Mathematical and Computational Applications
http://www.mdpi.com/journal/mca
Latest open access articles published in Math. Comput. Appl. at http://www.mdpi.com/journal/mca<![CDATA[MCA, Vol. 22, Pages 33: Surface Roughness Modeling Using Q-Sequence]]>
http://www.mdpi.com/2297-8747/22/2/33
Dynamical systems play a vital role in studying highly non-linear phenomena. One of the families of the dynamical systems is integer sequences. There is an integer sequence called Q-sequence: Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)); for n = 3, 4, …; and Q(1) = Q(2) = 1. It exhibits a unique chaotic-order that might help develop approximate models of highly nonlinear phenomena. We explore this possibility and show how to modify a segment of the Q-sequence so that the modified segment becomes an approximate model of surface roughness (a highly non-linear phenomena that results from the material removal processes (e.g., turning, milling, grinding, and so on). The Q-sequence-based models of surface roughness can be used to recreate the surface heights whenever necessary. As such, it is a helpful means for developing simulation systems for virtual manufacturing.Mathematical and Computational Applications2017-05-06222Article10.3390/mca22020033332297-87472017-05-06doi: 10.3390/mca22020033A.M.M. Ullah<![CDATA[MCA, Vol. 22, Pages 32: Topology on Soft Continuous Function Spaces]]>
http://www.mdpi.com/2297-8747/22/2/32
The concept of soft sets was initiated by Molodtsov. Then, some operations on soft sets were defined by Maji et al. Later on, the concept of soft topological space was introduced. In this paper, we introduce the concept of the pointwise topology of soft topological spaces. Finally, we investigate the properties of soft mapping spaces and the relationships between some soft mapping spaces.Mathematical and Computational Applications2017-04-27222Article10.3390/mca22020032322297-87472017-04-27doi: 10.3390/mca22020032Taha ÖztürkSadi Bayramov<![CDATA[MCA, Vol. 22, Pages 31: A New Smoothing Nonlinear Penalty Function for Constrained Optimization]]>
http://www.mdpi.com/2297-8747/22/2/31
In this study, a new smoothing nonlinear penalty function for constrained optimization problems is presented. It is proved that the optimal solution of the smoothed penalty problem is an approximate optimal solution of the original problem. Based on the smoothed penalty function, we develop an algorithm for finding an optimal solution of the optimization problems with inequality constraints. We further discuss the convergence of this algorithm and test this algorithm with three numerical examples. The numerical examples show that the proposed algorithm is feasible and effective for solving some nonlinear constrained optimization problems.Mathematical and Computational Applications2017-04-12222Article10.3390/mca22020031312297-87472017-04-12doi: 10.3390/mca22020031Touna YangNguyen BinhTran ThangDuong Hoa<![CDATA[MCA, Vol. 22, Pages 30: Broken Bar Diagnosis for Squirrel Cage Induction Motors Using Frequency Analysis Based on MCSA and Continuous Wavelet Transform]]>
http://www.mdpi.com/2297-8747/22/2/30
The importance of early fault detection in electric motors has attracted the attention of research groups, as the detection of incipient faults can prevent damage spreading and increase the lifetime of the motor. At present, studies have focused their attention on optimization procedures used for fault detection in induction machines to achieve a quick and easy-to-interpret assessment at an industrial level. This paper proposes an alternative approach based on the Continuous Wavelet Transform (CWT) for broken bar diagnosis in squirrel cage induction motors. This work uses the Motor Current Signature Analysis (MCSA) method to acquire the current signal of the induction motor. The novelty of this study lies in broken bar detection in electric machines operating at non-load by analyzing variations in the spectrum of the motor’s current signal. This way, the faults are presented as oscillations in the current signal spectrum. Additionally, a quantification of broken bars for the same type of motors operating at fullload is performed in this study. An experimental validation and the comparison with the Fast Fourier Transform (FFT) technique are provided to validate the proposed technique.Mathematical and Computational Applications2017-04-11222Article10.3390/mca22020030302297-87472017-04-11doi: 10.3390/mca22020030Danilo Granda ,Wilbert G. AguilarDiego Arcos-AvilesDanny Sotomayor<![CDATA[MCA, Vol. 22, Pages 29: Analysis of a Hand-Foot-Mouth Disease Model with Standard Incidence Rate and Estimation for Basic Reproduction Number]]>
http://www.mdpi.com/2297-8747/22/2/29
Firstly, an SEIR mathematical model with standard incidence rate is established to describe the transmission of Hand-Foot-Mouth disease (HFMD). The equilibrium of the nondimensionalized model is calculated and the basic reproduction number of the model is defined. In addition, the local stability of the equilibrium is analyzed via the characteristic roots of the Jacobian matrix at the equilibrium, respectively. Numerical simulations are given to confirm the theoretical results. Secondly, a formula aimed to estimate the basic reproduction number of the transmission of HFMD is deduced. As examples to make use of the formula, the basic reproduction number of the HFMD transmission of Singapore of years 2015 and 2016 is estimated based on the newly infected cases notified by the surveillance organizations, respectively. The formula can realize real time estimation for the basic reproduction number and does not need to estimate the transmission efficiency of HFMD between individuals.Mathematical and Computational Applications2017-04-08222Article10.3390/mca22020029292297-87472017-04-08doi: 10.3390/mca22020029Chunqing Wu<![CDATA[MCA, Vol. 22, Pages 28: An Algorithm Based on GSVD for Image Encryption]]>
http://www.mdpi.com/2297-8747/22/2/28
This paper represents a new image encryption algorithm based on modifying generalized singular value decomposition (GSVD) by decomposing the plain-image into two segments using GSVD with an exchanged key-image to produce the cipher-image. The exchanged key-image is used as an encrypting and decrypting image. Mathematically, this procedure is represented by transforming the plain-image’s matrix into two different matrices and applying the GSVD with the exchanged key-image’s matrix to obtain the cipher-image’s matrix. The two encoded segments can be kept in several places or assigned to a group of authorized persons. No one can obtain the information of the image easily without the knowledge of the decrypting key. This proposed algorithm is represented as one of the digital image encryption techniques used to enhance the security of images that have been sent between recipients.Mathematical and Computational Applications2017-03-28222Article10.3390/mca22020028282297-87472017-03-28doi: 10.3390/mca22020028Mohammed Al-KufiHayder HashimAmeer HusseinHind Mohammed<![CDATA[MCA, Vol. 22, Pages 27: Damping Characteristics of Viscoelastic Damping Structure under Coupled Condition]]>
http://www.mdpi.com/2297-8747/22/1/27
Temperature has an influence on damping characteristics of the viscoelastic damping structure. The change of the damping characteristics of the structure under the cycle load is a dynamic and coupled process. The hyperelastic-viscoelastic model was used to describe nonlinear deformation and viscoelasticity simultaneously. The temperature distribution and change of the damping characteristics under the coupled condition was analyzed by finite element method (FEM). The maximum value of the simulation results was in agreement with the one calculated by the formula in the literature. Dynamic stiffness and dissipated energy were obtained based on the hysteresis loop. Dynamic stiffness and dissipated energy gradually decreased with the increase of the temperature.Mathematical and Computational Applications2017-03-15221Article10.3390/mca22010027272297-87472017-03-15doi: 10.3390/mca22010027Jun WangDagang SunShizhong LiuXin Zhang<![CDATA[MCA, Vol. 22, Pages 26: Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique]]>
http://www.mdpi.com/2297-8747/22/1/26
The development of computational acoustics allows the simulation of sound generation and propagation in a complex environment. In particular, meshfree methods are widely used to solve acoustics problems through arbitrarily distributed field points and approximation smoothness flexibility. As a Lagrangian meshfree method, the smoothed particle hydrodynamics (SPH) method reduces the difficulty in solving problems with deformable boundaries, complex topologies, or multiphase medium. The traditional SPH method has been applied in acoustic simulation. This study presents the corrective smoothed particle method (CSPM), which is a combination of the SPH kernel estimate and Taylor series expansion. The CSPM is introduced as a Lagrangian approach to improve the accuracy when solving acoustic wave equations in the time domain. Moreover, a boundary treatment technique based on the hybrid meshfree and finite difference time domain (FDTD) method is proposed, to represent different acoustic boundaries with particles. To model sound propagation in pipes with different boundaries, soft, rigid, and absorbing boundary conditions are built with this technique. Numerical results show that the CSPM algorithm is consistent and demonstrates convergence with exact solutions. The main computational parameters are discussed, and different boundary conditions are validated as being effective for benchmark problems in computational acoustics.Mathematical and Computational Applications2017-03-15221Article10.3390/mca22010026262297-87472017-03-15doi: 10.3390/mca22010026Yong ZhangXu LiTao Zhang<![CDATA[MCA, Vol. 22, Pages 25: A Weakly Pareto Compliant Quality Indicator]]>
http://www.mdpi.com/2297-8747/22/1/25
In multi‐ and many‐objective optimization problems, the optimization target is to obtain a set of non‐dominated solutions close to the Pareto‐optimal front, well‐distributed, maximally extended and fully filled. Comparing solution sets is crucial in evaluating the performance of different optimization algorithms. The use of performance indicators is common in comparing those sets and, subsequently, optimization algorithms. Therefore, an effective performance indicator must encompass these features as a whole and, above all, it must be Pareto dominance compliant. Unfortunately, some of the known indicators often fail to properly reflect the quality of a solution set or cost a lot to compute. This paper demonstrates that the Degree of Approximation (DOA) quality indicator is a weakly Pareto compliant unary indicator that gives a good estimation of the match between the approximated front and the Pareto‐optimal front.Mathematical and Computational Applications2017-03-06221Article10.3390/mca22010025252297-87472017-03-06doi: 10.3390/mca22010025Emanuele DilettosoSanti RizzoNunzio Salerno<![CDATA[MCA, Vol. 22, Pages 24: Safeguarding against Inactivation Temperatures during Plasma Treatment of Skin: Multiphysics Model and Phase Field Method]]>
http://www.mdpi.com/2297-8747/22/1/24
One of the most appealing applications of cold plasmas is medical treatment of the skin. An important concern is the capability to safeguard the non-targeted cells against inactivation temperatures during the plasma treatment. Unfortunately, it is problematic to experimentally determine the highest transient temperatures in these cells during the plasma treatment. In the present work, a complete multiphysics model was built based on finite element analysis using phase field method coupled with heat transfer and fluid dynamics to study the discharge phenomenon of cold plasma with helium carrier gas ejected out of a tube for skin treatment. In such plasmas with carrier gas, the fractions of plasma constituents are small compared to the carrier gas, so thermofluid analysis is needed for the carrier gas as the major contributor to the fluid and heat flow. The phase field method has been used to capture the moving helium gas in air, which has enabled us to compute fluid dynamics parameters for each phase individually. In addition to computational fluid dynamic analyses, we have also considered heat transfer in the fluids and to the skin using the Fourier law of heat conduction, which led to a multiphysics system. In the present paper, various flow velocities and tube-to-target distances (TTDs) have been considered to reveal the dependence of the fluid discharge output parameters on the flow and efficiency of heat transfer to the skin and the surrounding environment. The built model is a useful tool for future development of plasma treatment devices and to safeguard the non-targeted cells against inactivation temperatures.Mathematical and Computational Applications2017-03-02221Article10.3390/mca22010024242297-87472017-03-02doi: 10.3390/mca22010024Mehrdad Shahmohammadi BeniKwan Yu<![CDATA[MCA, Vol. 22, Pages 23: Laplace Transform Homotopy Perturbation Method for the Two Dimensional Black Scholes Model with European Call Option]]>
http://www.mdpi.com/2297-8747/22/1/23
The Black Scholes model is a well-known and useful mathematical model in ﬁnancial markets. In this paper, the two-dimensional Black Scholes equation with European call option is studied. The explicit solution of this problem is carried out in the form of a Mellin–Ross function by using Laplace transform homotopy perturbation method. The solution example demonstrates that the proposed scheme is effective.Mathematical and Computational Applications2017-02-27221Article10.3390/mca22010023232297-87472017-02-27doi: 10.3390/mca22010023Kamonchat TrachooWannika SawangtongPanumart Sawangtong<![CDATA[MCA, Vol. 22, Pages 22: A Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme]]>
http://www.mdpi.com/2297-8747/22/1/22
