Search results for: the approximated method

Authors: Clarinda Vitorino Nhangumbe, Fredericks Ebrahim, Betuel Canhanga

Abstract:

The price of an option weather derivatives can be approximated as a solution of the two-dimensional convection-diffusion dominant partial differential equation derived from the Ornstein-Uhlenbeck process, where one variable represents the weather dynamics and the other variable represent the underlying weather index. With appropriate financial boundary conditions, the solution of the pricing equation is approximated using a nonstandard finite difference method. It is shown that the proposed numerical scheme preserves positivity as well as stability and consistency. In order to illustrate the accuracy of the method, the numerical results are compared with other methods. The model is tested for real weather data.

Keywords: nonstandard finite differences, Ornstein-Uhlenbeck process, partial differential equations approach, weather derivatives

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Authors: Haci Mehmet Guzey, Levent Acar

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In this paper, a novel decentralized controller is developed for linear systems with nonlinear unknown interconnections. A model linear decoupled system is assigned for each system. By using the difference actual and model state dynamics, the problem is formulated as inverse problem. Then, the interconnected dynamics are approximated by using Galerkin’s expansion method for inverse problems. Two different sets of orthogonal basis functions are utilized to approximate the interconnected dynamics. Approximated interconnections are utilized in the controller to cancel the interconnections and decouple the systems. Subsequently, the interconnected systems behave as a collection of decoupled systems.

Keywords: decentralized control, inverse problems, large scale systems, nonlinear interconnections, basis functions, system identification

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Authors: Fujio Akagi, Hiroaki Ito, Shin-Ichi Inage

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The aim of this study is to develop a combustion model that can be applied uniformly to laminar and turbulent premixed flames while considering the effect of the Lewis number (Le). The model considers the effect of Le on the transport equations of the reaction progress, which varies with the chemical species and temperature. The distribution of the reaction progress variable is approximated by a hyperbolic tangent function, while the other distribution of the reaction progress variable is estimated using the approximated distribution and transport equation of the reaction progress variable considering the Le. The validity of the model was evaluated under the conditions of propane with Le > 1 and methane with Le = 1 (equivalence ratios of 0.5 and 1). The estimated results were found to be in good agreement with those of previous studies under all conditions. A method of introducing a turbulence model into this model is also described. It was confirmed that conventional turbulence models can be expressed as an approximate theory of this model in a unified manner.

Keywords: combustion model, laminar flame, Lewis number, turbulent flame

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Authors: Mohammed Wadi, Bedri Kekezoglu, Mustafa Baysal, Mehmet Rida Tur, Abdulfetah Shobole

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This paper investigates the technical evaluation of the wind potential for present and future investments in Turkey taking into account the feasibility of sites, installments, operation, and maintenance. This evaluation based on the hourly measured wind speed data for the three years 2008–2010 at 30 m height for Çatalca district. These data were obtained from national meteorology station in Istanbul–Republic of Turkey are analyzed in order to evaluate the feasibility of wind power potential and to assure supreme assortment of wind turbines installing for the area of interest. Furthermore, the data are extrapolated and analyzed at 60 m and 80 m regarding the variability of roughness factor. Weibull bi-parameter probability function is used to approximate monthly and annually wind potential and power density based on three calculation methods namely, the approximated, the graphical and the energy pattern factor methods. The annual mean wind power densities were to be 400.31, 540.08 and 611.02 W/m² for 30, 60, and 80 m heights respectively. Simulation results prove that the analyzed area is an appropriate place for constructing large-scale wind farms.

Keywords: wind potential in Turkey, Weibull bi-parameter probability function, the approximated method, the graphical method, the energy pattern factor method, capacity factor

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Authors: Kamel Al-Khaled

Abstract:

In this paper, numerical solutions for the nonlinear coupled Korteweg-de Vries, (abbreviated as KdV) equations are calculated by Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. First, discretizing time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a $\theta-$weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method.

Keywords: Nonlinear coupled KdV equations, Soliton solutions, Sinc-collocation method, Sinc functions

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Authors: Rong-Tsorng Wang

Abstract:

In this paper, we derive a pricing formula for catastrophe equity put options (or CatEPut) with non-homogeneous loss and approximated compound distributions. We assume that the loss claims arrival process is a nonhomogeneous Poisson process (NHPP) representing the clustering occurrences of loss claims, the size of loss claims is a sequence of independent and identically distributed random variables, and the accumulated loss distribution forms a compound distribution and is approximated by a heavy-tailed distribution. A numerical example is given to calibrate parameters, and we discuss how the value of CatEPut is affected by the changes of parameters in the pricing model we provided.

