Journal of Applied Mathematics The latest articles from Hindawi © 2018 , Hindawi Limited . All rights reserved. Bridging the Gap between Economic Modelling and Simulation: A Simple Dynamic Aggregate Demand-Aggregate Supply Model with Matlab Tue, 16 Jan 2018 00:00:00 +0000 This paper aims to connect the bridge between analytical results and the use of the computer for numerical simulations in economics. We address the analytical properties of a simple dynamic aggregate demand and aggregate supply (AD-AS) model and solve it numerically. The model undergoes a bifurcation as its steady state smoothly interchanges stability depending on the relationship between the impact of real interest rate on demand for liquidity and how fast agents revise their expectations on inflation. Using code embedded into a unique function in Matlab, we plot the numerical solutions of the model and simulate different dynamic adjustments using different parameter values. The same function also accommodates the analysis of the impacts of fiscal and monetary policy and supplies side shocks on the steady state and the transition dynamics of the model. José M. Gaspar Copyright © 2018 José M. Gaspar. All rights reserved. Understanding Dengue Control for Short- and Long-Term Intervention with a Mathematical Model Approach Mon, 01 Jan 2018 09:56:18 +0000 A mathematical model of dengue diseases transmission will be discussed in this paper. Various interventions, such as vaccination of adults and newborns, the use of insecticides or fumigation, and also the enforcement of mechanical controls, will be considered when analyzing the best intervention for controlling the spread of dengue. From model analysis, we find three types of equilibrium points which will be built upon the dengue model. In this paper, these points are the mosquito-free equilibrium, disease-free equilibrium (with and without vaccinated compartment), and endemic equilibrium. Basic reproduction number as an endemic indicator has been found analytically. Based on analytical and numerical analysis, insecticide treatment, adult vaccine, and enforcement of mechanical control are the most significant interventions in reducing the spread of dengue disease infection caused by mosquitoes rather than larvicide treatment and vaccination of newborns. From short- and long-term simulation, we find that insecticide treatment is the best strategy to control dengue. We also find that, with periodic intervention, the result is not much significantly different with constant intervention based on reduced number of the infected human population. Therefore, with budget limitations, periodic intervention of insecticide strategy is a good alternative to reduce the spread of dengue. A. Bustamam, D. Aldila, and A. Yuwanda Copyright © 2018 A. Bustamam et al. All rights reserved. An Analysis of a Semelparous Population Model with Density-Dependent Fecundity and Density-Dependent Survival Probabilities Sun, 17 Dec 2017 00:00:00 +0000 A discrete age-structured semelparous Leslie matrix model where density dependence is included both in the fecundity and in the survival probabilities is analysed. Depending on strength of density dependence, we show in the precocious semelparous case that the nonstationary dynamics may indeed be rich, ranging from SYC (a dynamical state where the whole population is in one age class only) dynamics to cycles of low period where all age classes are populated. Quasiperiodic and chaotic dynamics have also been identified. Moreover, outside parameter regions where SYC dynamics dominates, we prove that the transfer from stability to instability goes through a supercritical Neimark−Sacker bifurcation, and it is further shown that when the population switches from possessing a precocious to a delayed semelparous life history both stability properties and the possibility of periodic dynamics become weaker. Arild Wikan Copyright © 2017 Arild Wikan. All rights reserved. Corrigendum to “Improved Combinatorial Benders Decomposition for a Scheduling Problem with Unrelated Parallel Machines” Sun, 17 Dec 2017 00:00:00 +0000 Francisco Regis Abreu Gomes and Geraldo Robson Mateus Copyright © 2017 Francisco Regis Abreu Gomes and Geraldo Robson Mateus. All rights reserved. Periodic Travelling Wave Solutions of Discrete Nonlinear Schrödinger Equations Sun, 26 Nov 2017 00:00:00 +0000 The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schrödinger equation (DNLS) on one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of interactions going beyond the usual nearest-neighbour interaction. The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem. Dirk Hennig Copyright © 2017 Dirk Hennig. All rights reserved. Hybrid Algorithm of Particle Swarm Optimization and Grey Wolf Optimizer for Improving Convergence Performance Thu, 16 Nov 2017 00:00:00 +0000 A newly hybrid nature inspired algorithm called HPSOGWO is presented with the combination of Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO). The main idea is to improve the ability of exploitation in Particle Swarm Optimization with the ability of exploration in Grey Wolf Optimizer to produce both variants’ strength. Some unimodal, multimodal, and fixed-dimension multimodal