In order to improve the flexibility of curves, a new five-point binary approximating subdivision scheme with two parameters is presented. The generating polynomial method is used to investigate the uniform convergence and C k -continuity of this scheme. In a special case, the five-point scheme changes into a four-point scheme, which can generate C 3 limit curves. The shape-preserving properties of the four-point scheme are analyzed, and a few examples are given to illustrate the efficiency and the shape-preserving effect of this special case.Mathematical and Computational Applications2017-02-24221Article10.3390/mca22010022222297-87472017-02-24doi: 10.3390/mca22010022Jieqing TanBo WangJun Shi<![CDATA[MCA, Vol. 22, Pages 21: An Initial Condition Optimization Approach for Improving the Prediction Precision of a GM(1,1) Model]]>
http://www.mdpi.com/2297-8747/22/1/21
Grey model GM(1,1) has attained excellent prediction accuracy with restricted data and has been broadly utilized in a range of areas. However, the GM(1,1) forecasting model sometimes yields large forecasting errors which directlyaffect the simulation and prediction precision directly. Therefore, the improvement of the GM(1,1) model is an essential issue, and the current study aims to enhance the prediction precision of the GM(1,1) model. Specifically, in order to improve the prediction precision of GM(1,1) model, it is necessary to consider improving the initial condition in the response function of the model. Consequently, the purpose of this paper is to put forward a new method to enhance the performance of the GM(1,1) model by optimizing its initial condition. The minimum sum of squared error was used to optimize the new initial condition of the model. The numerical outcomes show that the improved GM(1,1) model provides considerably better performance than traditional grey model GM(1,1) . The result demonstrates that the improved grey model GM(1,1) achieves the objective of minimizing the forecast errors.Mathematical and Computational Applications2017-02-22221Article10.3390/mca22010021212297-87472017-02-22doi: 10.3390/mca22010021Mahdi MadhiNorizan Mohamed<![CDATA[MCA, Vol. 22, Pages 20: Global Modulus-Based Synchronous Multisplitting Multi-Parameters TOR Methods for Linear Complementarity Problems]]>
http://www.mdpi.com/2297-8747/22/1/20
In 2013, Bai and Zhang constructed modulus-based synchronous multisplitting methods for linear complementarity problems and analyzed the corresponding convergence. In 2014, Zhang and Li studied the weaker convergence results based on linear complementarity problems. In 2008, Zhang et al. presented global relaxed non-stationary multisplitting multi-parameter method by introducing some parameters. In this paper, we extend Bai and Zhang’s algorithms and analyze global modulus-based synchronous multisplitting multi-parameters TOR (two parameters overrelaxation) methods. Moverover, the convergence of the corresponding algorithm in this paper are given when the system matrix is an H + -matrix.Mathematical and Computational Applications2017-02-21221Article10.3390/mca22010020202297-87472017-02-21doi: 10.3390/mca22010020Li-Tao ZhangTong-Xiang Gu<![CDATA[MCA, Vol. 22, Pages 18: Rape Plant Disease Recognition Method of Multi-Feature Fusion Based on D-S Evidence Theory]]>
http://www.mdpi.com/2297-8747/22/1/18
In view of the low accuracy and uncertainty of the traditional rape plant disease recognition relying on a single feature, this paper puts forward a rape plant disease recognition method based on Dempster-Shafer (D-S) evidence theory and multi-feature fusion. Firstly, color matrix and gray-level co-occurrence matrix are extracted as two kinds of features from rape plant images after processing. Then by calculating the Euclidean distance between the test samples and training samples, the basic probability assignment function can be constructed. Finally, the D-S combination rule of evidence is used to achieve fusion, and final recognition results are given by using the variance. This method is used to collect rape plant images for disease recognition, and recognition rate arrives at 97.09%. Compared with other methods, experimental results show that the method is more effective and with lower computational complexity.Mathematical and Computational Applications2017-02-15221Article10.3390/mca22010018182297-87472017-02-15doi: 10.3390/mca22010018Min HuXiangyu BuXiao SunZixi YuYaona Zheng<![CDATA[MCA, Vol. 22, Pages 19: On the Bias of the Maximum Likelihood Estimators of Parameters of the Weibull Distribution]]>
http://www.mdpi.com/2297-8747/22/1/19
Usually, the parameters of a Weibull distribution are estimated by maximum likelihood estimation. To reduce the biases of the maximum likelihood estimators (MLEs) of two-parameter Weibull distributions, we propose analytic bias-corrected MLEs. Two other common estimators of Weibull distributions, least-squares estimators and percentiles estimators, are also introduced. Based on a comparison of their performances in the simulation study, we strongly recommend the analytic bias-corrected MLEs for the parameters of Weibull distributions, especially when the sample size is small.Mathematical and Computational Applications2017-02-15221Article10.3390/mca22010019192297-87472017-02-15doi: 10.3390/mca22010019Man ChenZheng ZhangCen Cui<![CDATA[MCA, Vol. 22, Pages 17: New Scientiﬁc Contribution on the 2-D Subdomain Technique in Cartesian Coordinates: Taking into Account of Iron Parts]]>
http://www.mdpi.com/2297-8747/22/1/17
The most signiﬁcant assumptions in the subdomain technique (i.e., based on the formal resolution of Maxwell’s equations applied in subdomain) is deﬁned by: Theiron parts(i.e.,theteeth and the back-iron are considered to be inﬁnitely permeable, i.e., µiron → +∞, so that the saturation effect is neglected. In this paper, the authors present a new scientiﬁc contribution on improving of this method in two-dimensional (2-D) and in Cartesian coordinates by focusing on the consideration of iron. The subdomains connection is carried out in the two directions (i.e., x-andy-edges). Forexample,the improvement was performed by solving magnetostatic Maxwell’s equations for an air- or iron-cored coil supplied by a direct current. To evaluate the efﬁcacy of the proposed technique, the magnetic ﬂux density distributions have been compared with those obtained by the 2-D ﬁnite-element analysis (FEA). The semi-analytical results are in quite satisfying agreement with those obtained by the 2-D FEA, considering both amplitude and waveform.Mathematical and Computational Applications2017-02-10221Article10.3390/mca22010017172297-87472017-02-10doi: 10.3390/mca22010017Frédéric DubasKamel Boughrara<![CDATA[MCA, Vol. 22, Pages 16: On Existence and Uniqueness of Solutions to the Fuzzy Dynamic Equations on Time Scales]]>
http://www.mdpi.com/2297-8747/22/1/16
In this paper, we introduce a new metric on the space of fuzzy continuous functions on time scales by using the exponential function, e γ ( t , t 0 ) , where γ &gt; 0 is a constant. Then, we provide some conditions to prove an existence and uniqueness theorem for solutions to nonlinear fuzzy dynamic equations. Furthermore, we present three different examples including a practical example to illustrate the main results.Mathematical and Computational Applications2017-02-08221Article10.3390/mca22010016162297-87472017-02-08doi: 10.3390/mca22010016Omid Solaymani FardTayebeh Aliabdoli BidgoliAzim Rivaz<![CDATA[MCA, Vol. 22, Pages 15: An M/G/1 Retrial G-Queue with General Retrial Times and Working Breakdowns]]>
http://www.mdpi.com/2297-8747/22/1/15
This paper considers an M/G/1 retrial G-queue with general retrial times, in which the server is subject to working breakdowns and repairs. If the system is not empty during a normal service period, the arrival of a negative customer can cause the server breakdown, and the failed server still works at a lower service rate rather than stopping the service completely. Applying the embedded Markov chain, we obtain the necessary and sufﬁcient condition for the stability of the system. Using the supplementary variable method, we deal with the generating functions of the number of customers in the orbit. Various system performance measures are also developed. Finally, some numerical examples and a cost optimization analysis are presented.Mathematical and Computational Applications2017-02-02221Article10.3390/mca22010015152297-87472017-02-02doi: 10.3390/mca22010015Tao LiLiyuan Zhang<![CDATA[MCA, Vol. 22, Pages 14: Event Triggered Multi-Agent Consensus of DC Motors to Regulate Speed by LQR Scheme]]>
http://www.mdpi.com/2297-8747/22/1/14
In the presented paper, the leader-following consensus algorithm of a multi-agent system (MAS) is used along with the centralized event-triggering scheme to make the speed of the network-coupled multiple-motors synchronizable. In the proposed method, the updates for the controller are event-driven based on local information. Moreover, the basic consensus protocol is also revised such that the speed information of the motors is used in order to reach identical speed. The main beneﬁt of the planned event-triggered methodology is the energy saving by avoiding the continuous control of the system. As far as stability analysis of the system is concerned, a common Lyapunov function is incorporated to validate stability. The acquired results endorse the success of the proposed methodology.Mathematical and Computational Applications2017-01-26221Article10.3390/mca22010014142297-87472017-01-26doi: 10.3390/mca22010014Suhaib MasroorChen PengZain Ali<![CDATA[MCA, Vol. 22, Pages 13: New Unidimensional Indexes for China]]>
http://www.mdpi.com/2297-8747/22/1/13
A first principal component combines several indicators so as to maximize their internal consistency for measuring a construct. First principal components are extracted here from Swiss Economic Institute and World Bank datasets containing yearly societal indicators for China. These indicators are input to population-weighted regressions without recourse to survey sampling or probabilistic inference. The results demonstrate Chomskyan globalization and domestic credit as strong exogenous and endogenous predictors of Chinese per capita GDP. These encouraging findings, easily extendable to other nations, are brought by two new societal indexes with assured unidimensionality.Mathematical and Computational Applications2017-01-24221Article10.3390/mca22010013132297-87472017-01-24doi: 10.3390/mca22010013Gordon Bechtel <![CDATA[MCA, Vol. 22, Pages 12: Game Analysis of Low Carbonization for Urban Logistics Service Systems]]>
http://www.mdpi.com/2297-8747/22/1/12
To improve carbon efficiency for an urban logistics service system composed of a third-party logistics service provider (3PL) and an e-business enterprise, a low-carbon operation game between them was studied. Considering low carbon technology investment cost and sales expansion effect of low carbon level, profit functions for both players were constituted. Based on their different bargaining capabilities, in total, five types of game scenarios were designed. Through analytical solution, Nash Equilibria under different scenarios were obtained. By analyzing these equilibria, four major propositions were given, in which some key variables and the system performance indexes were compared. Results show that the best system yields could only be achieved under the fully cooperative situation. Limited cooperation only for carbon emission reduction does not benefit the system performance improvement. E-business enterprise-leading game’s performance overtook 3PL-leading ones.Mathematical and Computational Applications2017-01-22221Article10.3390/mca22010012122297-87472017-01-22doi: 10.3390/mca22010012Jidong GuoShugang Ma<![CDATA[MCA, Vol. 22, Pages 9: Analysis of Inflection and Singular Points on a Parametric Curve with a Shape Factor]]>
http://www.mdpi.com/2297-8747/22/1/9
The features of a class of cubic curves with a shape factor are analyzed by means of the theory of envelope and topological mapping. The effects of the shape factor on the cubic curves are made clear. Necessary and sufficient conditions are derived for the curve to have one or two inflection points, a loop or a cusp, or to be locally or globally convex. Those conditions are completely characterized by the relative position of the edge vectors of the control polygon and the shape factor. The results are summarized in a shape diagram, which is useful when the cubic parametric curves are used for geometric modeling. Furthermore, we discuss the influences of the shape factor on the shape diagram and the ability for adjusting the shape of the curve.Mathematical and Computational Applications2017-01-19221Article10.3390/mca2201000992297-87472017-01-19doi: 10.3390/mca22010009Zhi LiuChen LiJieqing TanXiaoyan Chen<![CDATA[MCA, Vol. 22, Pages 10: Quasi-Interpolation Operators for Bivariate Quintic Spline Spaces and Their Applications]]>
http://www.mdpi.com/2297-8747/22/1/10
Splines and quasi-interpolation operators are important both in approximation theory and applications. In this paper, we construct a family of quasi-interpolation operators for the bivariate quintic spline spaces S 5 3 ( Δ m n ( 2 ) ) . Moreover, the properties of the proposed quasi-interpolation operators are studied, as well as its applications for solving the two-dimensional Burgers’ equation and image reconstruction. Some numerical examples show that these methods, which are easy to implement, provide accurate results.Mathematical and Computational Applications2017-01-19221Article10.3390/mca22010010102297-87472017-01-19doi: 10.3390/mca22010010Rengui YuChungang ZhuXianmin HouLi Yin<![CDATA[MCA, Vol. 22, Pages 11: A New Approximation Method with High Order Accuracy]]>
http://www.mdpi.com/2297-8747/22/1/11
