A New Application of the Reproducing Kernel Hilbert Space Method to Solve MHD Jeffery-Hamel Flows Problem in Nonparallel Walls

Abstract

The present paper emphasizes Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is 2𝛼. 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 𝐻, 𝛼, 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. Introduction

1.1. Problem Formulation

Consider a system of cylindrical polar coordinates (𝑟,ℎ,𝑧), 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 𝑧-axis. It is assumed that there are no changes with respect to 𝑧, that the motion is purely in radial direction and merely depends on 𝑟 and 𝜃, and that there is no magnetic field along 𝑧-axis. Then the governing equations are given as [1]. 𝜌𝜕((𝑢()()𝑟𝜕𝑟𝑟𝑢𝑟,𝜃))=0,𝑟,𝜃𝜕𝑢𝑟,𝜃𝜕𝜕𝑟=𝑣²𝑢(𝑟,𝜃)𝜕𝑟²+1𝑟𝜕𝑢(𝑟,𝜃)+1𝜕𝑟𝑟²𝜕²𝑢(𝑟,𝜃)𝜕𝜃²−𝑢(𝑟,𝜃)𝑟²−𝜎𝐵²₀𝜌𝑟²𝑢()−1𝑟,𝜃𝜌𝜕𝑃,𝜕𝑟 (1)(2)1𝜌𝑟𝜕𝑃−𝜕𝜃2𝑣𝑟²𝜕𝑢(𝑟,𝜃)𝜕𝜃=0, (3) where 𝐵₀ is the electromagnetic induction, 𝜎 is the conductivity of the fluid, 𝑢(𝑟,𝜃) is the velocity along radial direction, 𝑃 is the fluid pressure, 𝑣 is the coefficient of kinematic viscosity, and 𝜌 is the fluid density. From (1) 𝑓(𝜃)=𝑟𝑢(𝑟,𝜃), (4) using dimensionless parameters 𝑓(𝑥)=𝑓(𝜃)𝑓_max𝜃,𝑥=𝛼, (5) where 𝛼 is the semiangle between the two inclined walls as shown in Figure 1. Substituting (5) into (2) and (3) and eliminating 𝑃, we obtain an ordinary differential equation for the normalized function profile 𝐹(𝑥) [2]: 𝐹^′′′(𝑥)+2𝛼Re𝐹(𝑥)𝐹^′(𝑥)+(4−𝐻)𝛼²𝐹^′(𝑥)=0, (6) with boundary conditions 𝐹(0)=1,𝐹^′(0)=0,𝐹(1)=0. (7)

(a)

(b)

Figure 1 

Geometry of the MHD Jeffery-Hamel flow in convergent cannel. (a) 2D view and (b) schematic setup of problem.

The Reynolds number is 𝑓Re=^max𝛼𝑣=𝑈_max𝑟𝛼𝑣=divergentchannel∶𝛼>0,𝑓_max>0convergentchannel∶𝛼<0,𝑓_max.<0 (8)

The Hartmann number is 𝐻=𝛼𝐵²₀.𝜌𝑣 (9)

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 [1] and Hamel et al. [2] 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 [3]. 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 [9]. 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 [10]. 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 𝑅𝐾𝐻𝑆𝑀 [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 𝑅𝐾𝐻𝑆𝑀 to investigate various scientific models. The 𝑅𝐾𝐻𝑆𝑀 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 𝑅𝐾𝐻𝑆𝑀 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 [15] applied the 𝑅𝐾𝐻𝑆𝑀 to handle the second-order boundary value problems. Wang et al. [24] investigated a class of singular boundary value problems by this method, and the obtained results were good. In [27], the method was used to solve nonlocal boundary value problems. Geng and Cui [18] investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the 𝑅𝐾𝐻𝑆𝑀. Recently, the method was appllied the fractional partial differential equations and multipoint boundary value problems [18–22]. For more details about 𝑅𝐾𝐻𝑆𝑀 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 𝑊⁴₂[0,1] 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. Preliminaries

2.1. Reproducing Kernel Spaces

In this section, we define some useful reproducing kernel spaces.

Definition 1 (reproducing kernel). Let 𝐸 be a nonempty abstract set. A function 𝐾∶𝐸×𝐸→𝐶 is a reproducing kernel of the Hilbert space 𝐻 if and only if ()𝜑(⋅)()(𝑡).∀𝑡∈𝐸,𝐾⋅,𝑡∈𝐻,∀𝑡∈𝐸,∀𝜑∈𝐻,,𝐾⋅,𝑡=𝜑 (10)
The last condition is called “the reproducing property”; the value of the function 𝜑 at the point 𝑡 is reproduced by the inner product of 𝜑 with 𝐾(⋅,𝑡).

