Search results for: reduced order method

Authors: Y. A. Yahaya, Ahmad Tijjani Asabe

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This paper focuses on the formulation of 3-step hybrid Adams type method for the solution of first order differential equation (ODE). The methods which was derived on both grid and off grid points using multistep collocation schemes and also evaluated at some points to produced Block Adams type method and Adams moulton method respectively. The method with the highest order was selected to serve as the corrector. The convergence was valid and efficient. The numerical experiments were carried out and reveal that hybrid Adams type methods performed better than the conventional Adams moulton method.

Keywords: adam-moulton type (amt), corrector method, off-grid, block method, convergence analysis

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Authors: Zubdahe Noor, Haseeb Athar

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In this article, first a relation for conditional expectation is developed and then is used to characterize a general class of distributions F(x) = 1-e^(-ah(x)) through conditional expectation of difference of pair of generalized order statistics. Some results are reduced for particular cases. In the end, a list of distributions is presented in the form of table that are compatible with the given general class.

Keywords: generalized order statistics, order statistics, record values, conditional expectation, characterization

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Authors: N. Gayathri Devi, K. Batri

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The advent of technology has increased the traffic hazards and the road accidents take place. Collision detection system in automobile aims at reducing or mitigating the severity of an accident. This project aims at avoiding Vehicle head on collision by means of collision detection algorithm. This collision detection algorithm predicts the collision and the avoidance or minimization have to be done within few seconds on confirmation. Under critical situation collision minimization is made possible by turning the vehicle to the desired turn radius so that collision impact can be reduced. In order to avoid the collision completely, the turning of the vehicle should be achieved at reduced speed in order to maintain the stability.

Keywords: collision avoidance system, time to collision, time to turn, turn radius

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Authors: Hassan Parandvar, Mehrdad Farid

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In this paper, a finite element modeling is presented for large amplitude vibration of functionally graded material (FGM) plates subjected to combined random pressure and thermal load. The material properties of the plates are assumed to vary continuously in the thickness direction by a simple power law distribution in terms of the volume fractions of the constituents. The material properties depend on the temperature whose distribution along the thickness can be expressed explicitly. The von Karman large deflection strain displacement and extended Hamilton's principle are used to obtain the governing system of equations of motion in structural node degrees of freedom (DOF) using finite element method. Three-node triangular Mindlin plate element with shear correction factor is used. The nonlinear equations of motion in structural degrees of freedom are reduced by using modal reduction method. The reduced equations of motion are solved numerically by 4th order Runge-Kutta scheme. In this study, the random pressure is generated using Monte Carlo method. The modeling is verified and the nonlinear dynamic response of FGM plates is studied for various values of volume fraction and sound pressure level under different thermal loads. Snap-through type behavior of FGM plates is studied too.

Keywords: nonlinear vibration, finite element method, functionally graded material (FGM) plates, snap-through, random vibration, thermal effect

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Authors: Te-Jen Chang, I-Hui Pan, Ping-Sheng Huang, Shan-Jen Cheng

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In this paper, a fast multiplication computing method utilizing the complement representation method and canonical recoding technique is proposed. By performing complements and canonical recoding technique, the number of partial products can be reduced. Based on these techniques, we propose an algorithm that provides an efficient multiplication method. On average, our proposed algorithm is to reduce the number of k-bit additions from (0.25k+logk/k+2.5) to (k/6 +logk/k+2.5), where k is the bit-length of the multiplicand A and multiplier B. We can therefore efficiently speed up the overall performance of the multiplication. Moreover, if we use the new proposes to compute common-multiplicand multiplication, the computational complexity can be reduced from (0.5 k+2 logk/k+5) to (k/3+2 logk/k+5) k-bit additions.

