The Optimal q-Homotopy Analysis Method (Oq-HAM)

In this paper, an optimal q-homotopy analysis method (Oq-HAM) is proposed. We present some examples to show the reliability and efficiency of the method. It is compared with the one-step optimal homotopy analysis method. The results reveal that the Oq-HAM has more accuracy to determine the convergence-control parameter than the one-step optimal HAM.


INTRODUCTION
The search for a better and easy to use tool for the solution of nonlinear equations illuminating the nonlinear phenomena of our life keeps continuing. A variety of methods therefore were proposed to find approximate solutions. Liao [10][11][12][13][14] employed the basic ideas of the homotopy in topology to propose a general analytic method for linear and nonlinear problems, namely homotopy analysis method (HAM). In recent years, this method has been successfully applied to solve many types of nonlinear problems in science and engineering [6,9,17,20].The HAM contains a certain auxiliary parameter which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. Moreover, by means of the so-called -curve, it is easy to determine the valid regions of to gain a convergent series solution. The use of the convergence-control parameter is indeed a great progress in the frame of the HAM. It seems that more "artificial" degrees of freedom imply larger possibility to gain better approximations by means of the homotopy analysis method. How to find a proper convergence-control parameter so as to gain a convergent series solution? A straight-forward way to check the convergence of a homotopy-series solution is to substitute it into original governing equations and boundary/initial conditions and then to check the corresponding squared residual integrated in the whole region.

The
-curves cannot tell us the best convergence-control parameter , which corresponds to the fastest convergent series. In 2007, Yabushita et al. [19] applied the HAM to solve two coupled nonlinear ODEs. They suggested the so-called "optimization method" to find out the two optimal convergence-control parameters by means of the minimum of the squared residual error of governing equations. In 2008, Akyildiz and Vajravelu [2] gained optimal convergence-control parameter by the minimum of squared residual of governing equation, and found that the corresponding homotopy-series solution converges very quickly, Marinca et al. [16,17] introduced the so-called "homotopy asymptotic method" which is similar to the homotopy analysis method, Niu et.al. [18] proposed a method namely one-step optimal homotopy analysis method, Liao [15] developed in an optimal HAM with only three convergence-control parameters. El-Tawil and Huseen [3,4] proposed a method namely q-homotopy analysis method (q-HAM) which is more general method of homotopy analysis method (HAM), The q-HAM contains an auxiliary parameter as well as such that the case of (q-HAM ; ) the standard homotopy analysis method (HAM) can be reached.
In this paper, an optimal q-homotopy analysis method is proposed. This optimal method contains only one convergencecontrol parameter and is computationally rather efficient.

BASIC IDEA OF THE OPTIMAL q-HOMOTOPY ANALYSIS METHOD (Oq-HAM)
Consider the following differential equation , (1) where N is a nonlinear operator, is an unknown function.
Let us construct the so-called zero-order deformation equation: Where is a nonzero auxiliary function , denotes the so-called embedded parameter, is an auxiliary linear operator.Choosing the function depends on the given problem. It is obvious that when equation (2) becomes: respectively. Thus as increases from 0 to , the solution varies from the initial guess to the solution .
Having the freedom to choose we can assume that all of them can be properly chosen so that the solution of equation (2) where .
Assume that are so properly chosen such that the series (4) converges at and (6) Defining the vector . Differentiating Equation (2) times with respect to and then setting and finally dividing them by we have the so-called order deformation equation: where N o v 2 0 , 2 0 1 3 (8) and (9) It should be emphasized that for is governed by the linear equation (7) with linear boundary conditions that come from the original problem. Let (10) where denote the square residual error of the -order appro-ximation of the equation (1) integrated in the whole domain , In theory if the square residual error tends to zero, then is a series solution of the original equation (1). Besides, at the given order of approximation, the minimum of the squared residual error corresponds to the optimal approximation, hence the optimal value of the convergence-control parameter that corresponds to the minimum of .
In the one-step optimal HAM, Niu and Wang [18] construct the zeroth-order deformation equation , where is an auxiliary linear operator , denotes the so-called embedded parameter an initial approximation of and the series converges at .
The order deformation equation is: Where and At the 1st-order of approximation is only dependent upon , so, the optimal value of is obtain by solving the nonlinear algebric equation At the 2nd-order, since is known, the square residual error is only dependent upon , thus we can gain the optimal value of by solving the nonlinear algebraic equation and so on.