Definition 2. We define the space 𝑊⁴₂ by 𝑊⁴₂[0,1]=𝑢∣𝑢,𝑢^′,𝑢^′′,𝑢^′′′,[]𝑢areabsolutelycontinuousin0,1₍₄₎∈𝐿²[][],𝑢(0)(1)0,1,𝑥∈0,1=0,𝑢=0,𝑢^′(0)=0.. (11)
The inner product and the norm in 𝑊⁴₂[0,1] are defined, respectively, by 𝑢,𝑔_𝑊42=³_𝑖=0𝑢^(𝑖)(0)𝑔^(𝑖)(0)+¹₀𝑢⁽⁴⁾(𝑥)𝑔⁽⁴⁾(𝑥)𝑑𝑥,𝑢,𝑔∈𝑊⁴₂[],‖𝑢‖0,1_𝑊42=⟨𝑢,𝑢⟩_𝑊42,𝑢∈𝑊⁴₂[0,1]. (12) The space 𝑊⁴₂[0,1] is a reproducing kernel space; that is, for each fixed 𝑦∈[0,1] and any 𝑢(𝑥)∈𝑊⁴₂[0,1], there exists a function 𝑅_𝑦(𝑥) such that𝑢𝑦=𝑢,𝑅_𝑦𝑊42. (13)

Definition 3. We define the space 𝑊²₂ by 𝑊²₂[0,1]=𝑢∣𝑢,𝑢^′[]𝑢areabsolutelycontinuousin0,1_′′∈𝐿²[0,1],𝑥∈[0,1].. (14) The inner product and the norm in 𝑊²₂[0,1] are defined, respectively, by 𝑢,𝑔_𝑊22=𝑢(0)𝑔(0)+𝑢^′(0)𝑔^′(0)+¹₀𝑢^′′(𝑥)𝑔^′′(𝑥)𝑑𝑥,𝑢,𝑔∈𝑊²₂[],‖𝑢‖0,1_𝑊22=⟨𝑢,𝑢⟩_𝑊²₂,𝑢∈𝑊²₂[0,1]. (15)(16) The space 𝑊²₂[0,1] is a reproducing kernel space and its reproducing kernel function 𝑇_𝑥 is given by 𝑇_𝑥𝑦=𝑦1+𝑥𝑦+2𝑥²−16𝑥³𝑥,𝑥≤𝑦,1+𝑥𝑦+2𝑦²−16𝑦³,𝑥>𝑦. (17)

Theorem 4. The space 𝑊⁴₂[0,1] is a complete reproducing kernel space; that is, for each fixed 𝑦∈[0,1], there exists 𝑢(𝑥)∈𝑊⁴₂[0,1], such that 𝑢𝑦=𝑢,𝑅_𝑦𝑊42 (18) for any 𝑢(𝑥)∈𝑊⁴₂[0,1]. The reproducing kernel 𝑅_𝑦 can be denoted by 𝑅_𝑦(𝑥)=⁸_𝑖∑₌₁𝑐_𝑖𝑦𝑥^𝑖−1,𝑥≤𝑦,_8𝑖∑₌₁𝑑_𝑖𝑦𝑥^𝑖−1,𝑥>𝑦, (19) where 𝑐₁𝑦𝑐=0,₂𝑦𝑐=0,₃𝑦=21𝑦5680⁵+1𝑦5680⁷−7𝑦1136⁴−7𝑦284³+2𝑦71²−7𝑦5680⁶,𝑐₄𝑦=7𝑦17040⁵+1𝑦51120⁷−7𝑦10224⁴+16𝑦639³−7𝑦284²−7𝑦51120⁶,𝑐₅𝑦=7𝑦68160⁵+1𝑦204480⁷−7𝑦40896⁴+4𝑦639³−7𝑦1136²−7𝑦204480⁶,𝑐₆𝑦=−7𝑦113600⁵−1𝑦340800⁷+7𝑦68160⁴+7𝑦17040³−1𝑦2130²+7𝑦340800⁶,𝑐₇𝑦=7𝑦340800⁵+1𝑦1022400⁷−7𝑦204480⁴−7𝑦51120³−7𝑦5680²−7𝑦1022400⁶+1𝑐720𝑦,₈𝑦=−1𝑦340800⁵−1𝑦7156800⁷+7𝑦204480⁴+1𝑦51120³+1𝑦5680²+1𝑦10224000⁶−1,𝑑5040₁𝑦=−1𝑦5040⁷,𝑑₂𝑦=1𝑦720⁶,𝑑₃𝑦=−1𝑦2130⁵+1𝑦5680⁷−7𝑦1136⁴−7𝑦284³+7𝑦21²−7𝑦5680⁶,𝑑₄𝑦=4𝑦639⁴+1𝑦17040⁵+7𝑦40896⁷+16𝑦639³−7𝑦284²−7𝑦51120⁶,𝑑₅𝑦=7𝑦68160⁵+1𝑦24480⁷−7𝑦40896⁴−7𝑦10224³−7𝑦1136²−7𝑦204480⁶,𝑑₆𝑦=−7𝑦113600⁵−1𝑦340800⁷+7𝑦68160⁴+7𝑦17040³+21𝑦5680²+7𝑦340800⁶,𝑑₇𝑦=7𝑦34080⁵+1𝑦1022400⁷−7𝑦204480⁴−7𝑦51120³−7𝑦5680²−7𝑦1022400⁶,𝑑₈𝑦=−1𝑦340800⁵−1𝑦7156800⁷+1𝑦204480⁴+1𝑦51120³+1𝑦5680²+7𝑦1022400⁶. (20)