Keywords: algorithm design, complexity analysis, canonical recoding, public key cryptography, common-multiplicand multiplication

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Authors: Qinglei Liu, Ali Charkhesht, Tiva Sharifi, Ashkan Bahadoran

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Graphene sheets are promising nanoscale building blocks as a support material for the dispersion of nanoparticles. In this work, a solvothermal method employed to directly grow Co1-xZnxFe2O4 ferrite nanospheres on graphene oxide support that is subsequently reduced to graphene. The samples morphology, structure and crystallography were investigated using field-emission scanning electron microscopy (FE-SEM) and powder X-ray diffraction (XRD). The influences of the Zn2+ content on photocatalytic activity, electrical conductivity and magnetic property of the samples are also investigated. The results showed that Co1-x Znx Fe2 O4 nanoparticles are dispersed on graphene sheets and obtained nanocomposites are soft magnetic materials. In addition the samples showed excellent photocatalytic activity under visible light irradiation.

Keywords: reduced graphene oxide, ferrite, magnetic nanocomposite, photocatalytic activity, solvothermal method

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Authors: Chanchal Biswas, Anrin Bhattacharyya, Gopes Chandra Das, Mahua Ghosh Chaudhuri, Rajib Dey

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Boiler coals with low fixed carbon and higher ash content have always challenged the metallurgists to develop a suitable method for their utilization. In the present study, an attempt is made to establish an energy effective method for the reduction of iron ore fines in the form of nuggets by using ‘Syngas’. By devolatisation (expulsion of volatile matter by applying heat) of boiler coal, gaseous product (enriched with reducing agents like CO, CO2, H2, and CH4 gases) is generated. Iron ore nuggets are reduced by this syngas. For that reason, there is no direct contact between iron ore nuggets and coal ash. It helps to control the minimization of the sulphur intake of the reduced nuggets. A laboratory scale devolatisation furnace designed with reduction facility is evaluated after in-depth studies and exhaustive experimentations including thermo-gravimetric (TG-DTA) analysis to find out the volatile fraction present in boiler grade coal, gas chromatography (GC) to find out syngas composition in different temperature and furnace temperature gradient measurements to minimize the furnace cost by applying one heating coil. The nuggets are reduced in the devolatisation furnace at three different temperatures and three different times. The pre-reduced nuggets are subjected to analytical weight loss calculations to evaluate the extent of reduction. The phase and surface morphology analysis of pre-reduced samples are characterized using X-ray diffractometry (XRD), energy dispersive x-ray spectrometry (EDX), scanning electron microscopy (SEM), carbon sulphur analyzer and chemical analysis method. Degree of metallization of the reduced nuggets is 78.9% by using boiler grade coal. The pre-reduced nuggets with lesser sulphur content could be used in the blast furnace as raw materials or coolant which would reduce the high quality of coke rate of the furnace due to its pre-reduced character. These can be used in Basic Oxygen Furnace (BOF) as coolant also.

Keywords: alternative ironmaking, coal gasification, extent of reduction, nugget making, syngas based DRI, solid state reduction

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Authors: Haniye Dehestani, Yadollah Ordokhani

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In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: collocation method, fractional partial differential equations, legendre-laguerre functions, pseudo-operational matrix of integration

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Authors: Guy Joseph Eyebe, Gambo Betchewe, Alidou Mohamadou, Timoleon Crepin Kofane

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In the present study, the dynamics of nanobeam resting on fractional order softening nonlinear viscoelastic pasternack foundations is studied. The Hamilton principle is used to derive the nonlinear equation of the motion. Approximate analytical solution is obtained by applying the standard averaging method. The Melnikov method is used to investigate the chaotic behaviors of device, the critical curve separating the chaotic and non-chaotic regions are found. It is shown that appearance of chaos in the system depends strongly on the fractional order parameter.

Keywords: chaos, fractional-order, Melnikov method, nanobeam

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Authors: Ayub Khan, Net Ram Garg, Geeta Jain

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In this paper, backstepping method is proposed to synchronize two fractional-order systems. The simulation results show that this method can effectively synchronize two chaotic systems.