In this paper, we propose a new multilevel univariate approximation method with high order accuracy using radial basis function interpolation and cubic B-spline quasi-interpolation. The proposed approach includes two schemes, which are based on radial basis function interpolation with less center points, and cubic B-spline quasi-interpolation operator. Error analysis shows that our method produces higher accuracy compared with other approaches. Numerical examples demonstrate that the proposed scheme is effective.Mathematical and Computational Applications2017-01-19221Article10.3390/mca22010011112297-87472017-01-19doi: 10.3390/mca22010011Ziwu Jiang<![CDATA[MCA, Vol. 22, Pages 8: A Sparse Representation Algorithm for Effective Photograph Retrieval]]>
http://www.mdpi.com/2297-8747/22/1/8
Searching through information based on a photograph, which may contain graphics and images, has become a popular trend, such as in electronic books, journals, and products. Although many context-based methods have been proposed to retrieve images, most work focuses on selecting appropriate features for different objects. In the present study, we apply sparse representation to simultaneously retrieve image and graphics from a photograph. The sparse vector can be regarded as the similarity between the query photograph and dataset. The image with the largest entry (or several largest entries) can be assigned as the retrieved result. In the sparse representation framework, the common image features are used. Experimental results demonstrate that if the similarity vector in photograph retrieval is sparse, feature extraction is no longer critical. Compared with similar works in photograph retrieval, the proposed method has better retrieval accuracy.Mathematical and Computational Applications2017-01-18221Article10.3390/mca2201000882297-87472017-01-18doi: 10.3390/mca22010008Hong-Bo ZhangQing LeiBi-Neng ZhongJi-Xiang DuDuan-Sheng Chen<![CDATA[MCA, Vol. 22, Pages 4: Design Optimization of the Lines of the Bulbous Bow of a Hull Based on Parametric Modeling and Computational Fluid Dynamics Calculation]]>
http://www.mdpi.com/2297-8747/22/1/4
To reduce the ship wave-making resistance, the lines of the bulbous bow of a hull are optimized by an automatic optimization platform at the ship design stage. Parametric modeling was applied to the hull by using non-uniform rational basis spline (NURBS). The Rankine-source panel method was used to calculate the wave-making resistance. A hybrid optimization strategy was applied to achieve the optimization goal. A Ro-Ro ship was taken as an example to illustrate the optimization method adopted, with the objective to minimize the wave-making resistance. The optimization results show that wave-making resistance obviously reduces and the wave-shape of the near bow becomes gentle after the lines of the bulbous bow of the hull are optimized, which demonstrates the validity of the proposed optimization design strategy.Mathematical and Computational Applications2017-01-16221Article10.3390/mca2201000442297-87472017-01-16doi: 10.3390/mca22010004Weilin LuoLinqiang Lan<![CDATA[MCA, Vol. 22, Pages 7: Spatiotemporal Dynamics of a Delayed and Diffusive Viral Infection Model with Logistic Growth]]>
http://www.mdpi.com/2297-8747/22/1/7
Viruses have important influences on human health: they not only cause some common diseases, but also cause serious illnesses. Moreover, the conventional medicines usually fail to prevent or treat them, and viral infections are hard to treat because viruses live inside the body’s cells. However, some mathematical models can help to understand the viral transmission mechanism and control viral diseases. In this paper, a delayed viral infection model with spatial diffusion and logistic growth is presented. The asymptotic stability of nonnegative uniform steady states is investigated by utilizing the linearized method and constructing the proper Lyapunov functional, respectively. The existence of Hopf bifurcation from the positive equilibrium point is established by analyzing the corresponding characteristic equation and the direction of bifurcation, and the properties of bifurcating periodic solutions are derived by the aid of the normal form theory for partial functional differential equations. Then, the cross-diffusion system is introduced. Furthermore, some numerical simulations are carried, out and discussions are given.Mathematical and Computational Applications2017-01-14221Article10.3390/mca2201000772297-87472017-01-14doi: 10.3390/mca22010007Kejun Zhuang<![CDATA[MCA, Vol. 22, Pages 6: Acknowledgement to Reviewers of MCA in 2016]]>
http://www.mdpi.com/2297-8747/22/1/6
The editors of MCA would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016.[...]Mathematical and Computational Applications2017-01-12221Editorial10.3390/mca2201000662297-87472017-01-12doi: 10.3390/mca22010006 MCA Editorial Office<![CDATA[MCA, Vol. 22, Pages 5: Optimal Hedging with Options and Futures against Price Risk and Background Risk]]>
http://www.mdpi.com/2297-8747/22/1/5
On the condition that both futures and options exist in the markets for hedging, this paper examines the optimal hedging strategy under price risk and background risk. Compared with the previous research, which has studied options hedging against basis risk and production risk being extended to options and futures hedging against price risk and background risk, we proposed a model and have taken the budget of buying options into consideration. The model is fairly general and some existing models are special cases of it. We firstly derive the necessary and sufficient conditions that guarantee the optimality of an under-hedge, a full-hedge and an over-hedge of futures for the risk-averse utility. Then, sufficient conditions are stipulated under which an over-hedge is optimal. Furthermore, we propose a program minimizing of tail conditional expectation (TCE), which is inherently equivalent to the risk measure of expected shortfall risk (ES) or the conditional VaR (CVaR) under the continuous-time framework. Finally, we find that ES, in our proposed model, is significantly smaller than the one in the model of options hedging only. Therefore, the results emphasize the need for combining futures hedging and options hedging, and it also shows that imposing background risk, whether it be additive or multiplicative, always has a great impact on the hedging efficiency. We also present some sensitivities of the relevant parameters to provide some suggestions for the investors.Mathematical and Computational Applications2017-01-04221Article10.3390/mca2201000552297-87472017-01-04doi: 10.3390/mca22010005Xing YuHongguo Sun<![CDATA[MCA, Vol. 22, Pages 3: Robust Object Tracking in Infrared Video via Adaptive Weighted Patches]]>
http://www.mdpi.com/2297-8747/22/1/3
With the quick development of computer and electronic techniques, infrared sensor-based object tracking has become a hot research topic in recent years. However, infrared object tracking is still a challenging task due to low resolution, lack of representing information, and occlusion. In this work, we present an adaptive weighted patch-based infrared object tracking scheme. First, the candidate local region is divided into non-overlapping sub regions, and a set of belief weights is set on these patches. After this, a particle filtering-based infrared object tracking system is realized. In the last, the belief weight of each patch is evaluated based on the linear discriminative analysis (LDA) and particle sampling scheme. Experimental results on challenging infrared sequences show that the proposed algorithm can effectively locate the tracking object.Mathematical and Computational Applications2016-12-30221Article10.3390/mca2201000332297-87472016-12-30doi: 10.3390/mca22010003Jiangtao WangJingai Zhang<![CDATA[MCA, Vol. 22, Pages 2: A Novel, Energy-Aware Task Duplication-Based Scheduling Algorithm of Parallel Tasks on Clusters]]>
http://www.mdpi.com/2297-8747/22/1/2
Increasing energy has become an important issue in high performance clusters. To balance the energy and performance, we proposed a novel, energy-aware duplication-based scheduling (NEADS). An existing energy-aware duplication-based algorithm replicates all qualified predecessor tasks in a bottom-up manner. Some tasks without direct relation may be replicated to the same processor, which cannot reduce the communication energy. Instead, the computation overhead may be increased. In contrast, the proposed algorithm only replicates the directly correlated predecessor tasks in the energy threshold range without lengthening the schedule length. The proposed algorithm is compared with the non-duplication algorithm and existing duplicated-based algorithm. Extensive experimental results show that the proposed algorithm can effectively reduce energy consumption in various applications. It has advantages over other algorithms on computation-intensive applications.Mathematical and Computational Applications2016-12-27221Article10.3390/mca2201000222297-87472016-12-27doi: 10.3390/mca22010002Aihua LiangYu Pang<![CDATA[MCA, Vol. 22, Pages 1: Optimization of Setting Take-Profit Levels for Derivative Trading]]>
http://www.mdpi.com/2297-8747/22/1/1
This paper develops an optimal stopping rule by characterizing the take-profit level. The optimization problem is modeled by geometric Brownian motion with two switchable regimes and solved by stochastic calculation. A closed-form profitability function for the trading strategies is given, and based on which the optimal take-profit level is numerically achievable with small cost of computational complexity.Mathematical and Computational Applications2016-12-22221Article10.3390/mca2201000112297-87472016-12-22doi: 10.3390/mca22010001Xiaodong RuiYue LiuAijun YangHongqiang YangChengcui Zhang<![CDATA[MCA, Vol. 21, Pages 50: Reliable Network Interdiction Models with Multiple Unit Costs]]>
http://www.mdpi.com/2297-8747/21/4/50
This paper proposes a reliable network interdiction model with multiple unit costs, which maximizes the minimum arrival cost of the invader to the sink by setting obstacles on some arcs with limited resources in the given network. In other words, given a graph with a source and a sink, several arcs will be selected with limited resources such that each path contains as many weights as possible. This model needs to be transferred into a bilevel program because its constraints can hardly be listed explicitly even for a graph with a moderate size, because the number of paths between any two given points increases exponentially according to the size of the graph. This bilevel model is equivalent to an integer model with a low degree number of constraints by converting the inner programming to a shortest path problem. We first prove that this problem is non-deterministic polynomial-time (NP)-hard. Secondly, we reduce the number of constraints to the first power from the exponential degree by using the dual technique. Lastly, the national railway network is used to show the feasibility of our method.Mathematical and Computational Applications2016-12-14214Article10.3390/mca21040050502297-87472016-12-14doi: 10.3390/mca21040050Jia ZhaoGang Sun<![CDATA[MCA, Vol. 21, Pages 48: Rational Spline Image Upscaling with Constraint Parameters]]>
http://www.mdpi.com/2297-8747/21/4/48
Image interpolation is one of key contents in image processing. We present an interpolation algorithm based on a rational function model with constraint parameters. Firstly, based on the construction principle of the rational function, the detection threshold is selected through contour analysis. The smooth and non-smooth areas are interpolated by bicubic interpolation and general rational interpolation, respectively. In order to enhance the contrast in non-smooth areas and preserve the details, the parameter optimization technique is applied to get optimal shape parameters. Experimental results on benchmark test images demonstrate that the proposed method achieves competitive performance with the state-of-the-art interpolation algorithms, especially in image details and texture features.Mathematical and Computational Applications2016-12-13214Article10.3390/mca21040048482297-87472016-12-13doi: 10.3390/mca21040048Xunxiang YaoYunfeng ZhangFangxun BaoCaiming Zhang<![CDATA[MCA, Vol. 21, Pages 49: VRP-GMRES(m) Iteration Algorithm for Fast Multipole Boundary Element Method]]>
http://www.mdpi.com/2297-8747/21/4/49
To solve large scale linear equations involved in the Fast Multipole Boundary Element Method (FM-BEM) efficiently, an iterative method named the generalized minimal residual method (GMRES(m)) algorithm with Variable Restart Parameter (VRP-GMRES(m)) algorithm is proposed. By properly changing a variable restart parameter for the GMRES(m) algorithm, the iteration stagnation problem resulting from improper selection of the parameter is resolved efficiently. Based on the framework of the VRP-GMRES(m) algorithm and the relevant properties of generalized inverse matrix, the projection of the error vector r m + 1 on r m is deduced. The result proves that the proposed algorithm is not only rapidly convergent but also highly accurate. Numerical experiments further show that the new algorithm can significantly improve the computational efficiency and accuracy. Its superiorities will be much more remarkable when it is used to solve larger scale problems. Therefore, it has extensive prospects in the FM-BEM field and other scientific and engineering computing.Mathematical and Computational Applications2016-12-13214Article10.3390/mca21040049492297-87472016-12-13doi: 10.3390/mca21040049Chunxiao YuCuihuan RenXueting Bai<![CDATA[MCA, Vol. 21, Pages 47: Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II)]]>
http://www.mdpi.com/2297-8747/21/4/47