APPLICATIONS
Example 3.1: Consider the nonlinear integro-differential Equation [8] With the boundary condition This problem solved by one-step optimal homotopy analysis method [8], so we will solve it by Oq-HAM and compare the results. We choose auxiliary linear operator , with the property where is constant.
We define the nonlinear operators We choose the initial approximations According to the zeroth-order deformation equation (2) and the -order deformation equation (7) with The solution of the -order deformation equation (7) for becomes: N o v 2 0 , 2 0 1 3 , where the constant determined by the initial condition Let . We now successively obtain: can be calculated similarly.Then the series solution expression by Oq-HAM can be written in the form: (12) Equation (12) is a family of approximation solutions to the problem (11) in terms of the convergence-control parameter . It is found that can be calculated similarly. The residual errors of one-step optimal HAM and Oq-HAM are shown in Table 1. It is obvious that, in this example, the Oq-HAM has more accuracy than the one-step optimal HAM.

Table (1): Comparison between residuals of one-step optimal HAM and optimal q-homotopy analysis method(Oq-HAM) for problem (11
With the boundary conditions . In order to prevent suffering from the strongly nonlinear term , we can use Taylor series expansion of Then (13) become (14) We choose auxiliary linear operator with the property where are integral constants. We define a nonlinear operator as We choose the initial approximations . According to the zeroth-order deformation equation (2) and the order deformation equation (7) with: N o v 2 0 , 2 0 1 3 The solution of the -order deformation equation (7) for becomes with the boundary conditions Let , . We now successively obtain: can be calculated similarly.Then the series solution expression by Oq-HAM can be written in the form: (15) Equation (15) is a family of approximation solutions to the problem (13) in terms of the convergence-control parameter . It is found that can be calculated similarly.
The 3 th order one-step optimal HAM approximation solution is The residual errors of one-step optimal HAM and Oq-HAM are shown in Table 2. Table 3 shows the comparison between of one-step optimal HAM and of Oq-HAM, with the exact solution which indicates that the speed of convergence for Oq-HAM is faster than the one-step optimal HAM.
With exact solutions and .
This problem solved by one-step optimal homotopy analysis [8], so we will solve it by Oq-HAM and compare the results.
We choose auxiliary linear operator with the property where are constants. We define the nonlinear operators We choose the initial approximations and . According to the zeroth-order deformation equation (2) and the -order deformation equation (7) with The solution of the -order deformation equation (7) for becomes , where the constants determined by the initial conditions .

Let
. We now successively obtain: , , can be calculated similarly.Then the series solution expression by Oq-HAM can be written in the form (17) Equation (17) is a family of approximation solutions to the problem (16) in terms of the convergence parameters . It is found that , , can be calculated similarly.
At the 4th-order of approximation, in order to determine the optimal value of , each of the equations in (18) is solved separately. So, the obtained values and corresponding square residual errors are for the first equation and for the second one. So, the minimum of the and is correspond to the optimal value of . Thus, is chosen. This procedure leads to the best approximate solution of the system. The 4 th order one-step optimal HAM approximation solutions are .
The residual errors of one-step optimal HAM and Oq-HAM are shown in Table 4. The comparison of , given by one-step optimal HAM and Oq-HAM with the exact solutions and is shown in  (1) and (2) show that the series solutions obtained by Oq-HAM converge faster than one-step optimal HAM.