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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 239454, 12 pages
A New Application of the Reproducing Kernel Hilbert Space Method to Solve MHD Jeffery-Hamel Flows Problem in Nonparallel Walls
1Department of Mathematics, Science Faculty, Fırat University, 23119 Elazığ, Turkey
2Department of Mathematics, Education Faculty, Dicle University, 21280 Diyarbakır, Turkey
3Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
4Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Malaysia
Received 9 February 2013; Accepted 28 February 2013
Academic Editor: Hassan Eltayeb
Copyright © 2013 Mustafa Inc 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.
The present paper emphasizes Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is 2. A new method called the reproducing kernel Hilbert space method (RKHSM) is briefly introduced. The validity of the reproducing kernel method is set by comparing our results with HAM, DTM, and HPM and numerical results for different values of , , and Re. The results show up that the proposed reproducing kernel method can achieve good results in predicting the solutions of such problems. Comparison between obtained results showed that RKHSM is more acceptable and accurate than other methods. This method is very useful and applicable for solving nonlinear problems.
1.1. Problem Formulation
Consider a system of cylindrical polar coordinates , where the steady two-dimensional flow of an incompressible conducting viscous fluid from a source or sink at channel walls lies in planes and intersects in -axis. It is assumed that there are no changes with respect to , that the motion is purely in radial direction and merely depends on and , and that there is no magnetic field along -axis. Then the governing equations are given as . where is the electromagnetic induction, is the conductivity of the fluid, is the velocity along radial direction, is the fluid pressure, is the coefficient of kinematic viscosity, and is the fluid density. From (1) using dimensionless parameters where is the semiangle between the two inclined walls as shown in Figure 1. Substituting (5) into (2) and (3) and eliminating , we obtain an ordinary differential equation for the normalized function profile : with boundary conditions
The Reynolds number is
The Hartmann number is
Internal flow between two plates is one of the most applicable cases in mechanics, civil and environmental engineering. In simple cases, the one-dimensional flow through tube and parallel plates, which is known as Couette-Poisseuille flow, has exact solution, but in general, like most of fluid mechanics equations, a set of nonlinear equations must be solved which make some problems for analytical solution. Many authors have shown interest in studying two-dimensional incompressible flow between two inclined plates. Jeffery  and Hamel et al.  were the first persons who discussed this problem, and so, it is known as Jeffery-Hamel problem. The incompressible viscous fluid flow through convergent and divergent channels is one of the most applicable cases in fluid mechanics, electrical, and bio- mechanical engineering. The MHD Jeffery-Hamel flows in nonparallel walls were investigated analytically for strongly nonlinear ordinary differential equations using homotopy analysis method (HAM). Results for velocity profiles in divergent and convergent channels were proffered for various values of Hartmann and Reynolds numbers in . The mathematical investigations of this problem were underresearched by [3, 4]. Jeffery-Hamel flows are of the Navier-Stokes equations in the particular case of two dimensional flow through a channel with inclined walls [3–13]. One of the most important examples of Jeffery-Hamel problems is this subjected to an applied magnetic field. The equations of magnetohydrodynamics have been solved exactly for the case of two-dimensional steady flow between nonparallel walls of a viscous, incompressible, electrically conducting fluid; this is a straightforward extension of the famous Jeffrey-Hamel problem in ordinary hydrodynamics . It has been indicated that for the Jeffrey-Hamel problem, the equations of magnetohydrodynamics can be curtailed to a set of three ordinary differential equations, two of which are linear and of first order . In addition, these kinds of problems have been well studied in literature [3–13]. Most recent problems such as Jeffery-Hamel flow and other fluid mechanic problems are inherently nonlinear. Except a limited number of these problems, most of them do not have analytical solutions. So, these nonlinear equations should be solved utilizing other methods.
In this paper, the [14–31] will be used to investigate MHD Jeffery-Hamel flows Problem. In recent years, a lot of attention has been devoted to the study of to investigate various scientific models. The which accurately computes the series solution is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that can be elegantly computed.
Recently, a lot of research work has been devoted to the application of to a wide class of stochastic and deterministic problems involving fractional differential equation, nonlinear oscillator with discontinuity, singular nonlinear two-point periodic boundary value problems, integral equations and nonlinear partial differential equations and so on [14–31]. The method is well suited to physical problems since it makes unnecessary restrictive methods.
The efficiency of the method was used by many authors to investigate several scientific applications. Cui and Lin  applied the to handle the second-order boundary value problems. Wang et al.  investigated a class of singular boundary value problems by this method, and the obtained results were good. In , the method was used to solve nonlocal boundary value problems. Geng and Cui  investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the . Recently, the method was appllied the fractional partial differential equations and multipoint boundary value problems [18–22]. For more details about and the modified forms and its effectiveness, see [14–31] and the references therein.
The paper is organized as follows. Section 2 is devoted to several reproducing kernel spaces. Solution representation in and a linear operator are introduced in Section 3. Section 4 provides the main results; the exact and approximate solution of system (34) and an iterative method are developed for the kind of problems in the reproducing kernel space. We have proved that the approximate solution converges to the exact solution uniformly. Numerical results are given in Section 5. The last Section is the conclusions.
2.1. Reproducing Kernel Spaces
In this section, we define some useful reproducing kernel spaces.
Definition 1 (reproducing kernel). Let be a nonempty abstract set. A function is a reproducing kernel of the Hilbert space if and only if
The last condition is called “the reproducing property”; the value of the function at the point is reproduced by the inner product of with .
