International Journal of Engineering Mathematics The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Rational-Like Solutions of a Differential-Difference Equation Related to Ablowitz-Ladik Spectral Problem Thu, 03 Sep 2015 14:00:43 +0000 By using the Casoratian technique, we construct the double Casoratian solutions whose entries satisfy matrix equation of a differential-difference equation related to the Ablowitz-Ladik spectral problem. Soliton solutions and rational-like solutions are obtained from taking special cases in general solutions. Jiang-ping Zhang, Qi Li, and Shou-ting Chen Copyright © 2015 Jiang-ping Zhang et al. All rights reserved. A Wavelet Algorithm for Fourier-Bessel Transform Arising in Optics Mon, 31 Aug 2015 13:00:10 +0000 The aim of the paper is to propose an efficient and stable algorithm that is quite accurate and fast for numerical evaluation of the Fourier-Bessel transform of order , using wavelets. The philosophy behind the proposed algorithm is to replace the part of the integral by its wavelet decomposition obtained by using CAS wavelets thus representing as a Fourier-Bessel series with coefficients depending strongly on the input function . The wavelet method indicates that the approach is easy to implement and thus computationally very attractive. Nagma Irfan and A. H. Siddiqi Copyright © 2015 Nagma Irfan and A. H. Siddiqi. All rights reserved. Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence Sun, 07 Dec 2014 07:38:14 +0000 In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result. Amine Bernoussi, Abdelilah Kaddar, and Said Asserda Copyright © 2014 Amine Bernoussi et al. All rights reserved. Influence of Variable Thermal Conductivity on MHD Boundary Layer Slip Flow of Ethylene-Glycol Based Cu Nanofluids over a Stretching Sheet with Convective Boundary Condition Thu, 06 Nov 2014 12:43:54 +0000 An analysis is carried out to investigate the influence of variable thermal conductivity and partial velocity slip on hydromagnetic two-dimensional boundary layer flow of a nanofluid with Cu nanoparticles over a stretching sheet with convective boundary condition. Using similarity transformation, the governing boundary layer equations along with the appropriate boundary conditions are transformed to a set of ordinary differential equations. Employing Runge-kutta fourth-order method along with shooting technique, the resultant system of equations is solved. The influence of various pertinent parameters such as nanofluid volume fraction parameter, the magnetic parameter, radiation parameter, thermal conductivity parameter, velocity slip parameter, Biot number, and suction or injection parameter on the velocity of the flow field and heat transfer characteristics is computed numerically and illustrated graphically. The present results are compared with the existing results for the case of regular fluid and found an excellent agreement. N. Bhaskar Reddy, T. Poornima, and P. Sreenivasulu Copyright © 2014 N. Bhaskar Reddy et al. All rights reserved. Aligned Magnetic Field, Radiation, and Rotation Effects on Unsteady Hydromagnetic Free Convection Flow Past an Impulsively Moving Vertical Plate in a Porous Medium Tue, 21 Oct 2014 09:06:38 +0000 We analyse the effects of aligned magnetic field, radiation, and rotation on unsteady hydromagnetic free convection flow of a viscous incompressible electrically conducting fluid past an impulsively moving vertical plate in a porous medium in presence of heat source. An exact solution of the governing equations in dimensionless form is obtained by Laplace transform technique in ramped temperature case. To compare the results obtained in this case with that of isothermal plate, the exact solution of the governing equations is also obtained for isothermal plate and results are discussed graphically in both ramped temperature and isothermal cases. Sandeep Naramgari, Sugunamma Vangala, and Mohankrishna Penem Copyright © 2014 Sandeep Naramgari et al. All rights reserved. Quadratic Prediction Models for the Performance Comparison of a Marine Engine Fuelled with Biodiesels B5 and B20 Tue, 30 Sep 2014 08:34:28 +0000 According to Thailand’s renewable energy development plan, biodiesel is one of the interesting alternative energies. In this research, biodiesels B5 and B20 are tested in a marine engine. The experimental results are then compared by using three different techniques including (1) the conventional technique, (2) average of the point-to-point comparisons, and (3) a comparison