Proof. By Definition 3, we have 𝑢,𝑅_𝑦𝑊42=³_𝑖=0𝑢^(𝑖)(0)𝑅_𝑦^(𝑖)(0)+¹₀𝑢⁽⁴⁾(𝑥)𝑅_𝑦⁽⁴⁾(𝑥)𝑑𝑥,𝑢,𝑅_𝑦∈𝑊⁴₂[0,1]. (21) Through several integrations by parts for (21) we have 𝑢(𝑥),𝑅_𝑦(𝑥)_𝑊42=³_𝑖=0𝑢^(𝑖)(0)𝑅_𝑦^(𝑖)(0)−(−1)^(3−𝑖)𝑅_𝑦^(7−𝑖)(0+_3𝑖=0)(−1)^(3−𝑖)𝑢^(𝑖)(1)𝑅_𝑦^(7−𝑖)(1)+¹₀𝑢(𝑥)𝑅_𝑦⁽⁸⁾(𝑥)𝑑𝑥. (22) Note that property of the reproducing kernel 𝑢,𝑅_𝑦𝑊42=𝑢𝑦, (23)𝑅_𝑦, is the solution of the following differential equation: 𝑅_𝑦⁽⁸⁾(𝑥)=𝛿𝑥−𝑦, (24) with the boundary conditions 𝑅_𝑦⁽³⁾(0)−𝑅_𝑦⁽⁴⁾(0)𝑅=0,_𝑦′′(0)+𝑅_𝑦⁽⁵⁾(0)𝑅=0,_𝑦(4)(1)𝑅=0,_𝑦(5)(1)𝑅=0,_𝑦(6)(1)=0, (25) when 𝑥≠𝑦, 𝑅_𝑦⁽⁸⁾(𝑥)=0, (26) therefore 𝑅_𝑦(𝑥)=⁸_𝑖∑₌₁𝑐_𝑖𝑦𝑥^𝑖−1,𝑥≤𝑦,_8𝑖∑₌₁𝑑_𝑖𝑦𝑥^𝑖−1,𝑥>𝑦. (27) Since 𝑅_𝑦⁽⁸⁾(𝑥)=𝛿𝑥−𝑦, (28) we have 𝜕^𝑘𝑅_𝑦+𝑦=𝜕^𝑘𝑅_𝑦−𝑦𝜕,𝑘=0,1,2,3,4,5,6,₇𝑅_𝑦+𝑦−𝜕⁷𝑅_𝑦−𝑦=1. (29)(30) Since 𝑅_𝑦(𝑥)∈𝑊⁴₂[0,1], it follows that 𝑅_𝑦(0)=0,𝑅^′_𝑦(0)=0,𝑅_𝑦(1)=0. (31) From (25)–(31), the unknown coefficients 𝑐_𝑖(𝑦) ve 𝑑_𝑖(𝑦)(𝑖=1,2,…,8) can be obtained. 𝑅_𝑦(𝑥)=21𝑥5680²𝑦⁵+1𝑥5680²𝑦⁷−7𝑥1136²𝑦⁴−7𝑥284²𝑦³+2𝑥71²𝑦²−7𝑥5680²𝑦⁶+7𝑥17040³𝑦⁵+1𝑥51120³𝑦⁷−7𝑥10224³𝑦⁴+16𝑥639³𝑦³−7𝑥284³𝑦²−7𝑥51120³𝑦⁶+7𝑥68160⁴𝑦⁵+1𝑥204480⁴𝑦⁷−7𝑥40896⁴𝑦⁴+4𝑥639⁴𝑦³−7𝑥1136⁴𝑦²−7𝑥204480⁴𝑦⁶−7𝑥113600⁵𝑦⁵−1𝑥340800⁵𝑦⁷+7𝑥68160⁵𝑦⁴+7𝑥17040⁵𝑦³−1𝑥2130⁵𝑦²+7𝑥340800⁵y⁶+7𝑥340800⁶𝑦⁵+1𝑥1022400⁶𝑦⁷−7𝑥204480⁶𝑦⁴−7𝑥51120⁶𝑦³−7𝑥5680⁶𝑦²−7𝑥1022400⁶𝑦⁶+1𝑥720⁶1𝑦−𝑥340800⁷𝑦⁵−1𝑥7156800⁷𝑦⁷+7𝑥204480⁷𝑦⁴+1𝑥51120⁷𝑦³+1𝑥5680⁷𝑦²+1𝑥10224000⁷𝑦⁶−𝑥⁷5040,𝑥≤𝑦21𝑦5680²𝑥⁵+1𝑦5680²𝑥⁷−7𝑦1136²𝑥⁴−7𝑦284²𝑥³+2𝑦71²𝑥²−7𝑦5680²𝑥⁶7𝑦17040³𝑥⁵+1𝑦51120³𝑥⁷−7𝑦10224³𝑥⁴+16𝑦639³𝑥³−7𝑦284³𝑥²−7𝑦51120³𝑥⁶7𝑦68160⁴𝑥⁵+1𝑦204480⁴𝑥⁷−7𝑦40896⁴𝑥⁴+4𝑦639⁴𝑥³−7𝑦1136⁴𝑥²−7𝑦204480⁴𝑥⁶−7𝑦113600⁵𝑥⁵−1𝑦340800⁵𝑥⁷+7𝑦68160⁵𝑥⁴+7𝑦17040⁵𝑥³−1𝑦2130⁵𝑥²+7𝑦340800⁵𝑥⁶+7𝑦340800⁶𝑥⁵+1𝑦1022400⁶𝑥⁷−7𝑦204480⁶𝑥⁴−7𝑦51120⁶𝑥³−7𝑦5680⁶𝑥²−7𝑦1022400⁶𝑥⁶+1𝑦720⁶1𝑥−𝑦340800⁷𝑥⁵−1𝑥7156800⁷𝑦⁷+7𝑦204480⁷𝑥⁴+1𝑦51120⁷𝑥³+1𝑦5680⁷𝑥²+1𝑦10224000⁷𝑥⁶−𝑦⁷5040,𝑥>𝑦. (32)