Keywords: backstepping method, fractional order, synchronization, chaotic system

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Authors: Ebrahim Izadi-Darbandi

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In order to investigation the effects of sowing methods, nitrogen and phosphorus application methods in wheat weeds management, an experiment was performed as split plot, based on randomized completely block design with three replications at Research Farm, Faculty of Agriculture, Ferdowsi University of Mashhad, in 2010. Treatments included, wheat sowing methods (single-row with 30 cm distance and twine row on 50 cm width ridges) as main plots and nitrogen and phosphorus application methods (Broadcast and Band) as sub plots. In this experiment, phosphorus and nitrogen sources for fertilization were super phosphate triple (150 kg ha-1) applied before wheat sowing and incorporated with soil and urea (200 kg ha-1) respectively, applied in 2 phases (pre-plant 50%) and near wheat shooting (50%). Results showed that the effect of fertilizers application methods and wheat sowing methods were significant (p≤0.01) on wheat yield increasing and reducing weed-wheat competition. Wheat twine row sowing method, reduced weeds biomass for 25% compared wheat single-row sowing method and increased wheat seed yield and biomass for 60% and 30% respectively. Phosphorus and nitrogen band application reduced weeds biomass for 46% and 53% respectively and increased wheat seed yield for 22% and 33% compared to their broadcast application. The effects of wheat sowing method plus phosphorus and nitrogen application methods interactions, showed that the fertilizers band application and wheat twine-row sowing method were the best methods in wheat yield improvement and reducing wheat-weeds interaction. These results shows that modifying of fertilization methods and wheat sowing method can have important role in fertilizers use efficiency and improving of weeds managements.

Keywords: competition, wheat yield, fertilizer management, biomass

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Authors: Jong-Yeon Kim, Kwang-Bok Shin, Tae-Hwan Ko

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The purpose of this paper is to suggest a weight-reduction design method for the aluminum extrusion carbody structure of a double deck high-speed train using size optimization method. The size optimization method was used to optimize thicknesses of skin and rib of the aluminum extrusion for the carbody structure. Thicknesses of 1st underframe, 2nd underframe, solebar and roof frame were selected by design variables in order to conduct size optimization. The results of the size optimization analysis showed that the weight of the aluminum extrusion could be reduced by 0.61 tons (5.60%) compared to the weight of the original carbody structure.

Keywords: double deck high-speed train, size optimization, weigh-reduction, aluminum extrusion

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Authors: Soyoon Bak, Sunyoung Bu, Philsu Kim

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In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration-free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results are in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes. Semi-Lagrangian method, iteration-free method, nonlinear advection-diffusion equation, second-order backward difference formula

Keywords: Semi-Lagrangian method, iteration free method, nonlinear advection-diffusion equation, second-order backward difference formula

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Authors: Sang Hoon Lee, Tae Il Lee, Su Jeong Lee, Jae Min Myoung

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In order to manufacture short gap single Si nanowire (NW) field effect transistor (FET) by imprinting and transferring method, we introduce the method using Al2O3 sacrificial layer. The diameters of cylindrical Si NW addressed between Au electrodes by dielectrophoretic (DEP) alignment method are controlled to 106, 128, and 148 nm. After imprinting and transfer process, cylindrical Si NW is embedded in PVP adhesive and dielectric layer. By curing transferred cylindrical Si NW and Au electrodes on PVP-coated p++ Si substrate with 200nm-thick SiO2, 3μm gap Si NW FET fabrication was completed. As the diameter of embedded Si NW increases, the mobility of FET increases from 80.51 to 121.24 cm2/V•s and the threshold voltage moves from –7.17 to –2.44 V because the ratio of surface to volume gets reduced.

Keywords: Al2O3 sacrificial transfer layer, cylindrical silicon nanowires, dielectrophorestic alignment, field effect transistor

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Authors: A. M. Sagir

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The aim of this paper is to investigate the performance of the developed linear multistep block method for solving first order initial value problem of Ordinary Differential Equations (ODEs). The method calculates the numerical solution at three points simultaneously and produces three new equally spaced solution values within a block. The continuous formulations enable us to differentiate and evaluate at some selected points to obtain three discrete schemes, which were used in block form for parallel or sequential solutions of the problems. A stability analysis and efficiency of the block method are tested on ordinary differential equations involving practical applications, and the results obtained compared favorably with the exact solution. Furthermore, comparison of error analysis has been developed with the help of computer software.