Definition 2. We define the space by
The inner product and the norm in are defined, respectively, by The space is a reproducing kernel space; that is, for each fixed and any , there exists a function such that
Definition 3. We define the space by The inner product and the norm in are defined, respectively, by The space is a reproducing kernel space and its reproducing kernel function is given by
Theorem 4. The space is a complete reproducing kernel space; that is, for each fixed , there exists , such that for any . The reproducing kernel can be denoted by where
Proof. By Definition 3, we have Through several integrations by parts for (21) we have Note that property of the reproducing kernel , is the solution of the following differential equation: with the boundary conditions when , therefore Since we have Since , it follows that From (25)–(31), the unknown coefficients ve can be obtained.
3. Solution Representation in
In this section, the solution of (34) is given in the reproducing kernel space .
On defining the linear operator as Model problem (6) changes the following problem: where
Theorem 5. The operator defined by (33) is a bounded linear operator.
4. The Structure of the Solution and the Main Results
In (33) it is clear that is a bounded linear operator. Put and , where is conjugate operator of . The orthonormal system of can be derived from Gram-Schmidt orthogonalization process of as
Theorem 6. For (33), if is dense on then is the complete system of and .
Proof. We have
The subscript by the operator indicates that the operator applies to the function of . Clearly, . For each fixed , let , which means that
Note that, is dense on , hence, . It follows that from the existence of . So the proof of Theorem 6 is complete.
Theorem 7. If is the exact solution of (34), then where is a dense set in .
Lemma 8. If , then there exists , such that where .
Lemma 9. If and is continuous for , then
Proof. Since , by Lemma 8, we know that is convergent uniformly to , therefore, the proof is complete.
Remark 10. (i) If (34) is linear, that is, , then the analytical solution of (34) can be obtained directly by (47).
(ii) If (34) is nonlinear; that is, depends on and then the solution of (34) can be obtained by the following iterative method.
We construct an iterative sequence , putting where
Next we will prove that given by the iterative formula (52) converges to the exact solution (47).
Theorem 11. Suppose that the following conditions are satisfied: (i) is bounded; (ii) is a dense in ; (iii) for any . Then in iterative formula (52) converges to the exact solution of (47) in and where is given by (53).
Proof. (i) First, we will prove the convergence of . By (52), we have
From the orthogonality of , it follows that
From boundedness of , we have
Let , in view of , it follows that Considering the completeness of , there exists , such that (ii) Second, we will prove that is the solution of (34).
By Lemma 8 and Theorem 11 (i), we know that converges uniformly to . It follows that, on taking limits in (52), Since it follows that
If , then
If , then
From (64) and (65), it is clear that
Furthermore, it is easy to see by induction that Notice that is dense on interval , for any , there exists subsequence , such that , as . Hence, by the convergence of and Lemma 9, we have that is, is the solution of (34) and where is given by (53).
Theorem 13. Assume that is the solution of (34) and is the error between the approximate solution and the exact solution . Then the error sequence is monotone decreasing in the sense of and .
5. Numerical Results
All computations are performed by Maple 15. Results obtained by the method are compared with the homotopy analysis method , three analytical methods , homotopy perturbation method , and a new spectral-homotopy analysis method . The does not require discretization of the variables, that is, time and space; it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. The accuracy of the for the MHD Jeffery-Hamel flows problem is controllable and absolute errors are small with present choice of (see Tables 1–5). The numerical results that we obtained justify the advantage of this methodology.
5.1. Result and Discussion
In this study the purpose is to apply the to obtain an approximate solution of the Jeffery-Hamel problem. The obtained results of solution and numerical ones are shown in the tables and figures. In Table 2 a comparison of the HAM and is shown. Tables 1 and 3 show the comparison between the numerical results and DTM, HPM, HAM, and solutions. Tables 4 and 5 indicate the errors of DTM, HPM, HAM, and for results. Our results further show that the fluid velocity increases with increasing Hartman numbers. Numerical simulations show that for fixed Hartmann numbers, the fluid velocity increases with Reynolds numbers in the case of convergent channels but decreases with Re in the case of divergent channels. Figure 2 indicates that increasing the Hartmann number leads to higher velocity which has a great effect on the performance of the system. In Figure 3 we give a comparison between the and the HAM solutions for several Re numbers at . In Figure 4 we can see a comparison between the DTM, HPM, and HAM solutions for the velocity profile and . There is a comparison between the DTM, HPM, , and HAM solutions for the velocity profile and in Figure 5. In Figure 6 we compare and SHAM solutions. We can see absolute error for and in Figure 7. The comparison of numerical results and solution for velocity in convergent channel for and is given with Figure 8. The solutions show that the results of the present method are in excellent agreement with those of the numerical ones. Moreover, has been used to investigate the effects of the parameters of the problem.
In this paper, we introduce an algorithm for solving the MHD Jeffery-Hamel flows problem with boundary conditions by using the . The approximate solution obtained by the present method is uniformly convergent. Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems. However, if the problem becomes nonlinear, then the does not require discretization or perturbation and it does not make closure approximation. Results show that the present method is an accurate and reliable analytical method for MHD Jeffery-Hamel flows problem with boundary conditions.
Conflict of Interests
The authors declare that they do not have any competing or conflict of interests.
The first and second authors acknowledge that this researchs supported by Fırat University Scientific Research Projects Unit, Turkey is under the Research University Grant Scheme FF.12.09.
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