The weight coefficients of the diaphragm spring depend on experiences in the traditional optimization. However, this method not only cannot guarantee the optimal solution but it is also not universal. Therefore, a new optimization target function is proposed. The new function takes the minimum of average compress force changing of the spring and the minimum force of the separation as total objectives. Based on the optimization function, the result of the clutch diaphragm spring in a car is analyzed by the non-dominated sorting genetic algorithm (NSGA-II) and the solution set of Pareto is obtained. The results show that the pressing force of the diaphragm spring is improved by 4.09% by the new algorithm and the steering separation force is improved by 6.55%, which has better stability and steering portability. The problem of the weight coefficient in the traditional empirical design is solved. The pressing force of the optimized diaphragm spring varied slightly during the abrasion range of the friction film, and the manipulation became remarkably light.Mathematical and Computational Applications2016-12-03214Article10.3390/mca21040047472297-87472016-12-03doi: 10.3390/mca21040047Junchao ZhouChun WangJunjun Zhu<![CDATA[MCA, Vol. 21, Pages 45: Solving a Fabrication Lot-Size and Shipping Frequency Problem with an Outsourcing Policy and Random Scrap]]>
http://www.mdpi.com/2297-8747/21/4/45
Operating in unstable and competitive globalized markets, management of today’s transnational enterprises continually searches for different alternatives to maintain product quality, streamline production activities, and reduce overall operating costs, particularly in their internal supply chain system. With the aim of revealing and offering insight information to support managerial decision making, this study explores the optimal replenishment lot-size and shipping frequency problem for an intra-supply chain system with a partial outsourcing policy and random scrap. In this study, the demand of a product is partially outsourced and partially fabricated by the production units, to release the workload of machine and smooth production schedule. During the fabrication process, a portion of random scrap items is produced, and finished products are distributed to sales locations using a multiple-shipment policy. The objective is to simultaneously determine an optimal fabrication lot-size and shipping frequency decisions that minimize the overall expected costs for such an intra-supply chain system. Mathematical modeling and optimization methods are used to solve the problem. Moreover, through the use of a numerical example and sensitivity analyses, various important insights with regard to the joint effects of the partial outsourcing policy and random scrap on the optimal solutions are revealed to support managerial decision making.Mathematical and Computational Applications2016-11-17214Article10.3390/mca21040045452297-87472016-11-17doi: 10.3390/mca21040045Yuan-Shyi ChiuGang-Ming LiangSinga Chiu<![CDATA[MCA, Vol. 21, Pages 46: A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems]]>
http://www.mdpi.com/2297-8747/21/4/46
A 5(4) pair of embedded explicit trigonometrically-fitted Runge–Kutta–Nyström (EETFRKN) methods especially designed for the numerical integration of second order initial value problems with oscillatory solutions is presented in this paper. Algebraic order analysis and the interval of absolute stability for the new method are also discussed. The new method is capable of integrating the test equation y ″ = − w 2 y . The new method is much more efficient than the other existing Runge–Kutta and Runge–Kutta–Nyström methods.Mathematical and Computational Applications2016-11-17214Article10.3390/mca21040046462297-87472016-11-17doi: 10.3390/mca21040046Musa DembaNorazak SenuFudziah Ismail<![CDATA[MCA, Vol. 21, Pages 44: Adjustable Bézier Curves with Simple Geometric Continuity Conditions]]>
http://www.mdpi.com/2297-8747/21/4/44
This paper aims to simplify the continuity conditions of Bézier curves. For this purpose, a special family of Bézier curves with three parameters, to be called adjustable Bézier curves, is constructed. They have the same structure as the quartic Bézier curves. The newly constructed curves possess some of the basic properties of Bézier curves, such as the convex hull property, symmetry, geometric invariance, etc., and they have shape adjustability. Moreover, under the geometric continuity of order 1 ( G 1 ) conditions of the usual Bézier curves, the adjustable Bézier curves can reach geometric continuity of order k ( G k ); here, k is one of the parameters of the newly constructed curves. The recursive evaluation algorithm of the new curves is provided. We also discuss how to construct the adjustable Bézier curves with a given tangent polygon. Numerical examples illustrate the correctness and validity of the proposed method.Mathematical and Computational Applications2016-11-17214Article10.3390/mca21040044442297-87472016-11-17doi: 10.3390/mca21040044Lanlan Yan<![CDATA[MCA, Vol. 21, Pages 43: Numerical Investigation of Copper-Water (Cu-Water) Nanofluid with Different Shapes of Nanoparticles in a Channel with Stretching Wall: Slip Effects]]>
http://www.mdpi.com/2297-8747/21/4/43
The present study is focused on the presentation of a numerical solution for copper-water nanofluid through a stretching channel with spherical and cylindrical shape nanoparticles. The analysis of nanofluid in a channel with stretching walls under slip effects is made by introducing the conservation equation of nanoparticle volume fraction into Hamilton-Crosser’s nanofluid model. Governing partial differential equations are transformed into nonlinear ordinary differential equations by applying similarity transformation and then solved with the help of shooting method. The effects of different physical parameters on the rheology of nanofluids’ particles are presented in tabulation and pictorial representation. The study reveals that the thermal boundary layer thickness increases by increasing the solid volume fraction.Mathematical and Computational Applications2016-10-31214Article10.3390/mca21040043432297-87472016-10-31doi: 10.3390/mca21040043Jawad RazaAzizah RohniZurni Omar<![CDATA[MCA, Vol. 21, Pages 42: Parameters Optimization of a Hydraulic Buffer System for Belt Arrestor in Downward Belt Conveyors]]>
http://www.mdpi.com/2297-8747/21/4/42
Hydraulic buffer systems play a significant role in energy absorption and improving belt arrest reliability in downward belt conveyors. In order give hydraulic buffer systems more preferable buffer properties, a parameters optimization method based on a reference model is proposed. Firstly, the working principle of a hydraulic buffer system for a belt arrestor is provided. Secondly, the mathematical model of the system is built and a reference model of buffer chamber pressure is constructed utilizing a second-order system. Furthermore, a genetic algorithm is introduced to optimize the system parameters. Finally, some simulation examples are carried out on the Simulink software. The simulation results show that the pressure peak in buffer process can drop down and that pressure fluctuation in buffer end processes decrease substantially after optimization. The parameters optimization method for hydraulic buffer systems is applicable to different structure parameters of the buffer cylinder.Mathematical and Computational Applications2016-10-19214Article10.3390/mca21040042422297-87472016-10-19doi: 10.3390/mca21040042Wenjun YinYinwei YangZhanyu WangJing Xu<![CDATA[MCA, Vol. 21, Pages 41: A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation]]>
http://www.mdpi.com/2297-8747/21/4/41
Traditional first order JWKB method ( = : ( J W K B ) 1 ) is a conventional semiclassical approximation method mainly used in quantum mechanical systems for accurate solutions. ( J W K B ) 1 general solution of the Time Independent Schrodinger’s Equation (TISE) involves application of the conventional asymptotic matching rules to give the accurate wavefunction in the Classically Inaccessible Region (CIR) of the related quantum mechanical system. In this work, Bessel Differential Equation of the first order ( = : ( B D E ) 1 ) is chosen as a mathematical model and its ( J W K B ) 1 solution is obtained by first transforming into the normal form via the change of independent variable. The ( J W K B ) 1 general solution for appropriately chosen initial values in both normal and standard form representations is analyzed via the generalized ( J W K B ) 1 asymptotic matching rules regarding the S ˜ i j matrix elements given in the literature. Instead of applying the common ( J W K B ) 1 asymptotic matching rules relying on the physical nature of the quantum mechanical system, i.e., a physically acceptable (normalizable) wavefunction, a pure semiclassical analysis is studied via the ( B D E ) 1 model mathematically. Finally, an application to a specific case of the exponential potential decorated quantum mechanical bound state problem is presented.Mathematical and Computational Applications2016-10-19214Article10.3390/mca21040041412297-87472016-10-19doi: 10.3390/mca21040041Coşkun Deniz<![CDATA[MCA, Vol. 21, Pages 39: A Model Proposal for a Multi-Objective and Multi-Criteria Vehicle Assignment Problem: An Application for a Security Organization]]>
http://www.mdpi.com/2297-8747/21/4/39
Law enforcement agencies have great importance to provide peace and prosperity for the community. If the quality level of policing services is high, stability within the country will also increase. The law enforcement authorities transfer their policing services to people by means of using tools and equipment. In this study, this subject has been studied in order to improve the service quality of motor vehicles to increase efficiency in the assignments area. For the assignment of the vehicle, four main criteria and fifteen sub-criteria are defined. The criteria’s weights achieving the desired goal are calculated using the Analytic Network Process (ANP). The obtained weights have been subjected to the evaluation of performance in terms of each vehicle and each region where it can be assigned. The decision model with four basic objectives, containing service, cost, time and usage of technical capacity to ensure the use of vehicles at optimum efficiency, is designed. After weighting of the criteria, a mathematical model aiming at maximizing the service and the effectiveness of using the technical capacity of vehicle and minimizing the time and cost has been developed. The results are compared with the current situation. The study has been tested with three different scenarios having different objective priorities.Mathematical and Computational Applications2016-09-26214Article10.3390/mca21040039392297-87472016-09-26doi: 10.3390/mca21040039Engin AcarHakan Aplak<![CDATA[MCA, Vol. 21, Pages 40: The Modeling and Calculation of the Heading Machine Based on Differential Geometry]]>
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The kinematic sketch of the heading machine’s cutting part is plotted and the kinematic relation is analyzed. The pose-attitude model of the cutting part is derived from the geometry method, and the velocity and acceleration relations are derived by the differential geometry method. According to the recurrence relation among the pose-attitude, the velocities and the accelerations, the numerical solving strategy is designed. The nonlinear part of the kinematics model is solved by the Newton iterative method. The kinematics model is simulated by MATLAB. The trigonometric functions are avoided by using the differential geometry method, and the derivation process and the results are simplified simultaneously. The simulation results give the curves of each kinematic parameter which verifies the validity of the kinematic model.Mathematical and Computational Applications2016-09-23214Article10.3390/mca21040040402297-87472016-09-23doi: 10.3390/mca21040040Long BaiLu-han MaXin-sheng Ge<![CDATA[MCA, Vol. 21, Pages 38: Cost Minimization for a Multi-Product Fabrication-Distribution Problem with Commonality, Postponement, and Quality Assurance]]>
http://www.mdpi.com/2297-8747/21/3/38
To gain more competitive advantages and attract more customers from the turbulent business environment, manufacturing firms today must offer a wide variety of products to marketplaces. The existence of component commonality in multi-product fabrication planning enables managers to reevaluate different production design alternatives to lower overall production relevant costs. Motivated by assisting managers of manufacturing firms in gaining competitive advantages, maximizing machine utilization, and reducing overall quality and fabrication-distribution costs, this study explores a multi-product fabrication-distribution problem with component commonality, postponement, and quality assurance. A two-stage single-machine production scheme with the reworking of repairable nonconforming items is proposed. The first stage fabricates common intermediate components for all products, and the second stage produces and distributes end products under a common cycle time policy. Mathematical modeling and optimization techniques are utilized to derive the optimal fabrication-distribution policy that minimizes the expected total system costs of the problem. Finally, we provide a numerical example with sensitivity analyses to not only show practical uses of the obtained results, but also demonstrate that the proposed production scheme is beneficial in terms of cost savings and cycle time reduction as compared to that in a single-stage production scheme. The research results enable manufacturers to gain more competitive advantages in the turbulent global business environment.Mathematical and Computational Applications2016-09-14213Article10.3390/mca21030038382297-87472016-09-14doi: 10.3390/mca21030038Singa ChiuJyun-Sian KuoVictoria ChiuYuan-Shyi Chiu<![CDATA[MCA, Vol. 21, Pages 37: Fuzzy Grey Prediction-Based Particle Filter for Object Tracking]]>
http://www.mdpi.com/2297-8747/21/3/37