Keywords: block method, first order ordinary differential equations, linear multistep, self-starting

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Authors: Norhashidah Hj Mohd Ali, Teng Wai Ping

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In this paper, the formulation of a new group explicit method with a fourth order accuracy is described in solving the two-dimensional Helmholtz equation. The formulation is based on the nine-point fourth-order compact finite difference approximation formula. The complexity analysis of the developed scheme is also presented. Several numerical experiments were conducted to test the feasibility of the developed scheme. Comparisons with other existing schemes will be reported and discussed. Preliminary results indicate that this method is a viable alternative high accuracy solver to the Helmholtz equation.

Keywords: explicit group method, finite difference, Helmholtz equation, five-point formula, nine-point formula

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Authors: Masoud Mahdavi

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During an earthquake, the structure is subjected to seismic loads that cause tension in the members of the building. The use of energy dissipation elements in the structure reduces the percentage of seismic forces on the main members of the building (especially the columns). Steel plate shear wall, as one of the most widely used types of energy dissipation element, has evolved today, and regular drilling of its inner plate is one of the common cases. In the present study, using a finite element method, the shear wall of the steel plate is designed as a floor (with dimensions of 447 × 6/246 cm) with Abacus software and in three different modes on which a cyclic load has been applied. The steel shear wall has a horizontal element (beam) with a reduced beam section (RBS). The hole in the interior plate of the models is created in such a way that it has the process of increasing the area, which makes the effect of increasing the surface area of the hole on the seismic performance of the steel shear wall completely clear. In the end, it was found that with increasing the opening level in the steel shear wall (with reduced cross-section beam), total displacement and plastic strain indicators increased, structural capacity and total energy indicators decreased and the Mises Monson stress index did not change much.

Keywords: steel plate shear wall with opening, cyclic loading, reduced cross-section beam, finite element method, Abaqus software

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Authors: A. H. Choudhury

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In this paper, a highly accurate numerical method for the solution of one-dimensional fourth-order wave equation is derived. This hyperbolic problem is solved by using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method, and the time variable is discretized by using Newmark schemes.

Keywords: hyperbolic problem, semidiscrete approximations, stability, Wavelet-Galerkin Method

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Authors: A. Hossein Madadi-Najafabadi, Masoud Nasiri

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The discrete element method is a powerful technique for numerical modeling the flow of granular materials such as direct reduced iron. It would enable us to study processes and equipment related to the production and handling of the material. However, the characteristics and properties of the granules have to be adjusted precisely to achieve reliable results in a DEM simulation. The main properties for DEM simulation are size distribution, density, Young's modulus, Poisson's ratio and the contact coefficients of restitution, rolling friction and sliding friction. In the present paper, the mentioned properties are determined for DEM simulation of DRI pellets. A reliable DEM simulation would contribute to optimizing the handling system of DRIs in an iron-making plant. Among the mentioned properties, Young's modulus is the most important parameter, which is usually hard to get for particulate solids. Here, an especial method is utilized to precisely determine this parameter for DRI.

Keywords: discrete element method, direct reduced iron, simulation parameters, granular material

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Authors: Shubham Jaiswal

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In the present article, a drive is taken to compute the solution of spatial fractional order advection-dispersion equation having source/sink term with given initial and boundary conditions. The equation is converted to a system of ordinary differential equations using second-kind shifted Chebyshev polynomials, which have finally been solved using finite difference method. The striking feature of the article is the fast transportation of solute concentration as and when the system approaches fractional order from standard order for specified values of the parameters of the system.

Keywords: spatial fractional order advection-dispersion equation, second-kind shifted Chebyshev polynomial, collocation method, conservative system, non-conservative system

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Authors: Mahdi Shamsi, Tohid Yousefi Rezaii, Siavash Eftekharifar

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Compressed sensing (CS) is a new powerful mathematical theory concentrating on sparse signals which is widely used in signal processing. The main idea is to sense sparse signals by far fewer measurements than the Nyquist sampling rate, but the reconstruction process becomes nonlinear and more complicated. Common dilemma in sparse signal recovery in CS is the lack of knowledge about sparsity order of the signal, which can be viewed as model order selection procedure. In this paper, we address the problem of sparsity order estimation in sparse signal recovery. This is of main interest in situations where the signal sparsity is unknown or the signal to be recovered is approximately sparse. It is shown that the proposed method also leads to some kind of signal denoising, where the observations are contaminated with noise. Finally, the performance of the proposed approach is evaluated in different scenarios and compared to an existing method, which shows the effectiveness of the proposed method in terms of order selection as well as denoising.

