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Approximate solutions for MHD squeezing fluid flow by a novel method

Mustafa Inc1 and Ali Akgül23*

Author Affiliations

1 Department of Mathematics, Science Faculty, Fırat University, Elazığ, 23119, Turkey

2 Department of Mathematics, Education Faculty, Dicle University, Diyarbakır, 21280, Turkey

3 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA

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Boundary Value Problems 2014, 2014:18  doi:10.1186/1687-2770-2014-18

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/18


Received:5 October 2013
Accepted:10 December 2013
Published:16 January 2014

© 2014 Inc and Akgül; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, a steady axisymmetric MHD flow of two-dimensional incompressible fluids has been investigated. The reproducing kernel Hilbert space method (RKHSM) has been implemented to obtain a solution of the reduced fourth-order nonlinear boundary value problem. Numerical results have been compared with the results obtained by the Runge-Kutta method (RK-4) and optimal homotopy asymptotic method (OHAM).

MSC: 46E22, 35A24.

Keywords:
reproducing kernel method; series solutions; squeezing fluid flow; magnetohydrodynamics; reproducing kernel space

1 Introduction

Squeezing flows have many applications in food industry, principally in chemical engineering [1-4]. Some practical examples of squeezing flow include polymer processing, compression and injection molding. Grimm [5] studied numerically the thin Newtonian liquids films being squeezed between two plates. Squeezing flow coupled with magnetic field is widely applied to bearing with liquid-metal lubrication [2,6-8].

In this paper, we use RKHSM to study the squeezing MHD fluid flow between two infinite planar plates. This problem has been solved by RKHSM and for comparison it has been compared with the OHAM and numerically with the RK-4 by using Maple 16.

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. The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [9] and Zhou et al.[10] applied the RKHSM to handle the second-order boundary value problems. Yao and Cui [11] and Wang et al.[12] investigated a class of singular boundary value problems by this method and the obtained results were good. Wang and Chao [13], Li and Cui [14], Zhou and Cui [15] independently employed the RKSHSM to variable-coefficient partial differential equations. Du and Cui [16] investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKM. Lv and Cui [17] presented a new algorithm to solve linear fifth-order boundary value problems. Cui and Du [18] obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the RKHSM. Wu and Li [19] applied iterative RKHSM to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. For more details about RKHSM and the modified forms and its effectiveness, see [9-37] and the references therein.

The paper is organized as follows. We give the problem formulation in Section 2. Section 3 introduces several reproducing kernel spaces. A bounded linear operator is presented in Section 4. In Section 5, we provide the main results, the exact and approximate solutions. An iterative method is developed for the kind of problems in the reproducing kernel space. We prove that the approximate solution converges to the exact solution uniformly. Some numerical experiments are illustrated in Section 6. There are some conclusions in the last section.

2 Problem formulation

Consider a squeezing flow of an incompressible Newtonian fluid in the presence of a magnetic field of a constant density ρ and viscosity μ squeezed between two large planar parallel plates separated by a small distance 2H and the plates approaching each other with a low constant velocity V, as illustrated in Figure 1, and the flow can be assumed to quasi-steady [1-3,39]. The Navier-Stokes equations [3,4] governing such flow in the presence of magnetic field, when inertial terms are retained in the flow, are given as [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M1">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M2">View MathML</a>

(2.2)

where u is the velocity vector, ∇ denotes the material time derivative, T is the Cauchy stress tensor,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M3">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M4">View MathML</a>

J is the electric current density, B is the total magnetic field and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M5">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M6">View MathML</a> represents the imposed magnetic field and b denotes the induced magnetic field. In the absence of displacement currents, the modified Ohm law and Maxwell’s equations (see [40] and the references therein) are given by [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M7">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M8">View MathML</a>

(2.4)

in which σ is the electrical conductivity, E is the electric field and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M9">View MathML</a> is the magnetic permeability.

thumbnailFigure 1. A steady squeezing axisymmetric fluid flow between two parallel plates[38].

The following assumptions are needed [38].

(a) The density ρ, magnetic permeability <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M9">View MathML</a> and electric field conductivity σ are assumed to be constant throughout the flow field region.

(b) The electrical conductivity σ of the fluid is considered to be finite.

(c) Total magnetic field B is perpendicular to the velocity field V and the induced magnetic field b is negligible compared with the applied magnetic field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M6">View MathML</a> so that the magnetic Reynolds number is small (see [40] and the references therein).

(d) We assume a situation where no energy is added or extracted from the fluid by the electric field, which implies that there is no electric field present in the fluid flow region.