test functions are used to check the solution quality and performance of HPSOGWO variant. The numerical and statistical solutions show that the hybrid variant outperforms significantly the PSO and GWO variants in terms of solution quality, solution stability, convergence speed, and ability to find the global optimum. Narinder Singh and S. B. Singh Copyright © 2017 Narinder Singh and S. B. Singh. All rights reserved. New Integrals Arising in the Samara-Valencia Heat Transfer Model in Grinding Tue, 14 Nov 2017 00:00:00 +0000 The Samara-Valencia model for heat transfer in grinding has been recently used for calculating nontabulated integrals. Based on these results, new infinite integrals can be calculated, involving the Macdonald function and the modified Struve function. J. L. González-Santander Copyright © 2017 J. L. González-Santander. All rights reserved. Generalized Hybrid One-Step Block Method Involving Fifth Derivative for Solving Fourth-Order Ordinary Differential Equation Directly Sun, 12 Nov 2017 07:30:29 +0000 A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated at while its fourth and fifth derivatives are collocated at all points ,, in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error. Mohammad Alkasassbeh and Zurni Omar Copyright © 2017 Mohammad Alkasassbeh and Zurni Omar. All rights reserved. Analysis of Economic Burden of Seasonal Influenza: An Actuarial Based Conceptual Model Wed, 11 Oct 2017 00:00:00 +0000 Analysing the economic burden of the seasonal influenza is highly essential due to the large number of outbreaks in recent years. Mathematical and actuarial models can be considered as management tools to understand the dynamical behavior, predict the risk, and compute it. This study is an attempt to develop conceptual model to investigate the economic burden due to seasonal influenza. The compartment SIS (susceptible-infected-susceptible) model is used to capture the dynamical behavior of influenza. Considering the current investment and future medical care expenditure as premium payment and benefit (claim), respectively, the insurance and actuarial based conceptual model is proposed to model the present economic burden due to the spread of influenza. Simulation is carried out to demonstrate the variation of the present economic burden with respect to model parameters. The sensitivity of the present economic burden is studied with respect to the risk of disease spread. The basic reproduction is used to identify the risk of disease spread. Impact of the seasonality is studied by introducing the seasonally varying infection rate. The proposed model provides theoretical background to investigate the economic burden of seasonal influenza. S. S. N. Perera Copyright © 2017 S. S. N. Perera. All rights reserved. Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform Wed, 27 Sep 2017 00:00:00 +0000 The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT. Mawardi Bahri and Muh. Saleh Arif Fatimah Copyright © 2017 Mawardi Bahri and Muh. Saleh Arif Fatimah. All rights reserved. Some New Volterra-Fredholm-Type Nonlinear Discrete Inequalities with Two Variables Involving Iterated Sums and Their Applications Wed, 27 Sep 2017 00:00:00 +0000 Some generalized discrete Volterra-Fredholm-type inequalities were developed, which can be used as effective tools in the qualitative analysis of the solution to difference equations. Run Xu Copyright © 2017 Run Xu. All rights reserved. A Greedy Clustering Algorithm Based on Interval Pattern Concepts and the Problem of Optimal Box Positioning Mon, 25 Sep 2017 00:00:00 +0000 We consider a clustering approach based on interval pattern concepts. Exact algorithms developed within the framework of this approach are unable to produce a solution for high-dimensional data in a reasonable time, so we propose a fast greedy algorithm which solves the problem in geometrical reformulation and shows a good rate of convergence and adequate accuracy for experimental high-dimensional data. Particularly, the algorithm provided high-quality clustering of tactile frames registered by Medical Tactile Endosurgical Complex. Stepan A. Nersisyan, Vera V. Pankratieva, Vladimir M. Staroverov, and Vladimir E. Podolskii Copyright © 2017 Stepan A. Nersisyan et al. All rights reserved. Extension of Wolfe Method for Solving Quadratic Programming with Interval Coefficients Thu, 14 Sep 2017 00:00:00 +0000 Quadratic programming with interval coefficients developed to overcome cases in classic quadratic programming where the coefficient value is unknown and must be estimated. This paper discusses the extension of Wolfe method. The extended Wolfe method can be used to solve quadratic programming with interval coefficients. The extension process of Wolfe method involves the transformation of the quadratic programming with interval coefficients model into linear programming with interval coefficients model. The next step is transforming linear programming with interval coefficients model into two classic linear programming models with special characteristics, namely, the optimum best and the worst optimum problem. Syaripuddin, Herry Suprajitno, and Fatmawati Copyright © 2017 Syaripuddin et al. All rights reserved. Analysis of a Heroin Epidemic Model with Saturated Treatment Function Thu, 31 Aug 2017 00:00:00 +0000 A mathematical model is developed that examines how heroin addiction spreads in society. The model is formulated to take into account the treatment of heroin users by incorporating a realistic functional form that “saturates” representing the limited availability of treatment. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the saturation parameter is larger than a fixed threshold. We are particularly interested in studying the model’s global stability. In the absence of backward bifurcations, Lyapunov functions can often be found and used to prove global stability. However, in the presence of backward bifurcations, such Lyapunov functions may not exist or may be difficult to construct. We make use of the geometric approach to global stability to derive a condition that ensures that the system is globally asymptotically stable. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users are mechanisms fuelling heroin epidemic proliferation. Isaac Mwangi Wangari and Lewi Stone Copyright © 2017 Isaac Mwangi Wangari and Lewi Stone. All rights reserved. Simulation of Wellbore Stability during Underbalanced Drilling Operation Tue, 15 Aug 2017 00:00:00 +0000 The wellbore stability analysis during underbalance drilling operation leads to avoiding risky problems. These problems include (1) rock failure due to stresses changes (concentration) as a result of losing the original support of removed rocks and (2) wellbore collapse due to lack of support of hydrostatic fluid column. Therefore, this paper presents an approach to simulate the wellbore stability by incorporating finite element modelling and thermoporoelastic environment to predict the instability conditions. Analytical solutions for stress distribution for isotropic and anisotropic rocks are presented to validate the presented model. Moreover, distribution of time dependent shear stresses around the wellbore is presented to be compared with rock shear strength to select appropriate weight of mud for safe underbalance drilling. Reda Abdel Azim Copyright © 2017 Reda Abdel Azim. All rights reserved. Gutman Index and Detour Gutman Index of Pseudo-Regular Graphs Tue, 15 Aug 2017 00:00:00 +0000 The Gutman index of a connected graph is defined as , where   and   are the degree of the vertices   and   and is the distance between vertices   and  . The Detour Gutman index of a connected graph is defined as , where is the longest distance between vertices   and  . In this paper, the Gutman index and the Detour Gutman index of pseudo-regular graphs are determined. S. Kavithaa and V. Kaladevi Copyright © 2017 S. Kavithaa and V. Kaladevi. All rights reserved. On the Solution of the Eigenvalue Assignment Problem for Discrete-Time Systems Thu, 10 Aug 2017 00:00:00 +0000 The output feedback eigenvalue assignment problem for discrete-time systems is considered. The problem is formulated first as an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to find a local solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a logarithmic barrier method is proposed for finding the local solution. The conjugate gradient method is further extended to tackle the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. The performance of the methods is illustrated through various test examples. El-Sayed M. E. Mostafa, Abdallah W. Aboutahoun, and Fatma F. S. Omar Copyright © 2017 El-Sayed M. E. Mostafa et al. All rights reserved. A Guide on Spectral Methods Applied to Discrete Data in One Dimension Mon, 24 Jul 2017 00:00:00 +0000 This paper provides an overview about the usage of the Fourier transform and its related methods and focuses on the subtleties to which the users must pay attention. Typical questions, which are often addressed to the data, will be discussed. Such a problem can be the origin of frequency or band limitation of the signal or the source of artifacts, when a Fourier transform is carried out. Another topic is the processing of fragmented data. Here, the Lomb-Scargle method will be explained with an illustrative example to deal with this special type of signal. Furthermore, the time-dependent spectral analysis, with which one can evaluate the point in time when a certain frequency appears in the signal, is of interest. The goal of this paper is to collect the important information about the common methods to give the reader a guide on how to use these for application on one-dimensional data. The introduced methods are supported by the spectral package, which has been published for the statistical environment prior to this article. Martin Seilmayer and Matthias Ratajczak Copyright © 2017 Martin Seilmayer and Matthias Ratajczak. All rights reserved. Solvability of the Brinkman-Forchheimer-Darcy Equation Thu, 20 Jul 2017 06:16:13 +0000 The nonlinear Brinkman-Forchheimer-Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy’s range. Piotr Skrzypacz and Dongming Wei Copyright © 2017 Piotr Skrzypacz and Dongming Wei. All rights reserved. Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis Sun, 16 Jul 2017 08:56:27 +0000 We examine an optimal way of eradicating rabies transmission from dogs into the human population, using preexposure prophylaxis (vaccination) and postexposure prophylaxis (treatment) due to public education. We obtain the disease-free equilibrium, the endemic equilibrium, the stability, and the sensitivity analysis of the optimal control model. Using the Latin hypercube sampling (LHS), the forward-backward sweep scheme and the fourth-order Range-Kutta numerical method predict that the global alliance for rabies control’s aim of working to eliminate deaths from canine rabies by 2030 is attainable through mass vaccination of susceptible dogs and continuous use of pre- and postexposure prophylaxis in humans. Joshua Kiddy K. Asamoah, Francis T. Oduro, Ebenezer Bonyah, and Baba Seidu Copyright © 2017 Joshua Kiddy K. Asamoah et al. All rights reserved. Nonlinear Waves in Rods and Beams of Power-Law Materials Thu, 13 Jul 2017 06:59:49 +0000 Some novel traveling waves and special solutions to the 1D nonlinear dynamic equations of rod and beam of power-law materials are found in closed forms. The traveling solutions represent waves of high elevation that propagates without change of forms in time. These waves resemble the usual kink waves except that they do not possess bounded elevations. The special solutions satisfying certain boundary and initial conditions are presented to demonstrate the nonlinear behavior of the materials. This note demonstrates the apparent distinctions between linear elastic and nonlinear plastic waves. Dongming Wei, Piotr Skrzypacz, and Xijun Yu Copyright © 2017 Dongming Wei et al. All rights reserved. Improved Combinatorial Benders Decomposition for a Scheduling Problem with Unrelated Parallel Machines Mon, 03 Jul 2017 06:15:01 +0000 This paper addresses the unrelated parallel machines scheduling problem with sequence and machine dependent setup times. Its goal is to minimize the makespan. The problem is solved by a combinatorial Benders decomposition. This method can be slow to converge. Therefore, three procedures are introduced to accelerate its convergence. The first procedure is a new method that consists of terminating the execution of the master problem when a repeated optimal solution is found. The second procedure is based on the multicut technique. The third procedure is based on the warm-start. The improved Benders decomposition scheme is compared to a mathematical formulation and a standard implementation of Benders decomposition algorithm. In the experiments, two test sets from the literature are used, with 240 and 600 instances with up to 60 jobs and 5 machines. For the first set the proposed method performs 21.85% on average faster than the standard implementation of the Benders algorithm. For the second set the proposed method failed to find an optimal solution in only 31 in 600 instances, obtained an average gap of 0.07%, and took an average computational time of 377.86 s, while the best results of the other methods were 57, 0.17%, and 573.89 s, respectively. Francisco Regis Abreu Gomes and Geraldo Robson Mateus Copyright © 2017 Francisco Regis Abreu Gomes and Geraldo Robson Mateus. All rights reserved. Finite Element Solution of an Unsteady MHD Flow through Porous Medium between Two Parallel Flat Plates Wed, 14 Jun 2017 10:56:28 +0000 Finite element solution of unsteady magnetohydrodynamics (MHD) flow of an electrically conducting, incompressible viscous fluid past through porous medium between two parallel plates is presented in the presence of a transverse magnetic field and Hall effect. The results obtained from some test cases are then compared with previous published work using the finite difference method (FDM). Numerical examples show that the finite element method (FEM) gives more accurate results in comparison with the finite difference method (FDM). AbdelLatif Sa’adAldin and Naji Qatanani Copyright © 2017 AbdelLatif Sa’adAldin and Naji Qatanani. All rights reserved. Urban Lead: Modeling Its Distribution and Effects on Children Sun, 11 Jun 2017 09:26:55 +0000 We model the transportation of lead from the atmosphere and from the surface of the soil simultaneously at the macroscale and mesoscale to study its health effects on children in Jersey City, NJ. We conceptualize Jersey City as an open system where lead is continuously emitted from a local smelting plant and a local power plant, deposited onto the surface soil of playgrounds, and ingested by children. The model is constructed using the diffusion-advection partial differential equation in three spatial dimensions and one temporal dimension with an initial condition and boundary conditions. The model is solved using the Crank-Nicolson numerical method at the macroscale to determine the deposition of lead from the smelting plant and the local power plant and at the mesoscale to refine the amount of lead deposition for the areas considered. We then determine the health consequences for the average child using the bioaccessibility of lead from soil to children, the bioavailability of ingested lead to the circulatory system, and the biological half-life of lead isotopes in the blood. The health effects on children from lead are directly proportional to the blood lead concentration. Zhixiong Chen, Yi Ding, Andrew Getz, and Bernard Lipat