3. Solution Representation in 𝑊⁴₂[0,1]

In this section, the solution of (34) is given in the reproducing kernel space 𝑊⁴₂[0,1].

On defining the linear operator 𝐿∶𝑊⁴₂[0,1]→𝑊²₂[0,1] as(𝐿𝑢)(𝑥)=𝑢^′′′(𝑥)+−2𝛼Re𝑥²−1+(4−𝐻)𝛼²𝑢^′(𝑥)−4𝛼𝑥Re𝑢(𝑥). (33) Model problem (6) changes the following problem: 𝐿𝑢=𝑓𝑥,𝑢,𝑢^′[],𝑢(0),𝑥∈0,1=0,𝑢^′(0)=0,𝑢(1)=0, (34) where 𝑓𝑥,𝑢,𝑢^′=−2𝛼Re𝑢(𝑥)𝑢^′(𝑥)−4𝛼Re𝑥³()𝛼−𝑥+24−𝐻²𝑢(𝑥)(𝑥)𝑥,=𝐹+𝑥²−1. (35)

Theorem 5. The operator 𝐿 defined by (33) is a bounded linear operator.

Proof. We only need to prove ‖𝐿𝑢‖²_𝑊22≤𝑀‖𝑢‖²_𝑊42, where 𝑀>0 is a positive constant. By (15) and (16), we have ‖𝐿𝑢‖²_𝑊22=⟨𝐿𝑢,𝐿𝑢⟩_𝑊²₂=[(𝐿𝑢)(0)]²+(𝐿𝑢)^′(0)²+¹₀(𝐿𝑢)^′′(𝑥)²𝑑𝑥. (36) By (18), we have 𝑢(𝑥)=𝑢,𝑅_𝑥𝑊42,(𝐿𝑢)(𝑥)=𝑢,𝐿𝑅_𝑥𝑊42,(𝐿𝑢)^′(𝑥)=𝑢,𝐿𝑅_𝑥^′_𝑊42, (37) so |(𝐿𝑢)(𝑥)|≤‖𝑢‖_𝑊⁴₂𝐿𝑅_𝑥𝑊42=𝑀₁‖𝑢‖_𝑊⁴₂,where𝑀₁,)>0isapositiveconstant(𝐿𝑢^′(𝑥)≤‖𝑢‖_𝑊⁴₂𝐿𝑅_𝑥^′_𝑊42=𝑀₂‖𝑢‖_𝑊⁴₂,where𝑀₂>0isapositiveconstant, (38) thus (𝐿𝑢)²(0)+(𝐿𝑢)^′(0)²≤𝑀²₁+𝑀²₂‖𝑢‖²_𝑊42. (39) Since (𝐿𝑢)^′′=𝑢,𝐿𝑅_𝑥^′′_𝑊42, (40) then (𝐿𝑢)^′′≤‖𝑢‖_𝑊⁴₂𝐿𝑅_𝑥^′′_𝑊42=𝑀₃‖𝑢‖_𝑊⁴₂,where𝑀₃>0isapositiveconstant, (41) so, we have (𝐿𝑢)^′′2≤𝑀²₃‖𝑢‖²_𝑊42,₁₀(𝐿𝑢)^′′(𝑥)²𝑑𝑥≤𝑀²₃‖𝑢‖²_𝑊42, (42) that is ‖𝐿𝑢‖²_𝑊22=[(𝐿𝑢)(0)]²+(𝐿𝑢)^′(0)²+¹₀(𝐿𝑢)^′′(𝑥)²≤𝑀𝑑𝑥₂₁+𝑀²₂+𝑀²₃‖𝑢‖²_𝑊42=𝑀‖𝑢‖²_𝑊42, (43) where 𝑀=(𝑀²₁+𝑀²₂+𝑀²₃)>0 is a positive constant.