Keywords: catastrophe equity put options, compound distributions, nonhomogeneous Poisson process, pricing model

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Authors: S. Aizikovich, L. Krenev, Y. Tokovyy, Y. C. Wang

Abstract:

An approximate analytical-numerical solution to the axisymmetric problem on thermo-mechanical indentation of a flat cylindrical punch into an arbitrarily non-homogeneous elastic half-space is constructed by making use of the bilateral asymptotic method. The key point of this method lies in evaluation of the ker¬nels in the obtained integral equations by making use of a numerical technique. Once the structure of the kernel is defined, it then is approximated by an analytical expression of special kind so that the solution of the integral equation can be achieved analytically. This fact allows for construction of the solution in an analytical form, which is convenient for analysis of the mechanical effects concerned with arbitrarily presumed non-homogeneity of the material.

Keywords: contact problem, circular punch, arbitrarily-nonhomogeneous halfspace

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Authors: Emmanuel Omokhuale

Abstract:

A mathematical model is presented to study free convective boundary layer flow in a semi-infinite vertical cylinder with heat sink effect in a porous medium. The governing dimensional governing partial differential equations (PDEs) with corresponding initial and boundary conditions are approximated and solved numerically employing finite difference method (FDM) the implicit type. Stability and convergence of the scheme are also established. Furthermore, the influence of significant physical parameters on the flow characteristics was analysed and shown graphically. The obtained results are benchmarked with previously published works in order to access the accuracy of the numerical method and found to be in good agreement.

Keywords: free convection flow, vertical cylinder, implicit finite difference method, heat sink and porous medium

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Authors: Mahdad M’hamed, Belaidi Idir

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This paper develops a meshless approach, called Element Free Galerkin (EFG) method, which is based on the weak form Moving Least Squares (MLS) of the partial differential governing equations and employs the interpolation to construct the meshless shape functions. The variation weak form is used in the EFG where the trial and test functions are approximated bye the MLS approximation. Since the shape functions constructed by this discretization have the weight function property based on the randomly distributed points, the essential boundary conditions can be implemented easily. The local weak form of the partial differential governing equations is obtained by the weighted residual method within the simple local quadrature domain. The spline function with high continuity is used as the weight function. The presently developed EFG method is a truly meshless method, as it does not require the mesh, either for the construction of the shape functions, or for the integration of the local weak form. Several numerical examples of two-dimensional static structural analysis are presented to illustrate the performance of the present EFG method. They show that the EFG method is highly efficient for the implementation and highly accurate for the computation. The present method is used to analyze the static deflection of beams and plate hole

Keywords: numerical computation, element-free Galerkin (EFG), moving least squares (MLS), meshless methods

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Authors: Itissam Abuiziah

Abstract:

In several hydraulic systems, it is necessary to calculate the head losses which depend on the resistance flow friction factor in Darcy equation. Computing the resistance friction is based on implicit Colebrook-White equation which is considered as the standard for the friction calculation, but it needs high computational cost, therefore; several explicit approximation methods are used for solving an implicit equation to overcome this issue. It follows that the relative error is used to determine the most accurate method among the approximated used ones. Steel, cast iron and polyethylene pipe materials investigated with practical diameters ranged from 0.1m to 2.5m and velocities between 0.6m/s to 3m/s. In short, the results obtained show that the suitable method for some cases may not be accurate for other cases. For example, when using steel pipe materials, Zigrang and Silvester's method has revealed as the most precise in terms of low velocities 0.6 m/s to 1.3m/s. Comparatively, Halland method showed a less relative error with the gradual increase in velocity. Accordingly, the simulation results of this study might be employed by the hydraulic engineers, so they can take advantage to decide which is the most applicable method according to their practical pipe system expectations.

Keywords: Colebrook–White, explicit equation, friction factor, hydraulic resistance, implicit equation, Reynolds numbers

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Authors: Autcha Araveeporn

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This paper is to compare the parameter estimation of the mean in normal distribution by Maximum Likelihood (ML), Bayes, and Markov Chain Monte Carlo (MCMC) methods. The ML estimator is estimated by the average of data, the Bayes method is considered from the prior distribution to estimate Bayes estimator, and MCMC estimator is approximated by Gibbs sampling from posterior distribution. These methods are also to estimate a parameter then the hypothesis testing is used to check a robustness of the estimators. Data are simulated from normal distribution with the true parameter of mean 2, and variance 4, 9, and 16 when the sample sizes is set as 10, 20, 30, and 50. From the results, it can be seen that the estimation of MLE, and MCMC are perceivably different from the true parameter when the sample size is 10 and 20 with variance 16. Furthermore, the Bayes estimator is estimated from the prior distribution when mean is 1, and variance is 12 which showed the significant difference in mean with variance 9 at the sample size 10 and 20.