Keywords: compressed sensing, data denoising, model order selection, sparse representation

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Authors: Ali Raheli, Rahim Habibifar, Behzad Mohammadi-Alasti, Mahdi Abbasgholipour

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This paper presents vibration behavior of a FGM micro-beam and its pull-in instability under a nonlinear electrostatic pressure. An exponential function has been applied to show the continuous gradation of the properties along thickness. Nonlinear integro-differential-electro-mechanical equation based on Euler–Bernoulli beam theory has been derived. The governing equation in the static analysis has been solved using Step-by-Step Linearization Method and Finite Difference Method. Fixed points or equilibrium positions and singular points have been shown in the state control space. In order to find the response to a step DC voltage, the nonlinear equation of motion has been solved using Galerkin-based reduced-order model and time histories and phase portrait for different applied voltages have been shown. The effects of electrostatic pressure on stability of FGM micro-beams having various amounts of the ceramic constituent have been investigated.

Keywords: FGM, MEMS, nonlinear vibration, electrical, dynamic pull-in voltage

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Authors: Aicha Bensouici, Assia Mili, Naouel Rdjem, Nacera Baali

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Copper oxide decorated reduced graphene oxide (GO) was obtained successfully using two steps route synthesis was used. Firstly, graphene oxide was obtained using a modified Hummers method by excluding sodium nitrate from starting materials. After washing-centrifugation routine pristine GO was decorated by copper oxide using a refluxation technique at 120°C during 2h, and an equal amount of GO and copper acetate was used. Three CuO-rGO nanocomposite samples types were obtained at 30min, 24h, and 7 day stirring time. TAC results show dose dependent behavior of CuO-rGO and confirm no influence of stirring time on antioxidant properties, 30min is considered as an optimal stirring condition.

Keywords: copper oxide, reduced graphene oxide, TAC, GO

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Authors: Johnson Oladele Fatokun, I. P. Akpan

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In this paper, the one-dimensional time dependent Schrödinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff ordinary differential equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10-4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

Keywords: Schrodinger’s equation, partial differential equations, method of lines (MOL), stiff ODE, trapezoidal-like integrator

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Authors: Tippawan Nasawan

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The objective of this study is to find the replenishment policy in Assembly-to-Order manufacturing (ATO) which some of the major components have lead-time longer than customer lead-time. The variety of products, independent component demand, and long component lead-time are the difficulty that has resulted in the overstock problem. In addition, the ordering cost is trivial when compared to the cost of material of the major component. A conceptual design of the Decision Supporting System (DSS) has introduced to assist the replenishment policy. Component replenishment by using the variable which calls Available to Promise (ATP) for making the decision is one of the keys. The Poisson distribution is adopted to realize demand patterns in order to calculate Safety Stock (SS) at the specified Customer Service Level (CSL). When distribution cannot identify, nonparametric will be applied instead. The test result after comparing the ending inventory between the new policy and the old policy, the overstock has significantly reduced by 46.9 percent or about 469,891.51 US-Dollars for the cost of the major component (material cost only). Besides, the number of the major component inventory is also reduced by about 41 percent which helps to mitigate the chance of damage and keeping stock.