Under these assumptions, the magnetohydrodynamic force involved in Eq. (2.2) can be put into the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M12">View MathML</a>

(2.5)

An axisymmetric flow in cylindrical coordinates r, θ, z with z-axis perpendicular to plates and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M13">View MathML</a> at the plates. Since we have axial symmetry, u is represented by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M14">View MathML</a>

when body forces are negligible, Navier-Stokes Eqs. (2.1)-(2.2) in cylindrical coordinates, where there is no tangential velocity (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M15">View MathML</a>), are given as [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M16">View MathML</a>

(2.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M17">View MathML</a>

(2.7)

where p is the pressure, and the equation of continuity is given by [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M18">View MathML</a>

(2.8)

The boundary conditions require

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M19">View MathML</a>

(2.9)

Let us introduce the axisymmetric Stokes stream function Ψ as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M20">View MathML</a>

(2.10)

The continuity equation is satisfied using Eq. (2.10). Substituting Eqs. (2.3)-(2.5) and Eq. (2.10) into Eqs. (2.7)-(2.8), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M21">View MathML</a>

(2.11)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M22">View MathML</a>

(2.12)

Eliminating the pressure from Eqs. (2.11) and (2.12) by the integrability condition, we get the compatibility equation as [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M23">View MathML</a>

(2.13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M24">View MathML</a>

The stream function can be expressed as [1,3]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M25">View MathML</a>

(2.14)

In view of Eq. (2.14), the compatibility equation (2.13) and the boundary conditions (2.9) take the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M26">View MathML</a>

(2.15)

subject to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M27">View MathML</a>

(2.16)

Non-dimensional parameters are given as [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M28">View MathML</a>

For simplicity omitting the ∗, the boundary value problem (2.15)-(2.16) becomes [38]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M29">View MathML</a>

(2.17)

with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M30">View MathML</a>

(2.18)

where Re is the Reynolds number and m is the Hartmann number.

3 Reproducing kernel spaces

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

Definition 3.1 (Reproducing kernel)

Let E be a nonempty abstract set. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M31">View MathML</a> is a reproducing kernel of the Hilbert space H if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M32">View MathML</a>

(3.1)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M33">View MathML</a>.

Definition 3.2 We define the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M35">View MathML</a>

The fifth derivative of u exists almost everywhere since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M36">View MathML</a> is absolutely continuous. The inner product and the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a> are defined respectively by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M38">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M39">View MathML</a>

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a> is a reproducing kernel space, i.e., for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M41">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M42">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M43">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M44">View MathML</a>

Definition 3.3 We define the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M46">View MathML</a>

The fourth derivative of u exists almost everywhere since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M47">View MathML</a> is absolutely continuous. The inner product and the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45">View MathML</a> are defined respectively by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M49">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M50">View MathML</a>

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45">View MathML</a> is a reproducing kernel space, i.e., for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M41">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M53">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M54">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M55">View MathML</a>

Theorem 3.1The space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a>is a reproducing kernel Hilbert space whose reproducing kernel function is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M57">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M59">View MathML</a>can be obtained easily by using Maple 16 and the proof of Theorem 3.1 is given in Appendix.

Remark 3.1 The reproducing kernel function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M54">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M45">View MathML</a> is given as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M62">View MathML</a>

This can be proved easily as the proof of Theorem 3.1.

4 Bounded linear operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a>

In this section, the solution of Eq. (2.17) is given in the reproducing kernel space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a>.

On defining the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M65">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M66">View MathML</a>

Model problem (2.17)-(2.18) changes the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M67">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M68">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M69">View MathML</a>

Theorem 4.1The operatorLdefined by (4.1) is a bounded linear operator.

Proof We only need to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M70">View MathML</a>

where P is a positive constant. By Definition 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M71">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M72">View MathML</a>

By the reproducing property, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M73">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M74">View MathML</a>

Therefore, by the Cauchy-Schwarz inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M75">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M76">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M77">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M78">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M79">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M80">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M81">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M82">View MathML</a> is a positive constant. This completes the proof. □

5 Analysis of the solution of (2.17)-(2.18)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M83">View MathML</a> be any dense set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M85">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M86">View MathML</a> is the adjoint operator of L and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M87">View MathML</a> is given by Remark 3.1. Furthermore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M88">View MathML</a>

Lemma 5.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M89">View MathML</a>is a complete system of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M42">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M92">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M93">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M83">View MathML</a> is the dense set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M95">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M96">View MathML</a>. Assume that (4.1) has a unique solution. Then L is one-to-one on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M98">View MathML</a>. This completes the proof. □

Lemma 5.2The following formula holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M99">View MathML</a>

where the subscriptηof the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M100">View MathML</a>indicates that the operatorLapplies to a function ofη.