Copyright © 2017 Zhixiong Chen et al. All rights reserved. The Implied Risk Neutral Density Dynamics: Evidence from the S&P TSX 60 Index Sun, 11 Jun 2017 09:09:37 +0000 The risk neutral density is an important tool for analyzing the dynamics of financial markets and traders’ attitudes and reactions to already experienced shocks by financial markets as well as the potential ones. In this paper, we present a new method for the extraction information content from option prices. By eliminating bias caused by daily variation of contract maturity through a completely nonparametric technique based on kernel regression, we allow comparing evolution of risk neutral density and extracting from time continuous indicators that detect evolution of traders’ attitudes, risk perception, and belief homogeneity. This method is useful to develop trading strategies and monetary policies. Nessim Souissi Copyright © 2017 Nessim Souissi. All rights reserved. A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality Sun, 04 Jun 2017 09:02:23 +0000 In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than , then the disease-free equilibrium is globally asymptotically stable and if it is greater than , then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results. Bakary Traoré, Boureima Sangaré, and Sado Traoré Copyright © 2017 Bakary Traoré et al. All rights reserved. Per-Spectral Characterizations of Bicyclic Networks Wed, 31 May 2017 09:03:09 +0000 Spectral techniques are used for the study of several network properties: community detection, bipartition, clustering, design of highly synchronizable networks, and so forth. In this paper, we investigate which kinds of bicyclic networks are determined by their per-spectra. We find that the permanental spectra cannot determine sandglass graphs in general. When we restrict our consideration to connected graphs or quadrangle-free graphs, sandglass graphs are determined by their permanental spectra. Furthermore, we construct countless pairs of per-cospectra bicyclic networks. Tingzeng Wu and Huazhong Lü Copyright © 2017 Tingzeng Wu and Huazhong Lü. All rights reserved. Comparing an Approximate Queuing Approach with Simulation for the Solution of a Cross-Docking Problem Sun, 28 May 2017 00:00:00 +0000 Cross-docking is a logistics management concept in which products are temporarily unloaded at intermediate facilities and loaded onto output trucks to be sent to their final destination. In this paper, we propose an approximate nonstationary queuing model to size the number of docks to receive the trucks, so that their unloading will be as short as possible at the receiving dock, thus making the cross-docking process more efficient. It is observed that the stochastic queuing process may not reach the steady equilibrium state. A type of modeling that does not depend on the stationary characteristics of the process developed is applied. In order to measure the efficiency, performance, and possible adjustments of the parameters of the algorithm, an alternative simulation model is proposed using the Arena® software. The simulation uses analytic tools to make the problem more detailed, which is not allowed in the theoretical model. The computational analysis compares the results of the simulated model with the ones obtained with the theoretical algorithm, considering the queue length and the average waiting time of the trucks. Based on the results obtained, the simulation represented very well the proposed problem and possible changes can be easily detected with small adjustments in the simulated model. Roberta Briesemeister and Antônio G. N. Novaes Copyright © 2017 Roberta Briesemeister and Antônio G. N. Novaes. All rights reserved. Computational Methods for Solving Linear Fuzzy Volterra Integral Equation Sun, 28 May 2017 00:00:00 +0000 Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results. Jihan Hamaydi and Naji Qatanani Copyright © 2017 Jihan Hamaydi and Naji Qatanani. All rights reserved. On a Bivariate Spectral Homotopy Analysis Method for Unsteady Mixed Convection Boundary Layer Flow, Heat, and Mass Transfer due to a Stretching Surface in a Rotating Fluid Tue, 09 May 2017 10:04:17 +0000 A bivariate spectral homotopy analysis method (BSHAM) is extended to solutions of systems of nonlinear coupled partial differential equations (PDEs). The method has been used successfully to solve a nonlinear PDE and is now tested with systems. The method is based on a new idea of finding solutions that obey a rule of solution expression that is defined in terms of the bivariate Lagrange interpolation polynomials. The BSHAM is used to solve a system of coupled nonlinear partial differential equations modeling the unsteady mixed convection boundary layer flow, heat, and mass transfer due to a stretching surface in a rotating fluid, taking into consideration the effect of buoyancy forces. Convergence of the numerical solutions was monitored using the residual error of the PDEs. The effects of the flow parameters on the local skin-friction coefficient, the Nusselt number, and the Sherwood number were presented in graphs. Sandile S. Motsa and Zodwa G. Makukula Copyright © 2017 Sandile S. Motsa and Zodwa G. Makukula. All rights reserved.