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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 136043, 7 pages
Approximate Solution of Nonlinear System of BVP Arising in Fluid Flow Problem
1Department of Mathematics, Faculty of Science, Hashemite University, Zarqa 13115, Jordan
2School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
3Department of Mathematics, Faculty of Science, Al-Balqa' Applied University, Salt 19117, Jordan
Received 15 December 2012; Accepted 7 April 2013
Academic Editor: Alexei Mailybaev
Copyright © 2013 A. K. Alomari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We extend for the first time the applicability of the Optimal Homotopy Asymptotic Method (OHAM) to find approximate solution of a system of two-point boundary-value problems (BVPs). The OHAM provides us with a very simple way to control and adjust the convergence of the series solution using the auxiliary constants which are optimally determined. Comparisons made show the effectiveness and reliability of the method.
Many real-world problems can be modelled by nonlinear differential equations. For example, fluid flow problems can give rise to boundary-value problems (BVPs) or systems of BVPs with conditions specified at two or more different points. Finding a reliable method for solving BVPs is of great interest. Noor and Mohyud-Din [1–3] presented approximate solutions of some classes of BVPs by using the variational iteration method (VIM), homotopy perturbation method (HPM), and variational iteration decomposition method (VIDM). Herisanu et al.  developed the so-called Optimal Homotopy Asymptotic Method (OHAM) for solving nonlinear problems. OHAM provides us with a very simple way to control and adjust the convergence of the series solution using the auxiliary constants which are optimally determined. Several promising applications of OHAM to problems in fluid dynamics have been presented [5–12]. Ali et al. [13, 14] solved several two-point and multipoint BVPs by OHAM. Very recently, Hashmi et al.  applied OHAM for finding the approximate solutions of a class of Volterra integral equations with weakly singular kernels.
The laminar fully developed combined free and forced magnetoconvection in a vertical channel with symmetric and asymmetric boundary heatings in the presence of viscous and Joulean dissipations was studied by Umavathi and Malashetty . The mathematical model describing the channel flow problem is governed by a system of nonlinear BVPs. Umavathi and Malashetty  employed the classical perturbation technique to solve the system of BVPs. The aim of the present work is thus to propose an accurate approach to the channel flow problem using an analytical technique, namely, OHAM. The efficiency of the procedure is based on the construction and determination of the auxiliary functions combined with a convenient way to optimally control the convergence of the solution.
2. The Model Equation
The system of BVPs modelling the channel flow problem as given in  is subject to where the parameter becomes one for asymmetric heating and zero for symmetric heating. The special case was solved exactly by Umavathi and Malashetty , and the exact solutions are Furthermore, when , solutions of (1)-(2) become where We remark that the general case of both and is very difficult to solve exactly. For this case, Umavathi and Malashetty  have given the standard perturbation solutions by assumingto be the small parameter in the expansion.
3. Basic Idea of OHAM
Consider the following differential equations: where is a linear operator, is a nonlinear operator, is an unknown function, denotes independent variable, is a known function, and is a boundary operator.
According to the basic idea of OHAM [4–6], we construct a homotopy which satisfies where and is an embedding parameter, is a nonzero auxiliary function for , and is an unknown function. Obviously, when and it holds that and , respectively. Thus, as varies from to , the solution approaches from to ,where is obtained from (7) for , and we have Next, we choose auxiliary function in the form where are constants to be determined, and can be expressed in many forms as reported in [4–7].
To get an approximate solution, we expand in Taylor’s series about in the following manner: Substituting (11) into (7) and equating the coefficient of the like powers of , we obtain the following linear equations. The zeroth-order problem is given by (9), and the first- and second-order problems are given as And the general governing equations for are given as where and is the coefficient of in the expansion of about the embedding parameter It has been observed that the convergence of the series (11) depends upon the auxiliary constants . If the series is convergent at , one has The results of the th-order approximations are Substituting (16) into (6) it results the following residual: If , then will be the exact solution. Generally this does not happen, especially in nonlinear problems. In order to find the optimal values of , , we first construct the functional and then minimizing it, we have where and are in the domain of the problem. With these constants known, the approximate solution (of order ) is well determined.
3.1. Application of OHAM
In this section, we apply OHAM for solving the nonlinear system of two-point BVP (1)-(2). By applying the proposed method, the zeroth-order deformation equation is subject to the boundary conditions Using the framework of OHAM the th-order where , where Now the zeroth-order problem is subject to the boundary conditions The solutions are Now the first-order problem is subject to the boundary conditions The second-order problem is subject to the boundary conditions The third-order problem is subject to the boundary conditions
Using the solution of (25)–(32) we obtain the following four-term approximate solutions for and by OHAM taking: The explicit expressions for the individual terms of the approximate solutions are not given here for brevity. Taking the residual errors the optimal values of ’s can be obtained. Table 1 shows some optimal values of for different values of and .
In Figure 1 we compare our approximate four-term solutions (34) against the exact solutions (3) for the special case and for several values of . The comparison of the special case is shown in Figure 2 for and several values of . It is observed that our four-term OHAM solutions agree very well with the exact solutions. The general case of both and admits no explicit analytical solution. So, in Figures 3 and 4 we plot the four-term approximate OHAM solutions for several values of and in the case for both the asymmetric and symmetric heating conditions, respectively. The residual errors corresponding to selected cases of the solutions depicted in Figures 1 and 2 are presented in Figures 5(a) and 5(b), respectively. Finally, the residual errors for a selected case of Figure 3 are shown in Figure 6. Clearly, all the residual error plots suggest that the OHAM approximate solutions are accurate enough.
In this paper we have extended the applicability of OHAM for the first time to solve a nonlinear system of two-point BVPs that arise in a fluid flow problem. OHAM is relatively simple to apply. It was shown that, with a few terms, the OHAM is capable of giving sufficient accuracy. OHAM can be a promising tool for solving strongly nonlinear systems of equations.
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