A particle filter is a powerful tool for object tracking based on sequential Monte Carlo methods under a Bayesian estimation framework. A major challenge for a particle filter in object tracking is how to allocate particles to a high-probability density area. A particle filter does not take into account the historical prior information on the generation of the proposal distribution and, thus, it cannot approximate posterior density well. Therefore, a new fuzzy grey prediction-based particle filter (called FuzzyGP-PF) for object tracking is proposed in this paper. First, a new prediction model which was based on fuzzy mathematics theory and grey system theory was established, coined the Fuzzy-Grey-Prediction (FGP) model. Then, the history state sequence is utilized as prior information to predict and sample a part of particles for generating the proposal distribution in the particle filter. Simulations are conducted in the context of two typical maneuvering motion scenarios and the results indicate that the proposed FuzzyGP-PF algorithm can exhibit better overall performance in object tracking.Mathematical and Computational Applications2016-08-23213Article10.3390/mca21030037372297-87472016-08-23doi: 10.3390/mca21030037Lian YangZhangping Lu<![CDATA[MCA, Vol. 21, Pages 36: On Generalized Double Statistical Convergence of Order α in Intuitionistic Fuzzy Normed Spaces]]>
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Our goal in this work is to introduce the notion V , λ ( I ) 2 -summability and ideal λ-double statistical convergence of order α with respect to the intuitionistic fuzzy norm μ , v . We also make some observations about these spaces and prove some inclusion relations.Mathematical and Computational Applications2016-08-10213Article10.3390/mca21030036362297-87472016-08-10doi: 10.3390/mca21030036Ekrem Savaş<![CDATA[MCA, Vol. 21, Pages 33: The Cubic α-Catmull-Rom Spline]]>
http://www.mdpi.com/2297-8747/21/3/33
By extending the definition interval of the standard cubic Catmull-Rom spline basis functions from [0,1] to [0,α], a class of cubic Catmull-Rom spline basis functions with a shape parameter α, named cubic α-Catmull-Rom spline basis functions, is constructed. Then, the corresponding cubic α-Catmull-Rom spline curves are generated based on the introduced basis functions. The cubic α-Catmull-Rom spline curves not only have the same properties as the standard cubic Catmull-Rom spline curves, but also can be adjusted by altering the value of the shape parameter α even if the control points are fixed. Furthermore, the cubic α-Catmull-Rom spline interpolation function is discussed, and a method for determining the optimal interpolation function is presented.Mathematical and Computational Applications2016-08-09213Article10.3390/mca21030033332297-87472016-08-09doi: 10.3390/mca21030033Juncheng LiSheng Chen<![CDATA[MCA, Vol. 21, Pages 34: A Comparison of Information Criteria in Clustering Based on Mixture of Multivariate Normal Distributions]]>
http://www.mdpi.com/2297-8747/21/3/34
Clustering analysis based on a mixture of multivariate normal distributions is commonly used in the clustering of multidimensional data sets. Model selection is one of the most important problems in mixture cluster analysis based on the mixture of multivariate normal distributions. Model selection involves the determination of the number of components (clusters) and the selection of an appropriate covariance structure in the mixture cluster analysis. In this study, the efficiency of information criteria that are commonly used in model selection is examined. The effectiveness of information criteria has been determined according to the success in the selection of the number of components and in the selection of an appropriate covariance matrix.Mathematical and Computational Applications2016-08-01213Article10.3390/mca21030034342297-87472016-08-01doi: 10.3390/mca21030034Serkan AkogulMurat Erisoglu<![CDATA[MCA, Vol. 21, Pages 35: On the O(1/n) Convergence Rate of the Auxiliary Problem Principle for Separable Convex Programming and Its Application to the Power Systems Multi-Area Economic Dispatch Problem]]>
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The auxiliary problem principle has been widely applied in power systems to solve the multi-area economic dispatch problem. Although the effectiveness and correctness of the auxiliary problem principle method have been demonstrated in relevant literatures, the aspect connected with accurate estimate of its convergence rate has not yet been established. In this paper, we prove the O ( 1 / n ) convergence rate of the auxiliary problem principle method.Mathematical and Computational Applications2016-07-29213Article10.3390/mca21030035352297-87472016-07-29doi: 10.3390/mca21030035Yaming RenZhongxian Chen<![CDATA[MCA, Vol. 21, Pages 31: Analytic Properties of the Sum B1(h, k)]]>
http://www.mdpi.com/2297-8747/21/3/31
In this paper, with the help of the Hardy and Dedekind sums we will give many properties of the sum B 1 ( h , k ) , which was defined by Cetin et al. Then we will give the connections of this sum with the other well-known finite sums such as the Dedekind sums, the Hardy sums, the Simsek sums Y ( h , k ) and the sum C 1 ( h , k ) . By using the Fibonacci numbers and two-term polynomial relation, we will also give a new property of the sum B 1 ( h , k ) .Mathematical and Computational Applications2016-07-21213Article10.3390/mca21030031312297-87472016-07-21doi: 10.3390/mca21030031Elif Cetin<![CDATA[MCA, Vol. 21, Pages 32: Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations]]>
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The purpose of this work is to introduce a new kind of finite difference formulation inspired from Fourier analysis, for reaction-diffusion equations. Compared to classical schemes, the proposed scheme is much more accurate and has interesting stability properties. Convergence properties and stability of the scheme are discussed. Numerical examples are provided to show better performance of the method, compared with other existing methods in the literature.Mathematical and Computational Applications2016-07-20213Article10.3390/mca21030032322297-87472016-07-20doi: 10.3390/mca21030032Utku ErdoğanKenan AkarbulutNeşet Tan<![CDATA[MCA, Vol. 21, Pages 30: A Two-Step Global Alignment Method for Feature-Based Image Mosaicing]]>
http://www.mdpi.com/2297-8747/21/3/30
Image mosaicing sits at the core of many optical mapping applications with mobile robotic platforms. As these platforms have been evolving rapidly and increasing their capabilities, the amount of data they are able to collect is increasing drastically. For this reason, the necessity for efficient methods to handle and process such big data has been rising from different scientific fields, where the optical data provides valuable information. One of the challenging steps of image mosaicing is finding the best image-to-map (or mosaic) motion (represented as a planar transformation) for each image while considering the constraints imposed by inter-image motions. This problem is referred to as Global Alignment (GA) or Global Registration, which usually requires a non-linear minimization. In this paper, following the aforementioned motivations, we propose a two-step global alignment method to obtain globally coherent mosaics with less computational cost and time. It firstly tries to estimate the scale and rotation parameters and then the translation parameters. Although it requires a non-linear minimization, Jacobians are simple to compute and do not contain the positions of correspondences. This allows for saving computational cost and time. It can be also used as a fast way to obtain an initial estimate for further usage in the Symmetric Transfer Error Minimization (STEMin) approach. We presented experimental and comparative results on different datasets obtained by robotic platforms for mapping purposes.Mathematical and Computational Applications2016-07-20213Article10.3390/mca21030030302297-87472016-07-20doi: 10.3390/mca21030030Armagan Elibol<![CDATA[MCA, Vol. 21, Pages 29: The Average Lower 2-Domination Number of Wheels Related Graphs and an Algorithm]]>
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The problem of quantifying the vulnerability of graphs has received much attention nowadays, especially in the field of computer or communication networks. In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the average lower 2-domination number of a graph is a measure of the graph vulnerability and it is defined by γ 2 a v ( G ) = 1 | V ( G ) | ∑ v ∈ V ( G ) γ 2 v ( G ) , where the lower 2-domination number, denoted by γ 2 v ( G ) , of the graph G relative to v is the minimum cardinality of 2-domination set in G that contains the vertex v. In this paper, the average lower 2-domination number of wheels and some related networks namely gear graph, friendship graph, helm graph and sun flower graph are calculated. Then, we offer an algorithm for computing the 2-domination number and the average lower 2-domination number of any graph G.Mathematical and Computational Applications2016-07-18213Article10.3390/mca21030029292297-87472016-07-18doi: 10.3390/mca21030029Tufan Turaci<![CDATA[MCA, Vol. 21, Pages 28: Correction: Kolkiran, A.; Agarwal, G.S. Amplitude Noise Reduction in a Nano-Mechanical Oscillator. Math. Comput. Appl. 2011, 16(1), 290–300]]>
http://www.mdpi.com/2297-8747/21/3/28
The authors would like to remove Girish S. Agarwal from the author list of the paper [1]. Aziz Kolkiran will therefore serve as the single author.[...]Mathematical and Computational Applications2016-07-08213Correction10.3390/mca21030028282297-87472016-07-08doi: 10.3390/mca21030028Aziz KolkiranGirish Agarwal<![CDATA[MCA, Vol. 21, Pages 26: New Nonlinear Metrics Model for Information of Individual Research Output and Its Applications]]>
http://www.mdpi.com/2297-8747/21/3/26
Evaluation on achievement of scientists plays an important role in efficiently mining information of human resources. A metrics model, which is employed to calculate the number of academic papers, research awards and scientific research projects, often significantly affects the degree of fairness as it is used to compare the achievements of more than one scientist. In particular, it often becomes difficult to quantify the achievement for each scientist if there are a lot of participants in the same research output. In this paper, a new nonlinear metrics model, called a credit function, is established to mine the information of the individual research outputs (IRO). An example is constructed to show that different credit functions may generate distinct ranking for the scientists. By the proposed nonlinear methods in this paper, the inequality relation of contribution in the same IRO can be quantified, and the obtained ranking on the scientists is more acceptable than the existing linear method available in the literature. Finally, the proposed metrics model is applied in solving three practical problems, especially combined with the technique for order preference by similarity to an ideal solution (TOPSIS).Mathematical and Computational Applications2016-06-30213Article10.3390/mca21030026262297-87472016-06-30doi: 10.3390/mca21030026Ming ChenZhong Wan<![CDATA[MCA, Vol. 21, Pages 27: Some Properties of a Function Originating from Geometric Probability for Pairs of Hyperplanes Intersecting with a Convex Body]]>
http://www.mdpi.com/2297-8747/21/3/27
In the paper, the authors derive an integral representation, present a double inequality, supply an asymptotic formula, find an inequality, and verify complete monotonicity of a function involving the gamma function and originating from geometric probability for pairs of hyperplanes intersecting with a convex body.Mathematical and Computational Applications2016-06-29213Short Note10.3390/mca21030027272297-87472016-06-29doi: 10.3390/mca21030027Feng QiMansour Mahmoud<![CDATA[MCA, Vol. 21, Pages 25: Fixed Order Controller for Schur Stability]]>
http://www.mdpi.com/2297-8747/21/2/25
If the characteristic polynomial of a discrete-time system has all its roots in the open unit disc of the complex plane, the system is called Schur stable. In this paper, the Schur stabilization problem of closed loop discrete-time system by affine compensator is considered. For this purpose, the distance function between the Schur stability region and the affine controller subset is investigated.Mathematical and Computational Applications2016-06-20212Article10.3390/mca21020025252297-87472016-06-20doi: 10.3390/mca21020025Taner Büyükköroğlu<![CDATA[MCA, Vol. 21, Pages 24: A Note on Some Solutions of Copper-Water (Cu-Water) Nanofluids in a Channel with Slowly Expanding or Contracting Walls with Heat Transfer]]>
http://www.mdpi.com/2297-8747/21/2/24
A study has been carried out to examine the occurrence of multiple solutions for Copper-Water nanofluids flows in a porous channel with slowly expanding and contracting walls. The governing equations are first transformed to similarity equations by using similarity transformation. The resulting equations are then solved numerically by using the shooting method. The effects of wall expansion ratio and solid volume fraction on velocity and temperature profile have been studied. Numerical results are presented graphically for the variations of different physical parameters. The study reveals that triple solutions exist only for the case of suction.Mathematical and Computational Applications2016-06-16212Article10.3390/mca21020024242297-87472016-06-16doi: 10.3390/mca21020024Jawad RazaAzizah RohniZurni Omar<![CDATA[MCA, Vol. 21, Pages 23: A Recommendation System for Execution Plans Using Machine Learning]]>
http://www.mdpi.com/2297-8747/21/2/23
Generating execution plans is a costly operation for the DataBase Management System (DBMS). An interesting alternative to this operation is to reuse the old execution plans, that were already generated by the optimizer for past queries, to execute new queries. In this paper, we present an approach for execution plan recommendation in two phases. We firstly propose a textual representation of our SQL queries and use it to build a Features Extractor module. Then, we present a straightforward solution to identify query similarity.This solution relies only on the comparison of the SQL statements. Next, we show how to build an improved solution enabled by machine learning techniques. The improved version takes into account the features of the queries’ execution plans. By comparing three machine learning algorithms, we find that the improved solution using Classification Based on Associative Rules (CAR) identifies similarity in 91 % of the cases.Mathematical and Computational Applications2016-06-15212Article10.3390/mca21020023232297-87472016-06-15doi: 10.3390/mca21020023Jihad ZahirAbderrahim El Qadi<![CDATA[MCA, Vol. 21, Pages 22: An Improved Interval-Valued Hesitant Fuzzy Multi-Criteria Group Decision-Making Method and Applications]]>