by using quadratic prediction models. This research aims to present the procedures of these techniques in-depth. The results show that the comparison by using quadratic prediction models can accurately predict ample amounts of results and make the comparison more logical. The results are compatible with those of the conventional technique, while the average of the point-to-point comparisons shows diverse results. These results are also explained on the fuel property basis, confirming that the quadratic prediction model and the conventional technique are practical, but the average of the point-to-point comparison technique presents an inaccurate result. The benefit of this research shows that the quadratic prediction model is more flexible for future science and engineering experimental design, thus reducing cost and time usage. The details of the calculation, results, and discussion are presented in the paper. Chedthawut Poompipatpong Copyright © 2014 Chedthawut Poompipatpong. All rights reserved. A New Approach to Solve Intuitionistic Fuzzy Optimization Problem Using Possibility, Necessity, and Credibility Measures Thu, 25 Sep 2014 13:03:03 +0000 Corresponding to chance constraints, real-life possibility, necessity, and credibility measures on intuitionistic fuzzy set are defined. For the first time the mathematical and graphical representations of different types of measures in trapezoidal intuitionistic fuzzy environment are defined in this paper. We have developed intuitionistic fuzzy chance constraints model (CCM) based on possibility and necessity measures. We have also proposed a new method for solving an intuitionistic fuzzy CCM using chance operators. To validate the proposed method, we have discussed three different approaches to solve the intuitionistic fuzzy linear programming (IFLPP) using possibility, necessity and credibility measures. Numerical and graphical representations of optimal solutions of the given example at different possibility and necessity, levels have been discussed. Dipankar Chakraborty, Dipak Kumar Jana, and Tapan Kumar Roy Copyright © 2014 Dipankar Chakraborty et al. All rights reserved. Design of Robust Output Feedback Guaranteed Cost Control for a Class of Nonlinear Discrete-Time Systems Wed, 10 Sep 2014 09:40:54 +0000 This paper investigates static output feedback guaranteed cost control for a class of nonlinear discrete-time systems where the delay in state vector is inconsistent with the delay in nonlinear perturbations. Based on the output measurement, the controller is designed to ensure the robust exponentially stability of the closed-loop system and guarantee the performance of system to achieve an adequate level. By using the Lyapunov-Krasovskii functional method, some sufficient conditions for the existence of robust output feedback guaranteed cost controller are established in terms of linear matrix inequality. A numerical example is provided to show the effectiveness of the results obtained. Yan Zhang, Yali Dong, and Tianrui Li Copyright © 2014 Yan Zhang et al. All rights reserved. Numerical Analysis of Vibration Isolation Using Pile Rows against the Vibration due to Moving Loads in a Viscoelastic Medium Mon, 08 Sep 2014 05:29:45 +0000 A numerical method for evaluating the vertical vibration isolation effect of pile rows embedded in a viscoelastic half space subjected to a moving load is developed in this paper on the basis of the Cole-Cole model and Muki’s method. Based on the proposed method, the influence of various parameters on the vibration isolation effect of pile rows embedded in the viscoelastic half space is investigated numerically. Bin Xu and Man-Qing Xu Copyright © 2014 Bin Xu and Man-Qing Xu. All rights reserved. A Modified Strip-Yield-Saturation-Induction Model Solution for Cracked Piezoelectromagnetic Plate Thu, 14 Aug 2014 00:00:00 +0000 A strip-yield-saturation-induction model is proposed for an impermeable crack embedded in piezoelectromagnetic plate. The developed slide-yield, saturation, and induction zones are arrested by distributing, respectively, mechanical, electrical, and magnetic loads over their rims. Two cases are considered: when saturation zone exceeds induction zone and vice-versa. It is assumed that developed slide-yield zone is the smallest because of the brittle nature of piezoelectromagnetic material. Fourier integral transform technique is employed to obtain the solution. Closed form analytic expressions are derived for developed zones lengths, crack