series (15) converges at q = 1 n , and one has We can deduced the governing equation from the zero order deformation equation by define the vector Differentiating (14) m-times with respect to q, then setting q = 0 and dividing them by m!, we have, using (16) , the so-called mth-order deformation equation where and 2.2. Basic idea of HATM. In this section, we introduce an approximate analytical method, namely the HATM, which is combination of the Laplace decomposition method (LDM) and the homotopy analysis method (HAM) [10,20,43,56,57]. This scheme is simple to apply to linear and nonlinear fractional differential equations and requires less computational effort compared with other exiting methods.
2.2.1. Laplace transform. Let f (t) be defined for 0 ≤ t < ∞. Then, when the improper integral exists, the Laplace transform F (s) of f (t), written symbolically as F (s) = L {f (t)}, is defined by where F (s) is the Laplace transform of f (t).
Apply the Laplace transform to the nonlinear differential operator N , we can obtain HATM solutions by the similar procedure with Oq-HAM but with n = 1.

Basic idea of VIM.
To illustrate the basic concept of the variational iteration method, we consider the following general nonlinear equation According to the variational iteration method [21,24,65], we can construct a correction functional in the form where u 0 (x, t) is an initial approximation with possible unknowns, λ(s) is a Lagrange multiplier which can be identified optimally via variational theory, the subscript m denotes the m-th approximation, andũ m is considered as a restricted variation [21,24], i.e. δũ m = 0 . It is shown that this method is very effective and easy for linear problems as its exact solution can be obtained by only one iteration because λ(s) can be exactly identified. To solve (23) by the VIM, we must first evaluate λ(s) that will be identified optimally via integration by parts. Then the successive approximation u m (x, t), m = 0, 1, · · · , of the solution u(x, t) will be readily obtained upon using λ(s) and u 0 (x, t). The zeroth approximation u 0 may be any function that satisfies at least the initial and boundary conditions with λ(s) determined.

Basic idea of ADM.
We present the basic idea of the ADM [26] in this section by considering the following nonlinear partial differential equation subject to the initial value where R is the remaining linear operator, which might include other fractional derivatives operator D ν (ν < α), N represent a nonlinear operator and g(x, t) is a given continuous function. Now, applying j α to both the sides of (25), we get We employ the Adomian decomposition method to solve equations (26)- (27). Let and where A m are Adomin polynomials which depend upon u. In view of Equations We set In order to determine the Adomian polynomials, we introduce a parameter λ and (29) becomes where In view of (34) and (35), we get Hence, (31)- (32) and (36) lead to the following recurrence relations
Substituting this value of the Lagrange multiplier into the solution (60)-(61), the variational iteration formula gives and Finally, the exact solution is obtained using and v(x, t) = lim n→∞ v n (x, t).

(75)
A m and B m are called Adomian polynomials. Furthermore, the components u m (x, t) and v m (x, t) of the solutions u(x, t) and v(x, t) are determined from the initial approximations and and the recurrence relations and Now if we take the initial values u 0 (x, 0) = u(x, 0) and v 0 (x, 0) = v(x, 0), we obtain the first iterates a sin(0.5(nπ))µ 2β n cos(0.5(nπ) − µ n x − βπ) b 2 sin(0.5(nπ)) sin(0.5(mπ)) cos(0.5(nπ) − µ n x) ab 2 sin(0.5(nπ)) sin(0.5(mπ)) sin(0.5(rπ)) 7. Comparison analysis. In this section we compare the solutions obtained above using the four methods for fractional differential equations with the numerical solutions of the fractional space-time STFCACS obtained using the command NDSolve of Mathematica 9. After substituting the initial values for u(x, 0) and v(x, 0) into the space-tiem fractional STFCACS (4)-(5), we obtain the first approximation of the VIM, which are the same as the first two terms of the Oq-HAM, HATM and ADM for (4)- (5). So the errors of each method are the same and we need more iterations to find differences between the methods. A comparison between numerical solutions and solutions obtained using the Oq-HAM, HATM, VIM and ADM methods are shown in Figures 1-4  We shall now compare the results using these approximate methods with numerical solutions as a function of x. Figures 5-6 show these comparisons. Due to the periodic initial conditions for our problem, the errors depend on x. In Figures  5-6(b), (c), (d) and (e), showing the ADM solution for 0 < t < 25, 0 < t < 70, 0 < t < 65 and 0 < t < 40 respectively. It can be seen that the ADM solution converges more rapidly than the Oq-HAM, HATM and VIM solutions. However, the errors displayed in 8. Conclusion. In this paper the Oq-HAM, HATM, VIM and ADM methods have been applied to efficiently obtain approximate solutions of the space-time fractional STFCACS. It was shonw that the Oq-HAM, HATM, VIM and ADM can be successfully applied to the STFCACS.
The main advantage of the four methods over mesh points methods is that they do not require discretization of the variables, i.e. time and space, and thus they are not affected by computation round off errors and one is not faced with the necessity of large computer memory and time. The four methods provide the solutions in terms of convergent series with easily computable components. The main disadvantage of these methods is they only give a good approximation of the true solution in a restricted region. The first two terms of the Oq-HAM, HATM and ADM solutions and the first approximation of the VIM are identical. Hence, we computed the first three terms of the Oq-HAM, HATM and ADM and the second approximation for the VIM. The efficiency and accuracy of these methods are clear from the comparisons with numerical solutions displayed in the figures. Comparisons of solutions obtained using the Oq-HAM, HATM, VIM and ADM methods with numerical results obtained using Mathematica show the efficiency of the methods. Finally, we

KHALED MOHAMMED SAAD AND EMAN HUSSAIN FAISSAL AL-SHARIF
found that the Oq-HAM has more rapid convergence than the HATM, VIM and ADM.