http://www.mdpi.com/2297-8747/21/2/22
The Bonferroni mean (BM) can be used in situations where the aggregated arguments are correlated. BM is very useful for solving decision-making problems. For describing fuzziness and vagueness more accurately, the interval-valued hesitant fuzzy set (IVHFS), which is a generalization of the hesitant fuzzy set (HFS), can be used to describe the membership degrees with interval numbers. The aim of this paper is to propose the interval-valued hesitant fuzzy Bonferroni mean (IVHFBM) for aggregating interval-valued hesitant fuzzy information. Furthermore, the weighted form of IVHFBM (IVHFWBM) is forwarded and, hereby, a multi-criteria group decision-making (MCGDM) method is established. A case study on the problem of evaluating research funding applications in China is analyzed. A comparison between the proposed method and existing ones demonstrates its practicability.Mathematical and Computational Applications2016-06-08212Article10.3390/mca21020022222297-87472016-06-08doi: 10.3390/mca21020022Zhenhua DingYingyu Wu<![CDATA[MCA, Vol. 21, Pages 21: Holographic Tachyon in Fractal Geometry]]>
http://www.mdpi.com/2297-8747/21/2/21
The search of a logical quantum gravity theory is one of the noteworthy issues in modern theoretical physics. It is known that most of the quantum gravity theories describe our universe as a dimensional flow. From this point of view, one can investigate whether and how these attractive properties are related with the ultraviolet-divergence problem. These important points motivated us to discuss the reconstruction of a scalar field problem in the fractal theory which is a well-known quantum theory of gravity. Making use of time-like fractal model and considering the holographic description of galactic dark energy, we implement a correspondence between the tachyon model of galactic dark energy effect and holographic energy. Such a connection gives us an opportunity to redefine the fractal dynamics of selected scalar field representation by considering the time-evolution of holographic energy.Mathematical and Computational Applications2016-05-31212Article10.3390/mca21020021212297-87472016-05-31doi: 10.3390/mca21020021Mustafa SaltiOktay Aydogdu<![CDATA[MCA, Vol. 21, Pages 20: Predictive Abilities of Bayesian Regularization and Levenberg–Marquardt Algorithms in Artificial Neural Networks: A Comparative Empirical Study on Social Data]]>
http://www.mdpi.com/2297-8747/21/2/20
The objective of this study is to compare the predictive ability of Bayesian regularization with Levenberg–Marquardt Artificial Neural Networks. To examine the best architecture of neural networks, the model was tested with one-, two-, three-, four-, and five-neuron architectures, respectively. MATLAB (2011a) was used for analyzing the Bayesian regularization and Levenberg–Marquardt learning algorithms. It is concluded that the Bayesian regularization training algorithm shows better performance than the Levenberg–Marquardt algorithm. The advantage of a Bayesian regularization artificial neural network is its ability to reveal potentially complex relationships, meaning it can be used in quantitative studies to provide a robust model.Mathematical and Computational Applications2016-05-24212Article10.3390/mca21020020202297-87472016-05-24doi: 10.3390/mca21020020Murat Kayri<![CDATA[MCA, Vol. 21, Pages 19: A Multiple Attribute Decision-Making Method Based on Exponential Fuzzy Numbers]]>
http://www.mdpi.com/2297-8747/21/2/19
Regarding the multi-attribute decision-making problem where both the attribute value and attribute weight of a scheme are exponential fuzzy numbers, this paper proposes a fuzzy multiple attribute decision-making method based on exponential fuzzy numbers. The expected value and variance of the exponential fuzzy number were calculated according to the definitions of expectation and variance in probability theory. By comprehensively considering factors such as expected value, variance, and attitude preference of the decision-maker, this paper gave the score function of the exponential fuzzy number, based on which the accurate attribute weight can be determined. Subsequently, based on the distance measure between exponential fuzzy numbers, the distance was calculated between each scheme and the positive/negative ideal scheme, respectively, and the relative closeness of each scheme was solved; based on this, sorting and prioritizing were conducted. Finally, the feasibility and effectiveness of the proposed method were verified through case analysis.Mathematical and Computational Applications2016-05-21212Article10.3390/mca21020019192297-87472016-05-21doi: 10.3390/mca21020019Sha FuGuo-bing Fan<![CDATA[MCA, Vol. 21, Pages 18: A Class of Cubic Trigonometric Automatic Interpolation Curves and Surfaces with Parameters]]>
http://www.mdpi.com/2297-8747/21/2/18
This paper presents the cubic trigonometric interpolation curves with two parameters generated over the space {1, sint, cost, sin2t, sin3t, cos3t}. The new curves can not only automatically interpolate the given data points without solving equation systems, but are also C2 and adjust their shape by altering values of the two parameters. The optimal interpolation curves can be determined by an energy optimization model. The corresponding interpolation surfaces have characteristics similar to the new curves.Mathematical and Computational Applications2016-05-20212Article10.3390/mca21020018182297-87472016-05-20doi: 10.3390/mca21020018Juncheng Li<![CDATA[MCA, Vol. 21, Pages 17: Three-Parameter Interval Grey Number Multi-Attribute Decision Making Method Based on Information Entropy]]>
http://www.mdpi.com/2297-8747/21/2/17
Aimed at the situation that the plan attribute value is a three-parameter interval grey number and that attribute weight is uncertain, a three-parameter interval grey number multi-attribute decision making method based on information entropy is proposed. In this study, we combine the three-parameter interval grey number decision-making problem with information entropy, build the distance entropy model of the three-parameter interval grey number, and provide a solution to the empowerment problem of the three-parameter interval grey number in grey decision-making problems. On this basis, we build the uncertainty decision making framework based on Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) thinking and obtain the comprehensive distances of plans from positive and negative ideal solutions through calculation. We thus determine the closeness, and, based on this value, organize these plans. Finally, we verify the feasibility and effectiveness of the proposed methods through case analysis.Mathematical and Computational Applications2016-05-11212Article10.3390/mca21020017172297-87472016-05-11doi: 10.3390/mca21020017Sha Fu<![CDATA[MCA, Vol. 21, Pages 16: Finite Volume Method for the Relativistic Burgers Model on a (1+1)-Dimensional de Sitter Spacetime]]>
http://www.mdpi.com/2297-8747/21/2/16
Several generalizations of the relativistic models of Burgers equations have recently been established and developed on different spacetime geometries. In this work, we take into account the de Sitter spacetime geometry, introduce our relativistic model by a technique based on the vanishing pressure Euler equations of relativistic compressible fluids on a (1+1)-dimensional background and construct a second order Godunov type finite volume scheme to examine numerical experiments within an analysis of the cosmological constant. Numerical results demonstrate the efficiency of the method for solutions containing shock and rarefaction waves.Mathematical and Computational Applications2016-05-10212Article10.3390/mca21020016162297-87472016-05-10doi: 10.3390/mca21020016Tuba CeylanBaver Okutmustur<![CDATA[MCA, Vol. 21, Pages 15: A New Algorithm for the Numerical Solution of Telegraph Equations by Using Fibonacci Polynomials]]>
http://www.mdpi.com/2297-8747/21/2/15
In this study, we present a numerical scheme to solve the telegraph equation by using Fibonacci polynomials. This method is based on the Fibonacci collocation method which transforms the equation into a matrix equation, and the unknown of this equation is a Fibonacci coefficients matrix. Some numerical examples with comparisons are included to demonstrate the validity and applicability of the proposed method. The results show the efficiency and accuracy of this paper.Mathematical and Computational Applications2016-05-10212Article10.3390/mca21020015152297-87472016-05-10doi: 10.3390/mca21020015Ayşe Kurt BahşıSalih Yalçınbaş<![CDATA[MCA, Vol. 21, Pages 14: Geometric Properties of Normalized Wright Functions]]>
http://www.mdpi.com/2297-8747/21/2/14
The purpose of the present paper is to investigate some characterizations for the Wright functions to be in the subclasses S * ( α , β ) and C ( α , β ) ( α , β ∈ [ 0 , 1 ) ) . Several sufficient conditions are obtained for the normalized Wright functions to be in these classes. Results obtained in this paper are new and their usefulness is put forth by several corollaries.Mathematical and Computational Applications2016-05-04212Article10.3390/mca21020014142297-87472016-05-04doi: 10.3390/mca21020014Nizami Mustafa<![CDATA[MCA, Vol. 21, Pages 13: Design of a Parameter-Dependent Optimal Vibration Control of a Non-Linear Vehicle Suspension System]]>
http://www.mdpi.com/2297-8747/21/2/13
This paper is concerned with the design of a parameter-dependent optimal controller for an active vibration attenuation problem of a non-linear vehicle system. A five degree-of-freedom vertical vibration model having an integrated vehicle seat, a non-linear vehicle suspension system, and a seated human body is presented to analyze ride comfort and safety requirements under different types of road disturbances. In the suspension system, the non-linear parts of spring and damper dynamics are considered as scheduling parameters, which are measurable and available for feedback. Then, a parameter-dependent optimal state-feedback controller design that minimizes L2 gain from disturbance to performance output for a linear parameter-varying (LPV) system is presented with linear matrix inequality (LMI) constraints. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed controller.Mathematical and Computational Applications2016-04-27212Article10.3390/mca21020013132297-87472016-04-27doi: 10.3390/mca21020013Hakan Yazici<![CDATA[MCA, Vol. 21, Pages 12: Direct Solution of Second-Order Ordinary Differential Equation Using a Single-Step Hybrid Block Method of Order Five]]>
http://www.mdpi.com/2297-8747/21/2/12
This paper proposes a new hybrid block method of order five for solving second-order ordinary differential equations directly. The method is developed using interpolation and collocation techniques. The use of the power series approximate solution as an interpolation polynomial and its second derivative as a collocation equation is considered in deriving the method. Properties of the method such as zero stability, order, consistency, convergence and region of absolute stability are investigated. The new method is then applied to solve the system of second-order ordinary differential equations and the accuracy is better when compared with the existing methods in terms of error.Mathematical and Computational Applications2016-04-12212Article10.3390/mca21020012122297-87472016-04-12doi: 10.3390/mca21020012Ra’ft AbdelrahimZurni Omar<![CDATA[MCA, Vol. 21, Pages 11: New Exact Solutions of the System of Equations for the Ion Sound and Langmuir Waves by ETEM]]>
http://www.mdpi.com/2297-8747/21/2/11
This manuscript focuses attention on new exact solutions of the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave. The extended trial equation method (ETEM), which is one of the analytical methods, has been handled for finding exact solutions of the system of equations for the ion sound wave and the Langmuir wave. By using this method, exact solutions including the rational function solution, traveling wave solution, soliton solution, Jacobi elliptic function solution, hyperbolic function solution and periodic wave solution of this system of equations have been obtained. In addition, by using Mathematica Release 9, some graphical simulations were done to see the behavior of these solutions.Mathematical and Computational Applications2016-04-02212Article10.3390/mca21020011112297-87472016-04-02doi: 10.3390/mca21020011Seyma Tuluce DemirayHasan Bulut<![CDATA[MCA, Vol. 21, Pages 10: Positive Almost Periodic Solutions for a Delayed Predator–Prey Model with Hassell-Varley Type Functional Response]]>
http://www.mdpi.com/2297-8747/21/2/10
By means of a fixed point theorem of coincidence degree theory, sufficient conditions are established for the existence of a positive almost periodic solution to a kind of delayed predator–prey model with Hassell-Varley type functional response. The method used in this paper offers a possible means to study the existence of positive almost periodic solutions to the models in biological populations. Finally, an example as well as numerical simulations are given to illustrate the feasibility and effectiveness of our results.Mathematical and Computational Applications2016-04-01212Article10.3390/mca21020010102297-87472016-04-01doi: 10.3390/mca21020010Tianwei ZhangLiyan PangYongzhi Liao<![CDATA[MCA, Vol. 21, Pages 9: The Regularization-Homotopy Method for the Two-Dimensional Fredholm Integral Equations of the First Kind]]>
http://www.mdpi.com/2297-8747/21/2/9