4. The Structure of the Solution and the Main Results

In (33) it is clear that 𝐿∶𝑊⁴₂[0,1]→𝑊²₂[0,1] is a bounded linear operator. Put 𝜑_𝑖=𝑇_𝑥𝑖 and 𝜓_𝑖=𝐿^∗𝜑_𝑖, where 𝐿^∗ is conjugate operator of 𝐿. The orthonormal system {Ψ_𝑖}^∞_𝑖=1 of 𝑊⁴₂[0,1] can be derived from Gram-Schmidt orthogonalization process of {𝜓_𝑖}^∞_𝑖=1 as 𝜓_𝑖(𝑥)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥),𝛽_𝑖𝑖>0,𝑖=1,2,…. (44)

Theorem 6. For (33), if {𝑥_𝑖}^∞_𝑖=1 is dense on [0,1] then {𝜓_𝑖}^∞_𝑖=1 is the complete system of 𝑊⁴₂[0,1] and 𝜓_𝑖(𝑥)=𝐿_𝑦𝑅_𝑥(𝑦)|_{𝑦=𝑥𝑖}.

Proof. We have 𝜓_𝑖(𝑥)=𝐿^∗𝜑_𝑖(𝑥)=𝐿^∗𝜑_𝑖𝑦,𝑅_𝑥𝑦=𝜑_𝑖𝑦,𝐿𝑦𝑅_𝑥𝑦=𝐿_𝑦𝑅_𝑥(𝑦)_{𝑦=𝑥𝑖}. (45)
The subscript 𝑦 by the operator 𝐿 indicates that the operator 𝐿 applies to the function of 𝑦. Clearly, 𝜓_𝑖(𝑥)∈𝑊⁴₂[0,1]. For each fixed 𝑢(𝑥)∈𝑊⁴₂[0,1], let ⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩=0,(𝑖=1,2,…), which means that 𝑢,𝐿^∗𝜑_𝑖=𝐿𝑢,𝜑_𝑖=𝐿𝑢,𝑇_𝑥𝑖=(𝐿𝑢)𝑥_𝑖=0. (46)
Note that, {𝑥_𝑖}^∞_𝑖=1 is dense on [0,1], hence, (𝐿𝑢)(𝑥)=0. It follows that 𝑢≡0 from the existence of 𝐿⁻¹. So the proof of Theorem 6 is complete.

Theorem 7. If 𝑢(𝑥) is the exact solution of (34), then 𝑢=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑢_𝑘,𝑢^′_𝑘Ψ_𝑖, (47) where {𝑥_𝑖}^∞_𝑖=1 is a dense set in [0,1].