Keywords: Assembly-to-Order, Decision Supporting System, Component replenishment , Poisson distribution

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Authors: Elida Gastelum-Martinez, Neith A. Pacheco-Lopez, Juan L. Morales-Landa

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Cashew agroindustry obtained from cashew apple crop (Anacardium occidentale L.) generates large amounts of unused waste in Campeche, Mexico. Despite having a high content of nutritional compounds such as ascorbic acid, carotenoids, fiber, carbohydrates, and minerals, it is not consumed due to its astringent sensation. The aim of this work was to develop a processing method for cashew apple waste in order to obtain a powder with reduced astringency able to be used as an additive in the food industry. The processing method consisted first in reducing astringency by inducing tannins from cashew apple peel to react and form precipitating complexes with a colloid rich in proline and histidine. Then cashew apples were processed to obtain a dry powder. Astringency reduction was determined by total phenolic content and evaluated by sensorial analysis in cashew-apple-powder based cookies. Total phenolic content in processed powders showed up to 72% lower concentration compared to control samples. The sensorial evaluation indicated that cookies baked using cashew apple powder with reduced astringency were 96.8% preferred. Sensorial characteristics like texture, color and taste were also well-accepted attributes. In conclusion, the method applied for astringency reduction is a viable tool to produce cashew apple powder with desirable sensorial properties to be used in the development of food products.

Keywords: astringency reduction, cashew apple waste, food industry, sensorial evaluation

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Authors: Adamu S. Salawu, Ibrahim O. Isah

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Direct solution to several forms of fourth-order ordinary differential equations is not easily obtained without first reducing them to a system of first-order equations. Thus, numerical methods are being developed with the underlying techniques in the literature, which seeks to approximate some classes of fourth-order initial value problems with admissible error bounds. Multistep methods present a great advantage of the ease of implementation but with a setback of several functions evaluation for every stage of implementation. However, hybrid methods conventionally show a slightly higher order of truncation for any k-step linear multistep method, with the possibility of obtaining solutions at off mesh points within the interval of solution. In the light of the foregoing, we propose the continuous form of a hybrid multistep method with Chebyshev polynomial as a basis function for the numerical integration of fourth-order initial value problems of ordinary differential equations. The basis function is interpolated and collocated at some points on the interval [0, 2] to yield a system of equations, which is solved to obtain the unknowns of the approximating polynomial. The continuous form obtained, its first and second derivatives are evaluated at carefully chosen points to obtain the proposed block method needed to directly approximate fourth-order initial value problems. The method is analyzed for convergence. Implementation of the method is done by conducting numerical experiments on some test problems. The outcome of the implementation of the method suggests that the method performs well on problems with oscillatory or trigonometric terms since the approximations at several points on the solution domain did not deviate too far from the theoretical solutions. The method also shows better performance compared with an existing hybrid method when implemented on a larger interval of solution.

Keywords: Chebyshev polynomial, collocation, hybrid multistep method, initial value problems, interpolation

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Authors: Hakiki Kheira, Belhamiti Omar

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In this paper, we propose a new compartmental fractional order model for the simulation of epidemic obesity dynamics. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. We also present some fractional differential illustrative examples to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

Keywords: Caputo derivative, epidemiology, Legendre wavelet method, obesity

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Authors: Siavash Soltani Hemmat

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Although 21st-century humanity is at the height of technology and has stepped toward the endless boundaries of knowledge, there are still people in many parts of the world who are deprived of even the most fundamental freedoms. Whereas without freedom, no possible meaning can be imagined for human life, none of the human talents will have the chance to flourish, and that man will be reduced to the level of an animal, removing the obstacles to human freedom, especially from the viewpoint of thoughts, is of utmost importance, in which the liberal ideas of the modern centuries have played an incomparable role. The aim of the present study is to introduce and explain the liberal ideas in the modern centuries and their role in the expansion of human freedoms in order to weaken and discredit the ideological and intellectual barriers to restricting the freedom of individuals and to pave the way for the liberation of humanity. A descriptive method has been employed in order to achieve the objectives of the research. Besides, for data collection, a library method has been conducted. In this study, three ideological teachings of the social contract , resistance against unjust governance and natural law were recognized as the foundations of the realization of fundamental freedoms of the people in the modern centuries and their content was explained and examined.

Keywords: freedom, natural law, social contract, resistance

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Authors: Reza Mohammadi

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Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods.

Keywords: second-order ordinary differential equation, singularly-perturbed, quartic spline, convergence analysis

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