sliding displacement, crack opening potential drop, crack opening induction drop, and energy release rate. Case study presented for BaTiO3–CoFe2O4 shows that crack arrest is possible under small-scale mechanical, electrical, and magnetic yielding. R. R. Bhargava and Pooja Raj Verma Copyright © 2014 R. R. Bhargava and Pooja Raj Verma. All rights reserved. Inviscid Uniform Shear Flow past a Smooth Concave Body Wed, 23 Jul 2014 00:00:00 +0000 Uniform shear flow of an incompressible inviscid fluid past a two-dimensional smooth concave body is studied; a stream function for resulting flow is obtained. Results for the same flow past a circular cylinder or a circular arc or a kidney-shaped body are presented as special cases of the main result. Also, a stream function for resulting flow around the same body is presented for an oncoming flow which is the combination of a uniform stream and a uniform shear flow. Possible fields of applications of this study include water flows past river islands, the shapes of which deviate from circular or elliptical shape and have a concave region, or past circular arc-shaped river islands and air flows past concave or circular arc-shaped obstacles near the ground. Abdullah Murad Copyright © 2014 Abdullah Murad. All rights reserved. Process Parameter Identification in Thin Film Flows Driven by a Stretching Surface Mon, 21 Jul 2014 10:30:31 +0000 The flow of a thin liquid film over a heated stretching surface is considered in this study. Due to a potential nonuniform temperature distribution on the stretching sheet, a temperature gradient occurs in the fluid which produces surface tension gradient at the free surface of the thin film. As a result, the free surface deforms and these deformations are advected by the flow in the stretching direction. This work focuses on the inverse problem of reconstructing the sheet temperature distribution and the sheet stretch rate from observed free surface variations. This work builds on the analysis of Santra and Dandapat (2009) who, based on the long-wave expansion of the Navier-Stokes equations, formulate a partial differential equation which describes the evolution of the thickness of a film over a nonisothermal stretched surface. In this work, we show that after algebraic manipulation of a discrete form of the governing equations, it is possible to reconstruct either the unknown temperature field on the sheet and hence the resulting heat transfer or the stretching rate of the underlying surface. We illustrate the proposed methodology and test its applicability on a range of test problems. Satyananda Panda, Mathieu Sellier, M. C. S. Fernando, and M. K. Abeyratne Copyright © 2014 Satyananda Panda et al. All rights reserved. Weak Nonlinear Double-Diffusive Magnetoconvection in a Newtonian Liquid under Temperature Modulation Sun, 06 Jul 2014 12:09:11 +0000 The present paper deals with a weak nonlinear theory of double-diffusive magnetoconvection in an electrically conducting Newtonian liquid, confined between two horizontal surfaces, under a constant vertical magnetic field, and subjected to imposed time-periodic thermal boundaries. The temperature of both walls is varied time periodic in this case. The disturbances are expanded in terms of power series of amplitude of convection, which is assumed to be small. Using nonautonomous Ginzburg-Landau equation, the Nusselt and Sherwood numbers obtained analytically and studied heat and mass transport in the system. Effect of various parameters on the heat and mass transport is discussed extensively. It is found that the effect of magnetic field is to stabilize the system. Further, it is also notified that the heat and mass transport can be controlled by suitably adjusting the external parameters of the system. B. S. Bhadauria and Palle Kiran Copyright © 2014 B. S. Bhadauria and Palle Kiran. All rights reserved. Jeffrey Fluid Flow through Porous Medium in the Presence of Magnetic Field in Narrow Tubes Wed, 25 Jun 2014 09:24:41 +0000 Jeffrey fluid flow in the presence of magnetic field through porous medium in tubes of small diameters is studied. It is assumed that the core region consists of a Jeffrey fluid and the peripheral region of a Newtonian fluid. Making the assumptions as in the work of Chaturani and Upadhya, the linearised equations of motion have been solved and analytical solution has been obtained. The influence of various pertinent parameters on the flow characteristics such as effective viscosity, core hematocrit, and mean