Keywords: Bayes method, Markov chain Monte Carlo method, maximum likelihood method, normal distribution

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Authors: H. C. Nguyen

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This contribution considers seismic effects on the stability of slope and footing resting on a slope. The seismic force is simply treated as static inertial force through the values of acceleration factor. All domains are assumed to be plasticity deformations approximated using node-based smoothed finite element method (NS-FEM). The failure mechanism and safety factor were then explored using numerical procedure based on upper bound approach in which optimization problem was formed as second order cone programming (SOCP). The data obtained confirm that upper bound procedure using NS-FEM and SOCP can give stable and rapid convergence results of seismic stability factors.

Keywords: upper bound analysis, safety factor, slope stability, footing resting on slope

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Authors: Marasovic Branka, Aljinovic Zdravka, Poklepovic Tea

Abstract:

Numerical methods like binomial and trinomial trees and finite difference methods can be used to price a wide range of options contracts for which there are no known analytical solutions. American options are the most famous of that kind of options. Besides numerical methods, American options can be valued with the approximation formulas, like Bjerksund-Stensland formulas from 1993 and 2002. When the value of American option is approximated by Bjerksund-Stensland formulas, the computer time spent to carry out that calculation is very short. The computer time spent using numerical methods can vary from less than one second to several minutes or even hours. However to be able to conduct a comparative analysis of numerical methods and Bjerksund-Stensland formulas, we will limit computer calculation time of numerical method to less than one second. Therefore, we ask the question: Which method will be most accurate at nearly the same computer calculation time?

Keywords: Bjerksund and Stensland approximations, computational analysis, finance, options pricing, numerical methods

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Authors: N. Ben-Mahmoud, M. Elfandi, A. Shallof

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This paper presents a methodology to design multivariable PID controllers for multi-input and multi-output systems. The proposed control strategy, which is centralized, combines of PID controllers. The proportional gains in the P controllers act as tuning parameters of (SISO) in order to modify the behavior of the loops almost independently. The design procedure consists of three steps: first, an ideal decoupler including integral action is determined. Second, the decoupler is approximated with PID controllers. Third, the proportional gains are tuned to achieve the specified performance. The proposed method is applied to representative processes.

Keywords: boiler turbine, MIMO, PID controller, control by decoupling, anti wind-up techniques

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Authors: Xuezhang Hou

Abstract:

In the present paper, a hybrid hyperbolic dynamic system formulated by partial differential equations with initial and boundary conditions is considered. First, the system is transformed to an abstract evolution system in an appropriate Hilbert space, and spectral analysis and semigroup generation of the system operator is discussed. Subsequently, a sliding model control problem is proposed and investigated, and an equivalent control method is introduced and applied to the system. Finally, a significant result that the state of the system can be approximated by an ideal sliding mode under control in any accuracy is derived and examined.

Keywords: hyperbolic dynamic system, sliding model control, semigroup of linear operators, partial differential equations

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Authors: Valeria Bondarenko

Abstract:

We proposed algorithms for: estimation of parameters fBm (volatility and Hurst exponent) and for the approximation of random time series by functional of fBm. We proved the consistency of the estimators, which constitute the above algorithms, and proved the optimal forecast of approximated time series. The adequacy of estimation algorithms, approximation, and forecasting is proved by numerical experiment. During the process of creating software, the system has been created, which is displayed by the hierarchical structure. The comparative analysis of proposed algorithms with the other methods gives evidence of the advantage of approximation method. The results can be used to develop methods for the analysis and modeling of time series describing the economic, physical, biological and other processes.

Keywords: mathematical model, random process, Wiener process, fractional Brownian motion

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Authors: P.-W. Tsai, C.-Y. Chen, C.-W. Chen

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In this paper, the stability analysis of a GA-Based adaptive fuzzy sliding model controller for a nonlinear system is discussed. First, a nonlinear plant is well-approximated and described with a reference model and a fuzzy model, both involving FLC rules. Then, FLC rules and the consequent parameter are decided on via an Evolved Bat Algorithm (EBA). After this, we guarantee a new tracking performance inequality for the control system. The tracking problem is characterized to solve an eigenvalue problem (EVP). Next, an adaptive fuzzy sliding model controller (AFSMC) is proposed to stabilize the system so as to achieve good control performance. Lyapunov’s direct method can be used to ensure the stability of the nonlinear system. It is shown that the stability analysis can reduce nonlinear systems into a linear matrix inequality (LMI) problem. Finally, a numerical simulation is provided to demonstrate the control methodology.