Proof. From (44) and uniqueness of solution of (34) we have 𝑢=^∞_𝑖=1𝑢,𝜓_𝑖𝑊42𝜓_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢,𝐿^∗𝑇_{𝑥𝑘𝑊42}𝜓_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢,𝑇_{𝑥𝑘𝑊22}𝜓_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓(𝑥,𝑢,𝑢^′),𝑇_{𝑥𝑘𝑊22}𝜓_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑢_𝑘,𝑢^′_𝑘𝜓_𝑖. (48) Now the approximate solution 𝑢_𝑛(𝑥) can be obtained by truncating the 𝑛-term of the exact solution 𝑢(𝑥)∶𝑢_𝑛=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑢_𝑘,𝑢^′_𝑘𝜓_𝑖. (49)

Lemma 8. If 𝑢∈𝑊⁴₂[0,1], then there exists 𝑀₁>0, such that ‖𝑢‖_𝐶²_[0,1]≤𝑀₁‖𝑢‖_𝑊⁴₂, (50) where ‖𝑢‖_𝐶²_[0,1]=max_𝑥∈[0,1]|𝑢(𝑥)|+max_𝑥∈[0,1]|𝑢^′(𝑥)|+max_𝑥∈[0,1]|𝑢^′′(𝑥)|.

Lemma 9. If ‖𝑢_𝑛−𝑢‖_𝑊42→0,𝑥_𝑛→𝑥,(𝑛→∞) and 𝑓(𝑥,𝑢,𝑢^′) is continuous for 𝑥∈0,1], then 𝑓𝑥_𝑛,𝑢_𝑛−1𝑥_𝑛,𝑢^′_𝑛−1𝑥_𝑛⟶𝑓𝑥,𝑢(𝑥),𝑢^′(𝑥)𝑎𝑠𝑛⟶∞. (51)

Proof. Since ‖𝑢_𝑛−𝑢‖_𝑊42→0(𝑛→∞), by Lemma 8, we know that 𝑢_𝑛 is convergent uniformly to 𝑢(𝑥), therefore, the proof is complete.

Remark 10. (i) If (34) is linear, that is, 𝑓(𝑥,𝑢)=𝑓(𝑥), then the analytical solution of (34) can be obtained directly by (47).
(ii) If (34) is nonlinear; that is, 𝑓 depends on 𝑢 and 𝑢^′ then the solution of (34) can be obtained by the following iterative method.
We construct an iterative sequence 𝑢_𝑛(𝑥), putting anyfixed𝑢₀(𝑥)∈𝑊⁴₂[],𝑢0,1_𝑛(𝑥)=^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥), (52) where 𝐴₁=𝛽₁₁𝑓𝑥₁,𝑢₀𝑥₁,𝑢^′₀𝑥₁,𝐴₂=²_𝑘=1𝛽_2𝑘𝑓𝑥_𝑘,𝑢_𝑘−1𝑥_𝑘,𝑢^′_𝑘−1𝑥_𝑘,⋮𝐴_𝑛=^𝑛_𝑘=1𝛽_𝑛𝑘𝑓𝑥_𝑘,𝑢_𝑘−1𝑥_𝑘,𝑢^′_𝑘−1𝑥_𝑘. (53)
Next we will prove that 𝑢_𝑛 given by the iterative formula (52) converges to the exact solution (47).

Theorem 11. Suppose that the following conditions are satisfied: (i)  ‖𝑢_𝑛‖_𝑊42 is bounded; (ii) {𝑥_𝑖}^∞_𝑖=1 is a dense in [0,1]; (iii)  𝑓(𝑥,𝑢,𝑢^′)∈𝑊²₂[0,1] for any 𝑢∈𝑊⁴₂[0,1]. Then 𝑢_𝑛 in iterative formula (52) converges to the exact solution of (47) in 𝑊⁴₂[0,1] and 𝑢=^∞_𝑖=1𝐴_𝑖𝜓_𝑖, (54) where 𝐴_𝑖 is given by (53).