Proof

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M101">View MathML</a>

This completes the proof. □

Remark 5.1 The orthonormal system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M102">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a> can be derived from the Gram-Schmidt orthogonalization process of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M104">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M105">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M106">View MathML</a> are orthogonal coefficients.

In the following, we give the representation of the exact solution of Eq. (2.17) in the reproducing kernel space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M34">View MathML</a>.

Theorem 5.1Ifuis the exact solution of (4.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M108">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M83">View MathML</a>is a dense set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M84">View MathML</a>.

Proof From (5.1) and the uniqueness of solution of (4.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M111">View MathML</a>

This completes the proof. □

Now the approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M112">View MathML</a> can be obtained by truncating the n-term of the exact solution u as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M113">View MathML</a>

Lemma 5.3 ([30])

Assume thatuis the solution of (4.1) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M114">View MathML</a>is the error between the approximate solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M112">View MathML</a>and the exact solutionu. Then the error sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M114">View MathML</a>is monotone decreasing in the sense of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M118">View MathML</a>.

6 Numerical results

In this section, comparisons of results are made through different Reynolds numbers Re and magnetic field effect m. All computations are performed by Maple 16. Figure 5.7 shows comparisons of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M119">View MathML</a> for a fixed Reynolds number with increasing magnetic field effect <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M120">View MathML</a>. From this figure, the velocity decreases due to an increase in m. Figure 5.8 shows comparisons of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M121">View MathML</a> for a fixed magnetic field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122">View MathML</a> with increasing Reynolds numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M123">View MathML</a>. It is observed that much increase in Reynolds numbers affects the results. The RKHSM does not require discretization of the variables, i.e., time and space, it is not affected by computation round of errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the MHD squeezing fluid flow is controllable and absolute errors are small with present choice of x (see Tables 1-6 and Figures 2-7). The numerical results we obtained justify the advantage of this methodology. Generally it is not possible to find the exact solution of these problems.

thumbnailFigure 2. Comparison RKHSM, OHAM and RK-4 solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M124">View MathML</a>.

thumbnailFigure 3. Comparison RKHSM, OHAM and RK-4 solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M125">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M126">View MathML</a>.

thumbnailFigure 4. AE and RE for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M127">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M128">View MathML</a>.

thumbnailFigure 5. AE and RE for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M127">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M130">View MathML</a>.

thumbnailFigure 6. Comparison of squeezing flow for a fixed Reynolds number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M126">View MathML</a>and increasing magnetic field effect<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M132">View MathML</a>.

thumbnailFigure 7. Comparison of squeezing flow for a fixed magnetic field effect<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M127">View MathML</a>and increasing Reynolds numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M134">View MathML</a>.

Table 1. Numerical results at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136">View MathML</a>

Table 2. Numerical results at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M137">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136">View MathML</a>

Table 3. Numerical results at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M139">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136">View MathML</a>

Table 4. Numerical results at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M136">View MathML</a>

Table 5. Numerical results at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M144">View MathML</a>

Table 6. Numerical results at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M122">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M146">View MathML</a>

7 Conclusion

In this paper, we introduced an algorithm for solving the MHD squeezing fluid flow. We applied a new powerful method RKHSM to the reduced nonlinear boundary value problem. 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 of numerical examples show that the present method is an accurate and reliable analytical method for this problem.

Appendix

Proof of Theorem 3.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M42">View MathML</a>. By Definition 3.2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M148">View MathML</a>

(A.1)

Through several integrations by parts for (A.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M149">View MathML</a>

(A.2)

Note the property of the reproducing kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M150">View MathML</a>

Now, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M151">View MathML</a>

(A.3)

then (A.2) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M152">View MathML</a>

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M153">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M154">View MathML</a>

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M155">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M156">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M157">View MathML</a>

(A.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M158">View MathML</a>

(A.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M159">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M160">View MathML</a>

(A.6)

From (A.3)-(A.6), the unknown coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/18/mathml/M163">View MathML</a>) can be obtained. This completes the proof. □

Competing interests

The authors declare that they do not have any competing or conflict of interests.

Authors’ contributions

Both authors contributed equally to this paper.

Acknowledgements

We presented this paper in the International Symposium on Biomathematics and Ecology Education Research in 2013. We would like to thank the organizers of this conference and the reviewers for their kind and helpful comments on this paper. Ali Akgül gratefully acknowledge that this paper was partially supported by the Dicle University and the Firat University. This paper is a part of PhD thesis of Ali Akgül.

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