Keywords: adaptive fuzzy sliding mode control, Lyapunov direct method, swarm intelligence, evolved bat algorithm

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Authors: Xuezhang Hou

Abstract:

A control problem of a flexible Lamina formulated by partial differential equations with viscoelastic boundary conditions is studied in this paper. The problem is written in standard form of linear infinite dimensional system in an appropriate energy Hilbert space. The semigroup approach of linear operators is adopted in investigating wellposedness of the closed loop system. A variable structural control for the system is proposed, and meanwhile an equivalent control method is applied to the thin plate system. A significant result on control theory that the thin plate can be approximated by ideal sliding mode in any accuracy in terms of semigroup approach is obtained.

Keywords: partial differential equations, flexible lamina, variable structural control, semigroup of linear operators

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Authors: Duy Cuong Nguyen, Ali Makke, Guillaume Montay

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This work presents an improved single fiber pull-out test for fiber/matrix interface characterization. This test has been used to study the Inter-Facial Shear Strength ‘IFSS’ of hemp fibers reinforced polypropylene (PP). For this aim, the fiber diameter has been carefully measured using a tomography inspired method. The fiber section contour can then be approximated by a circle or a polygon. The results show that the IFSS is overestimated if the circular approximation is used. The Influence of the molding temperature on the IFSS has also been studied. We find a molding temperature of 183°C leads to better interface properties. Above or below this temperature the interface strength is reduced.

Keywords: composite, hemp, interface, pull-out, processing, polypropylene, temperature

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Authors: Z. B. Ibrahim, N. Ismail, K. I. Othman

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Fuzzy Differential Equations (FDEs) play an important role in modelling many real life phenomena. The FDEs are used to model the behaviour of the problems that are subjected to uncertainty, vague or imprecise information that constantly arise in mathematical models in various branches of science and engineering. These uncertainties have to be taken into account in order to obtain a more realistic model and many of these models are often difficult and sometimes impossible to obtain the analytic solutions. Thus, many authors have attempted to extend or modified the existing numerical methods developed for solving Ordinary Differential Equations (ODEs) into fuzzy version in order to suit for solving the FDEs. Therefore, in this paper, we proposed the development of a fuzzy version of three-point block method based on Block Backward Differentiation Formulas (FBBDF) for the numerical solution of first order FDEs. The three-point block FBBDF method are implemented in uniform step size produces three new approximations simultaneously at each integration step using the same back values. Newton iteration of the FBBDF is formulated and the implementation is based on the predictor and corrector formulas in the PECE mode. For greater efficiency of the block method, the coefficients of the FBBDF are stored at the start of the program. The proposed FBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with the existing fuzzy version of the Modified Simpson and Euler methods in terms of the accuracy of the approximated solutions. The numerical results show that the FBBDF method performs better in terms of accuracy when compared to the Euler method when solving the FDEs.

Keywords: block, backward differentiation formulas, first order, fuzzy differential equations

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Authors: Yoshifumi Hino

Abstract:

An infinitely repeated prisoner’s dilemma is a typical model that represents teamwork situation. If both players choose costly actions and contribute to the team, then both players are better off. However, each player has an incentive to choose a selfish action. We analyze the game under costly observation. Each player can observe the action of the opponent only when he pays an observation cost in that period. In reality, teamwork situations are often costly observation. Members of some teams sometimes work in distinct rooms, areas, or countries. In those cases, they have to spend their time and money to see other team members if they want to observe it. The costly observation assumption makes the cooperation difficult substantially because the equilibrium must satisfy the incentives not only on the action but also on the observational decision. Especially, it is the most difficult to cooperate each other when the stage-game is prisoner's dilemma because players have to communicate through only two actions. We examine whether or not players can cooperate each other in prisoner’s dilemma under costly observation. Specifically, we check whether symmetric Pareto efficient payoff vectors in repeated prisoner’s dilemma can be approximated by sequential equilibria or not (efficiency result). We show the efficiency result without any randomization device under certain circumstances. It means that players can cooperate with each other without any randomization device even if the observation is costly. Next, we assume that public randomization device is available, and then we show that any feasible and individual rational payoffs in prisoner’s dilemma can be approximated by sequential equilibria under a specific situation (folk theorem). It implies that players can achieve asymmetric teamwork like leadership situation when public randomization device is available.