Proof. (i) First, we will prove the convergence of 𝑢_𝑛(𝑥). By (52), we have 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+𝐴_𝑛+1𝜓_𝑛+1(𝑥). (55) From the orthogonality of {Ψ_𝑖(𝑥)}^∞_𝑖=1, it follows that 𝑢_𝑛+1²_𝑊42=𝑢_𝑛²_𝑊42+𝐴_𝑛+1²=𝑢_𝑛−1²_𝑊42+𝐴_𝑛²+𝐴_𝑛+1²=⋯=^𝑛+1_𝑖=1𝐴_𝑖². (56) From boundedness of ‖𝑢_𝑛‖_𝑊42, we have ^∞_𝑖=1𝐴_𝑖²<∞, (57) that is, 𝐴_𝑖∈𝑙²(𝑖=1,2,…). (58)
Let 𝑚>𝑛, in view of (𝑢_𝑚−𝑢_𝑚−1)⟂(𝑢_𝑚−1−𝑢_𝑚−2)⟂⋯⟂(𝑢_𝑛+1−𝑢_𝑛), it follows that 𝑢_𝑚−𝑢_𝑛²_𝑊42=𝑢_𝑚−𝑢_𝑚−1+𝑢_𝑚−1−u_𝑚−2+⋯+𝑢_𝑛+1−𝑢_𝑛²_𝑊42≤𝑢_𝑚−𝑢_𝑚−1²_𝑊42+⋯+𝑢_𝑛+1−𝑢_𝑛²_𝑊42=^𝑚_𝑖=𝑛+1𝐴_𝑖²⟶0(𝑚,𝑛⟶∞). (59) Considering the completeness of 𝑊⁴₂[0,1], there exists 𝑢(𝑥)∈𝑊⁴₂[0,1], such that 𝑢_𝑛(𝑥)⟶^{‖⋅‖𝑊42}𝑢(𝑥),as𝑛⟶∞. (60) (ii) Second, we will prove that 𝑢(𝑥) is the solution of (34).
By Lemma 8 and Theorem 11 (i), we know that 𝑢_𝑛 converges uniformly to 𝑢. It follows that, on taking limits in (52), 𝑢=^∞_𝑖=1𝐴_𝑖𝜓_𝑖. (61) Since (𝐿𝑢)𝑥_𝑗=^∞_𝑖=1𝐴_𝑖𝐿Ψ_𝑖(𝑥),𝜑_𝑗(𝑥)_𝑊22=^∞_𝑖=1𝐴_𝑖Ψ_𝑖(𝑥),𝐿^∗𝜑_𝑗(𝑥)_𝑊42=^∞_𝑖=1𝐴_𝑖Ψ_𝑖(𝑥),Ψ_𝑗(𝑥)_𝑊42, (62) it follows that ^𝑛_𝑗=1𝛽_𝑛𝑗(𝐿𝑢)𝑥_𝑗=^∞_𝑖=1𝐴_𝑖Ψ_𝑖(𝑥),^𝑛_𝑗=1𝛽_𝑛𝑗Ψ_𝑗(𝑥)_𝑊42=^∞_𝑖=1𝐴_𝑖Ψ_𝑖(𝑥),Ψ_𝑛(𝑥)_𝑊42=𝐴_𝑛. (63)
If 𝑛=1, then (𝐿𝑢)𝑥₁=𝑓𝑥₁,𝑢₀𝑥₁,𝑢^′₀𝑥₁. (64)
If 𝑛=2, then 𝛽₂₁(𝐿𝑢)𝑥₁+𝛽₂₂(𝐿𝑢)𝑥₂=𝛽₂₁𝑓𝑥₁,𝑢₀𝑥₁,𝑢^′₀𝑥₁+𝛽₂₂𝑓𝑥₂,𝑢₁𝑥₂,𝑢^′₁𝑥₂. (65)
From (64) and (65), it is clear that (𝐿𝑢)𝑥₂=𝑓𝑥₂,𝑢₁𝑥₂,𝑢^′₁𝑥₂. (66)
Furthermore, it is easy to see by induction that (𝐿𝑢)𝑥_𝑗=𝑓𝑥_𝑗,𝑢_𝑗−1𝑥_𝑗,𝑢^′_𝑗−1𝑥_𝑗. (67) Notice that {𝑥_𝑖}^∞_𝑖=1 is dense on interval [0,1], for any 𝑦∈[0,1], there exists subsequence {𝑥_𝑛𝑗}, such that 𝑥_𝑛𝑗→𝑦, as 𝑗→∞. Hence, by the convergence of 𝑢_𝑛(𝑥) and Lemma 9, we have (𝐿𝑢)𝑦=𝑓𝑦,𝑢𝑦,𝑢^′𝑦, (68) that is, 𝑢(𝑥) is the solution of (34) and 𝑢=^∞_𝑖=1𝐴_𝑖𝜓_𝑖, (69) where 𝐴_𝑖 is given by (53).

Corollary 12. Assume that the conditions of Theorem 11 hold; then 𝑢_𝑛 in (52) satisfies ‖𝑢_𝑛−𝑢‖_𝐶²_[0,1]→0,𝑛→∞, where 𝑢 is the solution of (34).

Theorem 13. Assume that 𝑢 is the solution of (34) and 𝑟_𝑛 is the error between the approximate solution 𝑢_𝑛 and the exact solution 𝑢. Then the error sequence 𝑟_𝑛 is monotone decreasing in the sense of ‖⋅‖_𝑊⁴₂ and ‖𝑟_𝑛(𝑥)‖_𝑊42→0.