In this work, we consider two-dimensional linear and nonlinear Fredholm integral equations of the first kind. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method is used to find a solution to the equation. The application of this method is based upon converting the first kind of equation to the second kind by applying the regularization method. Then the homotopy perturbation method is employed to the resulting second kind of equation to obtain a solution. A few examples including linear and nonlinear equations are provided to show the validity and applicability of this approach.Mathematical and Computational Applications2016-03-30212Article10.3390/mca2102000992297-87472016-03-30doi: 10.3390/mca21020009Ahmet Altürk<![CDATA[MCA, Vol. 21, Pages 8: New Analytic Solutions for the (N + 1)-Dimensional Generalized Boussinesq Equation]]>
http://www.mdpi.com/2297-8747/21/2/8
In this paper, the generalized Jacobi elliptic functions expansion method with computerized symbolic computation are employed to investigate explicitly analytic solutions of the (N + 1)-dimensional generalized Boussinesq equation. The exact solutions to the equation are constructed analytically under certain circumstances, some of these solutions are degenerated to soliton-like solutions and trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic function solutions tends to 0 and 1, which shows that the applied method is more powerful and will be used in further works to establish more entirely new exact solutions for other kinds of higher-dimensional nonlinear partial differential equations in mathematical physics.Mathematical and Computational Applications2016-03-25212Article10.3390/mca2102000882297-87472016-03-25doi: 10.3390/mca21020008Baojian HongDianchen Lu<![CDATA[MCA, Vol. 21, Pages 7: Estimating Variances in Weighted Least-Squares Estimation of Distributional Parameters]]>
http://www.mdpi.com/2297-8747/21/2/7
Many estimation methods have been proposed for the parameters of statistical distribution. The least squares estimation method, based on a regression model or probability plot, is frequently used by practitioners since its implementation procedure is extremely simple in complete and censoring data cases. However, in the procedure, heteroscedasticity is present in the used regression model and, thus, the weighted least squares estimation or alternative methods should be used. This study proposes an alternative method for the estimation of variance, based on a dependent variable generated via simulation, in order to estimate distributional parameters using the weighted least squares method. In the estimation procedure, the variances or weights are expressed as a function of the rank of the data point in the sample. The considered weighted estimation method is evaluated for the shape parameter of the log-logistic and Weibull distributions via a simulation study. It is found that the considered weighted estimation method shows better performance than the maximum likelihood, least-squares, and certain other alternative estimation approaches in terms of mean square error for most of the considered sample sizes. In addition, a real-life example from hydrology is provided to demonstrate the performance of the considered method.Mathematical and Computational Applications2016-03-25212Article10.3390/mca2102000772297-87472016-03-25doi: 10.3390/mca21020007Yeliz Mert Kantar<![CDATA[MCA, Vol. 21, Pages 6: New Complex Hyperbolic Function Solutions for the (2+1)-Dimensional Dispersive Long Water–Wave System]]>
http://www.mdpi.com/2297-8747/21/2/6
In this paper, new algorithms called the “Modified exp(−Ω)-expansion function method” and “Improved Bernoulli sub-equation function method” have been proposed. The first algorithm is based on the exp(−Ω(ξ))-expansion method; the latter is based on the Bernoulli sub-Ordinary Differential Equation method. The methods proposed have been expressed comprehensively in this manuscript. The analytical solutions and application results are presented by drawing the two- and three-dimensional surfaces of solutions such as hyperbolic, complex, trigonometric and exponential solutions for the (2+1)-dimensional dispersive long water–wave system. Finally, a conclusion has been presented by mentioning the important discoveries in this manuscript.Mathematical and Computational Applications2016-03-24212Article10.3390/mca2102000662297-87472016-03-24doi: 10.3390/mca21020006Hasan BulutHaci Baskonus<![CDATA[MCA, Vol. 21, Pages 5: Elasto-Plastic Stress Analysis in a Ductile Adhesive & Aluminum Adherends]]>
http://www.mdpi.com/2297-8747/21/1/5
In this study, an elasto-plastic stress analysis is proposed in an aluminum adherend and a ductile adhesive. Elasto-plastic analysis was carried out for the aluminum adherend and DP460 ductile adhesive in a double-lap joint. The analytical solution was compared with the finite element solution. ANSYS 12 was used in the stress analysis. The analytical solution was performed for the one-dimensional case in the adhesive and the adherends. A FEM solution was given for the two-dimensional case. Similar results were obtained for both. In addition, the solution was carried out for brittle and ductile materials. The mechanical properties of the loctite were nearly the same as the ductile DP460 adhesive. It was observed that the ductile adhesive increased the strength of the structure due to the small shear stresses at the free ends of the adhesive.Mathematical and Computational Applications2016-03-09211Article10.3390/mca2101000552297-87472016-03-09doi: 10.3390/mca21010005Mustafa AkdagTuran BatarFuat OkumusOnur SaymanVolkan Arikan<![CDATA[MCA, Vol. 21, Pages 4: Variant of Constants in Subgradient Optimization Method over Planar 3-Index Assignment Problems]]>
http://www.mdpi.com/2297-8747/21/1/4
A planar 3-index assignment problem (P3AP) of size n is an NP-complete problem. Its global optimal solution can be determined by a branch and bound algorithm. The efficiency of the algorithm depends on the best lower and upper bound of the problem. The subgradient optimization method, an iterative method, can provide a good lower bound of the problem. This method can be applied to the root node or a leaf of the branch and bound tree. Some conditions used in this method may result in one of those becoming optimal. The formulas used in this method contain some constants that can be evaluated by computational experiments. In this paper, we show a variety of initial step length constants whose values have an effect on the lower bound of the problem. The results show that, for small problem sizes, when n &lt; 20, the most suitable constants are best chosen in the interval [0.1, 1]. Meanwhile, the interval [0.05, 0.1] is the best interval chosen for the larger problem sizes, when n ≥ 20.Mathematical and Computational Applications2016-03-08211Article10.3390/mca2101000442297-87472016-03-08doi: 10.3390/mca21010004Sutitar Maneechai<![CDATA[MCA, Vol. 21, Pages 3: Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory]]>
http://www.mdpi.com/2297-8747/21/1/3
In this study, the non-local Euler-Bernoulli beam theory was employed in the nonlinear free and forced vibration analysis of a nanobeam resting on an elastic foundation of the Pasternak type. The analysis considered the effects of the small-scale of the nanobeam on the frequency. By utilizing Hamilton’s principle, the nonlinear equations of motion, including stretching of the neutral axis, are derived. Forcing and damping effects are considered in the analysis. The linear part of the problem is solved by using the first equation of the perturbation series to obtain the natural frequencies. The multiple scale method, a perturbation technique, is applied in order to obtain the approximate closed solution of the nonlinear governing equation. The effects of the various non-local parameters, Winkler and Pasternak parameters, as well as effects of the simple-simple and clamped-clamped boundary conditions on the vibrations, are determined and presented numerically and graphically. The non-local parameter alters the frequency of the nanobeam. Frequency-response curves are drawn.Mathematical and Computational Applications2016-03-07211Article10.3390/mca2101000332297-87472016-03-07doi: 10.3390/mca21010003Necla TogunSüleyman Bağdatlı<![CDATA[MCA, Vol. 21, Pages 2: Numerical Simulation Study on the Flow Field Out of a Submerged Abrasive Water Jet Nozzle]]>
http://www.mdpi.com/2297-8747/21/1/2
In order to optimize the parameters of pre-mixed abrasive water jet cutting technology, make it more efficient in the coal mine gas environment and solve the problem of hard coal and the difficulty of rock drilling, FLUENT software was used to get the isothermal, incompressible, steady flow field out of a submerged abrasive water jet nozzle through numerical simulation, with different particle sizes and different confining pressures under submerged conditions. The results show that, under submerged conditions, the maximum velocity of the abrasive particle outside the pre-mixed abrasive water jet nozzle is about 6 mm far away from the nozzle; the abrasive particle diameter has little influence on the velocity outside the nozzle. The external confining pressure of the nozzle has an important influence on the velocity, which is that the jet velocity of the same position decreases with the increase of confining pressure and the relationship between the confining pressure of different distance from the nozzle exit and the abrasive velocity is exponential function. The results of the simulation laid the foundation for optimizing the nozzle structure, improving efficiency and developing the abrasive water jet nozzle.Mathematical and Computational Applications2016-03-04211Article10.3390/mca2101000222297-87472016-03-04doi: 10.3390/mca21010002Junqing MengQingen WeiYechao Ma<![CDATA[MCA, Vol. 21, Pages 1: A New Perturbation Approach to Optimal Polynomial Regression]]>
http://www.mdpi.com/2297-8747/21/1/1
A new approach to polynomial regression is presented using the concepts of orders of magnitudes of perturbations. The data set is normalized with the maximum values of the data first. The polynomial regression of arbitrary order is then applied to the normalized data. Theorems for special properties of the regression coefficients as well as some criteria for determining the optimum degrees of the regression polynomials are posed and proven. The new approach is numerically tested, and the criteria for determining the best degree of the polynomial for regression are discussed.Mathematical and Computational Applications2016-03-04211Article10.3390/mca2101000112297-87472016-03-04doi: 10.3390/mca21010001Mehmet Pakdemirli<![CDATA[MCA, Vol. 20, Pages 217-227: Study on Water Seepage Law of Raw Coal During Loading Process]]>
http://www.mdpi.com/2297-8747/20/3/217
In order to reveal the water seepage law of raw coal during different loading process, the gravity constant load seepage experimental system is developed and used in this paper. The water seepage law of raw coal during different loading process was tested. The mathematical model of axial strain-damage-permeability coefficient during the loading process is proposed based on the Wei-bull distribution of coal damage. According to the water seepage experiments and results analysis, the following conclusions are gotten. Under the same experimental conditions, with the increasing of axial pressure, the permeability coefficient of coal sample has a distinct decrease trend, and then an increase trend when reached the extreme point. The same trend of permeability coefficient-strain curves and stress-strain curve of the raw coal samples under different axial pressure are got-ten, that mean the permeability coefficient is closely related to the damage evolution process. The water seepage law of raw coal during different loading process can be described by the mathematical model of axial strain-damage-permeability coefficient, and its parameters can be obtained easily. This research is important for revealing the mechanism of coal seam water seepage, guiding field coal seam water infusion, controlling mine water hazard, preventing coal mine disasters.Mathematical and Computational Applications2015-12-01203Article10.3390/mca200102272172272297-87472015-12-01doi: 10.3390/mca20010227Meng JunqingNie Baisheng<![CDATA[MCA, Vol. 20, Pages 200-216: Stability and Bifurcation Analysis of a Pipe Conveying Pulsating Fluid with Combination Parametric and Internal Resonances]]>
http://www.mdpi.com/2297-8747/20/3/200
The stability and bifurcations of a hinged-hinged pipe conveying pulsating fluid with combination parametric and internal resonances are studied with both analytical and numerical methods. The system has geometric cubic nonlinearity. Three types of critical points for the bifurcation response equations are considered. These points are characterized by a double zero and two negative eigenvalues, double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues, respectively. With the aid of normal form theory, the expressions for the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. Possible bifurcations leading to 2-D tori are also investigated. Numerical simulations confirm the analytical results.Mathematical and Computational Applications2015-12-01203Article10.3390/mca200102162002162297-87472015-12-01doi: 10.3390/mca20010216Liangqiang ZhouFangqi ChenYushu Chen<![CDATA[MCA, Vol. 20, Pages 189-199: Multiple Attribute Decision-Making Model of Grey Target Based on Positive and Negative Bull's-Eye]]>
http://www.mdpi.com/2297-8747/20/3/189
Aiming at complexity and uncertainty of actual decision-making environment, this study proposes a multiple attribute decision-making model of grey target based on positive and negative bull’s-eye. Firstly, it defines that the optimal effect vector and the worst effect vector of grey target decision are respectively positive, negative bull’s-eye of the grey target; secondly, comprehensively considering the space projection distance between various schemes and the positive and negative bull’s-eye, it takes bull’s-eye distance as the basis for space analysis and obtains a new integrated bull’s-eye distance; then, in accordance with the comprehensive guidelines to minimize the bull’s-eye distance, it constructs goal programming model for goal function, and thus solves the index weight. Finally, through case studies of selective purchase of information system, it verifies feasibility and effectiveness of the proposed grey target decision-making model.Mathematical and Computational Applications2015-12-01203Article10.3390/mca200101991891992297-87472015-12-01doi: 10.3390/mca20010199Sha Fu<![CDATA[MCA, Vol. 20, Pages 174-188: Spinning Flow of Casson Fluid near an Infinite Rotating Disk]]>