hematocrit has been studied and discussed through graphs. It is found that the effective viscosity and mean hematocrit decrease with Jeffrey parameter and Darcy number but increase with tube hematocrit and tube radius. Also, the core hematocrit decreases with Jeffrey parameter, Darcy number, tube hematocrit, and tube radius. Further, it is noticed that the flow exhibits the anomalous Fahraeus-Lindquist effect. Santhosh Nallapu and G. Radhakrishnamacharya Copyright © 2014 Santhosh Nallapu and G. Radhakrishnamacharya. All rights reserved. Analytical Solutions of Some Fully Developed Flows of Couple Stress Fluid between Concentric Cylinders with Slip Boundary Conditions Tue, 17 Jun 2014 06:26:41 +0000 We establish, in this paper, the closed form analytical solutions of steady fully developed flows of couple stress fluid between two concentric cylinders, generated due to the constant pressure gradient or the translatory motion of the outer cylinder or both, using the slip boundary conditions. The classical solutions for Newtonian fluid in the hydrodynamic case appear as a limiting case of our solutions. The velocity profiles of the flows are presented and the effect of various parameters on velocity is discussed. The results indicate that the presence of couple stresses decreases the velocity of the fluid. M. Devakar, D. Sreenivasu, and B. Shankar Copyright © 2014 M. Devakar et al. All rights reserved. Inversion of Fourier Transforms by Means of Scale-Frequency Series Tue, 27 May 2014 00:00:00 +0000 We report on inversion of the Fourier transform when the frequency variable can be scaled in a variety of different ways that improve the resolution of certain parts of the frequency domain. The corresponding inverse Fourier transform is shown to exist in the form of two dual scale-frequency series. Upon discretization of the continuous scale factor, this Fourier transform series inverse becomes a certain nonharmonic double series, a discretized scale-frequency (DSF) series. The DSF series is also demonstrated, theoretically and practically, to be rate-optimizable with respect to its two free parameters, when it satisfies, as an entropy maximizer, a pertaining recursive nonlinear programming problem incorporating the entropy-based uncertainty principle. Nassar H. S. Haidar Copyright © 2014 Nassar H. S. Haidar. All rights reserved. Controller Design for an Observer-Based Modified Repetitive-Control System Tue, 13 May 2014 11:33:54 +0000 This paper presents a method of designing a state-observer based modified repetitive-control system that provides a given level of disturbance attenuation for a class of strictly proper linear plants. Since the time delay in a repetitive controller can be treated as a kind of disturbance, we convert the system design problem into a standard state-feedback control problem for a linear time-invariant system. The Lyapunov functional and the singular-value decomposition of the output matrix are used to derive a linear-matrix-inequality (LMI) based design algorithm for the parameters of the feedback controller and the state-observer. A numerical example demonstrates the validity of the method. Lan Zhou, Jinhua She, Shaowu Zhou, and Qiwei Chen Copyright © 2014 Lan Zhou et al. All rights reserved. An Application of Filtered Renewal Processes in Hydrology Mon, 05 May 2014 07:58:36 +0000 Filtered renewal processes are used to forecast daily river flows. For these processes, contrary to filtered Poisson processes, the time between consecutive events is not necessarily exponentially distributed, which is more realistic. The model is applied to obtain one- and two-day-ahead forecasts of the flows of the Delaware and Hudson Rivers, both located in the United States. Better results are obtained than with filtered Poisson processes, which are often used to model river flows. Mario Lefebvre and Fatima Bensalma Copyright © 2014 Mario Lefebvre and Fatima Bensalma. All rights reserved. Axially Symmetric Vibrations of Composite Poroelastic Spherical Shell Mon, 28 Apr 2014 00:00:00 +0000 This paper deals with axially symmetric vibrations of composite poroelastic spherical shell consisting of two spherical shells (inner one and outer one), each of which retains its own distinctive properties. The frequency equations for pervious and impervious surfaces are obtained within the framework of Biot’s theory of wave propagation in poroelastic solids. Nondimensional frequency against the ratio of outer and inner radii is computed for two types of