Proof. From (47) and (49), it follows that 𝑟_𝑛𝑊42=^∞𝑖_{𝑖=𝑛+1𝑘=1}𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑢_𝑘,𝑢^′_𝑘𝜓_𝑖(𝑥)_𝑊42=^∞_𝑖=𝑛+1^𝑖_𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑢_𝑘,𝑢^′_𝑘². (70) Equation (70) shows that the error 𝑟_𝑛 is decreasing in the sense of ‖⋅‖_𝑊⁴₂.

5. Numerical Results

All computations are performed by Maple 15. Results obtained by the method are compared with the homotopy analysis method [3], three analytical methods [5], homotopy perturbation method [6], and a new spectral-homotopy analysis method [8]. The 𝑅𝐾𝐻𝑆𝑀 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 𝑅𝐾𝐻𝑆𝑀 for the MHD Jeffery-Hamel flows problem is controllable and absolute errors are small with present choice of 𝑥 (see Tables 1–5). The numerical results that we obtained justify the advantage of this methodology.

Table 1 

The comparison between the numerical results and DTM, HPM, HAM, and  RKHSM  solutions for Re=110,𝛼=3,and𝐻=0.

Table 2 

The numerical results for Re=50,𝐻=1000.

Table 3 

The comparison between the numerical results and DTM, HPM, HAM, and  RKHSM  solutions for Re=80,𝛼=−5, and 𝐻=0.

Table 4 

The errors of DTM, HPM, HAM, and RKHSM for 𝐹(𝑥) results when Re=110,𝛼=3, and 𝐻=0.

Table 5 

The errors of DTM, HPM, HAM, and  RKHSM  for 𝐹(𝑥) results when for Re=80,𝛼=−5, and 𝐻=0.

5.1. Result and Discussion

In this study the purpose is to apply the 𝑅𝐾𝐻𝑆𝑀 to obtain an approximate solution of the Jeffery-Hamel problem. The obtained results of 𝑅𝐾𝐻𝑆𝑀 solution and numerical ones are shown in the tables and figures. In Table 2 a comparison of the HAM and 𝑅𝐾𝐻𝑆𝑀 is shown. Tables 1 and 3 show the comparison between the numerical results and DTM, HPM, HAM, and 𝑅𝐾𝐻𝑆𝑀 solutions. Tables 4 and 5 indicate the errors of DTM, HPM, HAM, and 𝑅𝐾𝐻𝑆𝑀 for 𝐹(𝑥) 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 𝑅𝐾𝐻𝑆𝑀 and the HAM solutions for several Re numbers at 𝐻=1000. In Figure 4 we can see a comparison between the DTM, HPM, 𝑅𝐾𝐻𝑆𝑀 and HAM solutions for the velocity profile Re=80 and 𝐻=0. There is a comparison between the DTM, HPM, 𝑅𝐾𝐻𝑆𝑀, and HAM solutions for the velocity profile Re=110 and 𝐻=0 in Figure 5. In Figure 6 we compare 𝑅𝐾𝐻𝑆𝑀 and SHAM solutions. We can see absolute error for Re=50 and 𝐻=1000 in Figure 7. The comparison of numerical results and 𝑅𝐾𝐻𝑆𝑀 solution for velocity in convergent channel for Re=50 and 𝐻=1000 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, 𝑅𝐾𝐻𝑆𝑀 has been used to investigate the effects of the parameters of the problem.

Figure 2 

A comparison between increasing Hartmann numbers for the velocity profile Re=100.

Figure 3 

A comparison between the increasing values of Re for the velocity profile 𝐻=1000.

Figure 4 

A comparison between the DTM, HPM, 𝑅𝐾𝐻𝑆𝑀, and HAM solutions for the velocity profile Re=80 and 𝐻=0.

Figure 5 

A comparison between the DTM, HPM, 𝑅𝐾𝐻𝑆𝑀, and HAM solutions for the velocity profile Re=110 and 𝐻=0.

Figure 6 

A comparison between the 𝑅𝐾𝐻𝑆𝑀 and SHAM solutions for the velocity profile Re=50,𝛼=5, and 𝐻=0.

Figure 7 

Absolute error for Re=50 and 𝐻=1000.

Figure 8 

A comparison between different values of 𝛼 for velocity in convergent channel for Re=50 and 𝐻=1000.

6. Conclusion

In this paper, we introduce an algorithm for solving the MHD Jeffery-Hamel flows problem with boundary conditions by using the 𝑅𝐾𝐻𝑆𝑀. 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 𝑅𝐾𝐻𝑆𝑀 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.

Acknowledgment

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.