http://www.mdpi.com/2297-8747/20/3/174
This paper presents an investigation of the spinning flow of a non-Newtonian Casson fluid over a rotating disk. The model established for the governing problem in the form of partial differential equations has been converted to ordinary differential equations with the use of suitable similarity transformation. The analytical approximation has been made with the most likely analytical method, homotopy analysis method (HAM). The convergence region of the obtained solution is determined and plotted. The velocity profiles are shown and the influence of Casson parameter is discussed in detail. Also comparison has been made with the Newtonian fluid as the special case of considered problem.Mathematical and Computational Applications2015-12-01203Article10.3390/mca200101881741882297-87472015-12-01doi: 10.3390/mca20010188Najeeb KhanZehra Husain<![CDATA[MCA, Vol. 20, Pages 160-173: Bernstein Collocation Method for Solving the First Order Nonlinear Differential Equations with the Mixed Non-Linear Conditions]]>
http://www.mdpi.com/2297-8747/20/3/160
In this study, we present the Bernstein matrix method to solve the first order nonlinear ordinary differential equations with the mixed non-linear conditions. By using this method, we obtain the approximate solutions in form of the Bernstein polynomials [1,2,16,17]. The method reduces the problem to a system of the nonlinear algebraic equations by means of the required matrix relations of the solutions form. By solving this system, the approximate solution is obtained. Finally, the method will be illustrated on the examples.Mathematical and Computational Applications2015-12-01203Article10.3390/mca200101731601732297-87472015-12-01doi: 10.3390/mca20010173Salih YalçınbaşHuriye Gürler<![CDATA[MCA, Vol. 20, Pages 151-159: Derivation of a Six-Step Block Method for Direct Solution of Second Order Ordinary Differential Equations]]>
http://www.mdpi.com/2297-8747/20/3/151
A new six-step block method for solving second order initial value problems of ordinary differential equations is proposed using interpolation and collocation strategies. In developing this method, the power series adopted as an approximate solution is employed as interpolation equation while its second derivative is used as collocation equation. In addition, the stability properties of the developed method are also established. The numerical results reveal that the new method produces better accuracy if compared to existing methods when solving the same problems.Mathematical and Computational Applications2015-12-01203Article10.3390/mca200101591511592297-87472015-12-01doi: 10.3390/mca20010159J. KuboyeZ. Omar<![CDATA[MCA, Vol. 20, Pages 137-150: Solution of Quadratic Nonlinear Problems with Multiple Scales Lindstedt-Poincare Method]]>
http://www.mdpi.com/2297-8747/20/2/137
A recently developed perturbation algorithm namely the multiple scales Lindstedt-Poincare method (MSLP) is employed to solve the mathematical models. Three different models with quadratic nonlinearities are considered. Approximate solutions are obtained with classical multiple scales method (MS) and the MSLP method and they are compared with the numerical solutions. It is shown that MSLP solutions are better than the MS solutions for the strongly nonlinear case of the considered models.Mathematical and Computational Applications2015-08-01202Article10.3390/mca200101501371502297-87472015-08-01doi: 10.3390/mca20010150Mehmet PakdemirliGözde Sarı<![CDATA[MCA, Vol. 20, Pages 121-136: Co-Evolution Analysis on Coal-Power Industries Cluster Ecosystem Based on the Lotka-Volterra Model: A Case Study of China]]>
http://www.mdpi.com/2297-8747/20/2/121
This contradiction caused by differences in coal-electricity industry market forces is “market for coal, plans for electricity". The traditional coal and power enterprises cause serious pollution problems and ecological problems in the production process, restricting the sustainable socio-economic development. The coal-electricity industry cluster ecosystem forms may effectively mediate organizations conflict existing in the development of industrial clusters, improving resource utilization, reducing energy consumption, to achieve harmony with the natural environment. This paper combines with the theory of industrial ecology, with the Lotka-Volterra model, to explore co-evolution mechanism of coal–electricity industry cluster at the micro, meso and macro levels. In the end, this paper adapted the "Henon mapping" method to describe the chaos phenomenon existed in the development. This paper put forward specific development model for industrial clusters in lower, steady and advanced stages. The co-evolution coal-electricity industry cluster ecosystem is a good solution to numerous conflicts.Mathematical and Computational Applications2015-08-01202Article10.3390/mca200101361211362297-87472015-08-01doi: 10.3390/mca20010136Cui HeruiPeng XuZhao Yuqi<![CDATA[MCA, Vol. 20, Pages 111-120: Visual and Computational Comparison of Functionals Used in Density Functional Theory]]>
http://www.mdpi.com/2297-8747/20/2/111
This work presents the visual and quantitative comparison of Density Functional Theory (DFT) exchange-correlation energy Exc functionals with Coupled Cluster with Single and Double excitations (CCSD) calculations (and experiment where possible). The Exc functional is an approximate term which is a component of the total energy of a molecule. This comparison is based on visualizing the differences of computed properties, such as the charge density, geometry and other molecular properties between the functional and a CCSD calculation. In this work, this visual comparison for a set of functionals using a set of small molecules is presented to elucidate the method. Specifically, this visual comparison of the local molecular properties includes the charge density and electron localization function and global molecular properties such as molecular geometry for each DFT functional compared with a CCSD calculation. Note, that the differences of the particular computed properties are computed visually.Mathematical and Computational Applications2015-08-01202Article10.3390/mca200101201111202297-87472015-08-01doi: 10.3390/mca20010120K. Flurchick<![CDATA[MCA, Vol. 20, Pages 106-110: A Weakly Convergence Result on Lp(x) Spaces]]>
http://www.mdpi.com/2297-8747/20/2/106
In this paper with 1 &lt; p- ≤ p+ &lt;∞ condition we prove a weak convergence result under pointwise convergence and bounded of the sequence. Our theorem is an extension of classical result to variable exponent setting.Mathematical and Computational Applications2015-08-01202Article10.3390/mca200101101061102297-87472015-08-01doi: 10.3390/mca20010110Yasin Kaya<![CDATA[MCA, Vol. 20, Pages 94-105: Fermat Collocation Method for Nonlinear System of First Order Boundary Value Problems]]>
http://www.mdpi.com/2297-8747/20/2/94
In this study, a numerical approach is proposed to obtain approximate solutions of nonlinear system of first order boundary value problem. This technique is essentially based on the truncated Fermat series and its matrix representations with collocation points. Using the matrix method, we reduce the problem to a system of nonlinear algebraic equations. Numerical examples are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and produces accurate results.Mathematical and Computational Applications2015-08-01202Article10.3390/mca20010105941052297-87472015-08-01doi: 10.3390/mca20010105Dilek TaştekinSalih Yalçınbaş<![CDATA[MCA, Vol. 20, Pages 76-93: Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations]]>
http://www.mdpi.com/2297-8747/20/2/76
This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented.Mathematical and Computational Applications2015-08-01202Article10.3390/mca2001009376932297-87472015-08-01doi: 10.3390/mca20010093Abdollah BorhanifarKhadijeh Sadri<![CDATA[MCA, Vol. 20, Pages 62-75: A Method of Hybrid Multiple Attributes Group Decision Making with Risk Considering Decision-Makers' Confidence]]>
http://www.mdpi.com/2297-8747/20/1/62
With respect to different confidence on evaluation result from different decision-makers to the hybrid multi-attribute with risk, an approach to group decision- making based on prospect theory and projection theory is proposed. Firstly, a tuple is established to record the evaluation result and the hybrid decision information. Then the element of the tuple is changed into a single triangular fuzzy number by the transformation rule. Considering the confidence degree from single decision-maker to evaluation information, the group decision information is aggregated and the weights of attributes are calculated based on the intuitionistic fuzzy set theory. The improved projection method and prospect theory are proposed to rank the alternatives respectively. Finally, an application case is given to demonstrate the effectiveness and feasibility of the proposed approach.Mathematical and Computational Applications2015-04-01201Article10.3390/mca2001007562752297-87472015-04-01doi: 10.3390/mca20010075Xingzi WenJunlin OuyangYoujin Liu<![CDATA[MCA, Vol. 20, Pages 50-61: Homotopic Approximate Solutions for the General Perturbed Nonlinear Schrödinger Equation]]>
http://www.mdpi.com/2297-8747/20/1/50
In this work, a class of perturbed nonlinear Schrödinger equation is studied by using the homotopy perturbation method. Firstly, we obtain some Jacobi-like elliptic function solutions of the corresponding typical general undisturbed nonlinear Schrödinger equation through the mapping deformation method, and secondly, a homotopic mapping transform is constructed, then the approximate solution with arbitrary degree of accuracy for the perturbed equation is researched, it is pointed out that the series of approximate solution is convergent. Finally, the efficiency and accuracy of the approximate solution is also discussed by using the fixed point theorem.Mathematical and Computational Applications2015-04-01201Article10.3390/mca2001006150612297-87472015-04-01doi: 10.3390/mca20010061Baojian HongDianchen LuYangzheng Liu<![CDATA[MCA, Vol. 20, Pages 39-49: Analysis of an M (λ1,λ2) / M /1 / WV Queue with Controlled Vacation Interruption and Variable Arrival Rate]]>
http://www.mdpi.com/2297-8747/20/1/39
This paper studies a Markovian queue with multiple working vacations and controlled vacation interruption. If there are at least N customers waiting upon completion of a service at a lower rate, the vacation is interrupted and the server returns to the system to resume the normal working level. Otherwise, the server continues the vacation until the system is non-empty after a vacation ends or there are at least N customers after a service ends. Moreover, the variable arrival rate of the customers is taken into account. Under such assumptions, by using the quasi-birth-and-death process, the matrix- geometric method and the difference equation theory, the steady-state queue length distribution along with various performance measures are derived. Additionally, under a certain cost structure, the optimal threshold N* that minimizes the long-run expected cost function per unit time is numerically determined.Mathematical and Computational Applications2015-04-01201Article10.3390/mca2001004939492297-87472015-04-01doi: 10.3390/mca20010049Wenqing WuYinghui Tang<![CDATA[MCA, Vol. 20, Pages 25-38: A Difference-Index Based Ranking Method of Trapezoidal Intuitionistic Fuzzy Numbers and Application to Multiattribute Decision Making]]>
http://www.mdpi.com/2297-8747/20/1/25
The order relation of fuzzy number is important in decision making and optimization modeling, and ranking fuzzy numbers is difficult in nature. Ranking trapezoidal intuitionistic fuzzy numbers (TrIFNs) is more difficult due to the fact that the TrIFNs are a generalization of the fuzzy numbers. The aim of this paper is to develop a new methodology for ranking TrIFNs. We define the value-index and ambiguity-index based on the value and ambiguity of the membership and non-membership functions, and then propose a difference-index based ranking method, which is applied to multiattribute decision making (MADM) problems. The proposed method is compared to show its advantages and applicability.Mathematical and Computational Applications2015-04-01201Article10.3390/mca2001003825382297-87472015-04-01doi: 10.3390/mca20010038Deng-Feng LiJie Yang<![CDATA[MCA, Vol. 20, Pages 15-24: Dtm-Bf Method for the Flow and Heat Transfer Over a Nonlinearly Stretching Sheet With Nanofluids]]>
http://www.mdpi.com/2297-8747/20/1/15
This paper investigates an analytical analysis for the flow and heat transfer in a viscous fluid over a nonlinear stretching sheet. The governing partial differential equations are transformed into coupled nonlinear differential equations by introducing a siφmilarity transformation. The asymptotic analytical solutions are obtained by using differential transform method-basic functions (DTM-BF). Four types of nanofluids, namely Cu-water, Ag-water, Al2O3 -water and TiO2 -water were studied. The influence of the nanoparticle volume fraction φ, the nonlinear stretching parameter n , Prandtl number Pr, Eckert number Ec and different nanoparticles on the velocity and temperature are discussed and shown graphically. The comparison with the numerical results is presented and it is found to be in excellent agreement.Mathematical and Computational Applications2015-04-01201Article10.3390/mca2001002415242297-87472015-04-01doi: 10.3390/mca20010024Cao LimeiSun YinaShen YananSi Xinhui