sandstone spherical shells and the results are presented graphically. From the graphs, nondimensional frequency values are periodic in nature, but in the case of ring modes, frequency values increase with the increase of the ratio. The nondimensional phase velocity as a function of wave number is also computed for two types of sandstone spherical shells and for the spherical bone implanted with titanium. In the case of sandstone shells, the trend is periodic and distinct from the case of bone. In the case of bone, when the wave number lies between 2 and 3, the phase velocity values are periodic, and when the wave number lies between 0.1 and 1, the phase velocity values decrease. Rajitha Gurijala and Malla Reddy Perati Copyright © 2014 Rajitha Gurijala and Malla Reddy Perati. All rights reserved. On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions Tue, 01 Apr 2014 07:21:21 +0000 The th-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the th-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the th-order nonlinearity of a Boolean function is very complicated problem, especially when . This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form for all , , where , where , , and are integers such that and , and , where is a positive integer such that and . Brajesh Kumar Singh Copyright © 2014 Brajesh Kumar Singh. All rights reserved. Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions Mon, 24 Feb 2014 11:54:01 +0000 Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme. Ishfaq Ahmad Ganaie, Shelly Arora, and V. K. Kukreja Copyright © 2014 Ishfaq Ahmad Ganaie et al. All rights reserved. Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear Equations Sun, 23 Feb 2014 09:20:23 +0000 In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing third- and fourth-order methods. Anuradha Singh and J. P. Jaiswal Copyright © 2014 Anuradha Singh and J. P. Jaiswal. All rights reserved. Numerical Performance of Higher-Order Semicompact Scheme for Arbitrary Triangular Cavity Flow Tue, 04 Feb 2014 10:41:09 +0000 An efficient fourth-order semicompact finite difference scheme has been developed to solve steady incompressible Navier-Stokes (N-S) equations in stream function and vorticity formulation in a triangular cavity of arbitrary geometry. The governing equations are transformed into curvilinear coordinates by a simple linear transformation to handle the nonregular geometry of the problem. The main feature of the new higher-order semicompact scheme is that it can calculate a triangle flow with arbitrary shape for high Reynolds numbers. It is found that the solutions obtained with the present scheme are in good agreement with the analytical results or with the existing results depending on the availability. Xiaofeng Wang and Dongyang Shi Copyright © 2014 Xiaofeng Wang and Dongyang Shi. All rights reserved. Developing Buoyancy Driven Flow of a Nanofluid in a Vertical Channel Subject to Heat Flux Sun, 02 Feb 2014 08:43:22 +0000 The developing natural convective flow of a nanofluid in an infinite vertical channel with impermeable bounding walls has been investigated. It is assumed that the nanofluid is dominated by two specific slip mechanisms and that the channel walls are subject to constant heat flux and isothermal temperature, respectively. The governing nonlinear partial differential equations coupling different transport processes have been solved numerically. The variations of velocity, temperature, and nanoparticles concentration have been discussed in relation to a number of physical parameters. It is seen that the approach to the steady-state profiles of velocity and temperature in the present work is different from the ones reported in a previous study corresponding to isothermal wall conditions. Nirmal C. Sacheti, Pallath Chandran, Ashok K. Singh, and Beer S. Bhadauria Copyright © 2014 Nirmal C. Sacheti et al. All rights reserved. A Study of I-Function of Several Complex Variables Mon, 27 Jan 2014 12:29:36 +0000 The aim of this paper is to introduce a natural generalization of the well-known, interesting, and useful Fox H-function into generalized function of several variables, namely, the I-function of ‘‘’’ variables. For , we get the I-function introduced and studied by Arjun Rathie (1997) and, for , we get I-function of two variables introduced very recently by ShanthaKumari et al. (2012). Convergent conditions, elementary properties, and special cases have also been given. The results presented in this paper generalize the results of H-function of ‘‘’’ variables available in the literature. Prathima Jayarama, Vasudevan Nambisan Theke Madam, and Shantha Kumari Kurumujji Copyright © 2014 Prathima Jayarama et al. All rights reserved. Non-Darcy Mixed Convection in a Doubly Stratified Porous Medium with Soret-Dufour Effects Thu, 09 Jan 2014 08:23:33 +0000 This paper presents the nonsimilarity solutions for mixed convection heat and mass transfer along a semi-infinite vertical plate embedded in a doubly stratified fluid saturated porous medium in the presence of Soret and Dufour effects. The flow in the porous medium is described by employing the Darcy-Forchheimer based model. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless forms and then solved numerically. The influence of pertinent parameters on dimensionless velocity, temperature, concentration, heat, and mass transfer in terms of the local Nusselt and Sherwood numbers is discussed and presented graphically. D. Srinivasacharya and O. Surender Copyright © 2014 D. Srinivasacharya and O. Surender. All rights reserved. Effects of Mass Transfer, Radiation, Joule Heating, and Viscous Dissipation on Steady MHD Marangoni Convection Flow over a Flat Surface with Suction and Injection Sat, 14 Dec 2013 12:38:08 +0000 The combined effects of radiation and mass transfer on a steady MHD two-dimensional Marangoni convection flow over a flat surface in presence of Joule heating and viscous dissipation under influence of suction and injection is studied numerically. The general governing partial differential equations are transformed into a set of nonlinear ordinary differential equations by using unique similarity transformation. Numerical solutions of the similarity equations are obtained using the Runge-Kutta method along with shooting technique. The effects of governing parameters on velocity, temperature, and concentration as well as interface velocity, the surface temperature gradient, and the surface concentration gradient were presented in graphical and tabular forms. Comparisons with previously published work are performed and the results are found to be in excellent agreement. S. Mohammed Ibrahim Copyright © 2013 S. Mohammed Ibrahim. All rights reserved. Investigation of Through-Thickness Stresses in Composite Laminates Using Layerwise Theory Thu, 12 Dec 2013 15:36:23 +0000 In this study, an analytical method is developed to exactly obtain the interlaminar stresses near the free edges of laminated composite plates under the bending moment based on the reduced form of elasticity displacement field for a long laminate. The analytical and numerical studies were performed based on the Reddy’s layerwise theory for the boundary layer stresses within cross-ply, symmetric, angle-ply, and general composite laminates. Finally, a variety of numerical results are presented for the interlaminar normal and shear stresses along the interfaces and through thickness of laminates near the free edges. The results showed high stress gradient of interlaminar normal and shear stresses near the edges of laminates. Hamidreza Yazdani Sarvestani and Ali Naghashpour Copyright © 2013 Hamidreza Yazdani Sarvestani and Ali Naghashpour. All rights reserved. Sufficient Conditions of Asymptotic Stability of the Time-Varying Descriptor Systems Tue, 10 Dec 2013 18:35:07 +0000 We discuss the time-varying descriptor systems. Firstly, a sufficient condition of asymptotic stability and impulse-free is derived based on Riccati equation. Secondly, we design a state feedback controller to make the close-loop system asymptotically stable and impulse-free. Finally, a numerical example demonstrates the proposed results. Xiaoming Su and Yali Zhi Copyright © 2013 Xiaoming Su and Yali Zhi. All rights reserved. Asymptotic Solution for a Water Quality Model in a Uniform Stream Thu, 28 Nov 2013 10:32:34 +0000 We employ approximate analytical method, namely, Optimal Homotopy Asymptotic Method (OHAM), to investigate a one-dimensional steady advection-diffusion-reaction equation with variable inputs arises in the mathematical modeling of dispersion of pollutants in water is proposed. Numerical values are obtained via Runge-Kutta-Fehlberg fourth-fifth order method for comparison purpose. It was found that OHAM solution agrees well with the numerical solution. An example is included to demonstrate the efficiency, accuracy, and simplicity of the proposed method. Fazle Mabood and Nopparat Pochai Copyright © 2013 Fazle Mabood and Nopparat Pochai. All rights reserved.