Keywords: cost observation, efficiency, folk theorem, prisoner's dilemma, private monitoring, repeated games.

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Authors: Rita Rani, Ramesh Kumar

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The intend of the current study is to date an old grand Peepal tree that is still alive. The tree is situated in Kalibard village, Sector 9, Chandigarh (India). Due to its huge structure, it has got the status of ‘Heritage tree.’ Optically Stimulated Luminescence of sediments beneath the roots is used to determine the age of the tree. Optical dating is preferred over conventional dating methods due to more precession. The methodology includes OSL of quartz grain using SAR protocol for accumulated dose measurement. The age determination of an alive tree using sedimentary quartz is in close agreement with the approximated age provided by the related agency. This is the first attempt at using optically stimulated luminescence in the age determination of alive trees in this region. The study concludes that the Luminescence dating of alive trees is the nondestructive and more precise method.

Keywords: luminescence, dose rate, optical dating, sediments

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Authors: Neha Bhardwaj

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In this paper, we consider positive linear operators and study the Voronovskaya type result of the operator then obtain an error estimate in terms of the higher order modulus of continuity of the function being approximated and its A-statistical convergence. Also, we compute the corresponding rate of A-statistical convergence for the linear positive operators.

Keywords: Poisson distribution, Voronovskaya, modulus of continuity, a-statistical convergence

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Authors: Zehai Yu, Hui Zhu, Linglong Lin, Huawei Liang, Biao Yu, Weixin Huang

Abstract:

With the development of intelligent vehicle systems, a high-precision road map is increasingly needed in many aspects. The automatic lane lines extraction and modeling are the most essential steps for the generation of a precise lane-level road map. In this paper, an automatic lane-level road map generation system is proposed. To extract the road markings on the ground, the multi-region Otsu thresholding method is applied, which calculates the intensity value of laser data that maximizes the variance between background and road markings. The extracted road marking points are then projected to the raster image and clustered using a two-stage clustering algorithm. Lane lines are subsequently recognized from these clusters by the shape features of their minimum bounding rectangle. To ensure the storage efficiency of the map, the lane lines are approximated to cubic polynomial curves using a Bayesian estimation approach. The proposed lane-level road map generation system has been tested on urban and expressway conditions in Hefei, China. The experimental results on the datasets show that our method can achieve excellent extraction and clustering effect, and the fitted lines can reach a high position accuracy with an error of less than 10 cm.

Keywords: curve fitting, lane-level road map, line recognition, multi-thresholding, two-stage clustering

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Authors: Ainul Haque, Ameeye Kumar Nayak

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Flow enhancement and species transport in a slit hydrophobic microchannel is studied for non-Newtonian fluids with the externally imposed electric field and pressure gradient. The incompressible Poisson-Nernst-Plank equations and the Navier-Stokes equations are approximated by lubrication theory to quantify the flow structure due to hydrophobic and hydrophilic surfaces. The analytical quantification of velocity and pressure of electroosmotic flow (EOF) is made with the numerical results due to the staggered grid based finite volume method for flow governing equations. The resistance force due to fluid friction and shear force along the surface are decreased by the hydrophobicity, enables the faster movement of fluid particles. The resulting flow enhancement factor Ef is increased with the low viscous fluid and provides maximum species transport. Also, the analytical comparison of EOF with pressure driven EOF justifies the flow enhancement due to hydrophobicity and shear impact on flow variation.

Keywords: electroosmotic flow, hydrophobic surface, power-law fluid, shear effect

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Authors: Gregor Kosec

Abstract:

Simple and robust numerical approach for solving flow problems is presented, where involved physical fields are represented through the local approximation functions, i.e., the considered field is approximated over a local support domain. The approximation functions are then used to evaluate the partial differential operators. The type of approximation, the size of support domain, and the type and number of basis function can be general. The solution procedure is formulated completely through local computational operations. Besides local numerical method also the pressure velocity is performed locally with retaining the correct temporal transient. The complete locality of the introduced numerical scheme has several beneficial effects. One of the most attractive is the simplicity since it could be understood as a generalized Finite Differences Method, however, much more powerful. Presented methodology offers many possibilities for treating challenging cases, e.g. nodal adaptivity to address regions with sharp discontinuities or p-adaptivity to treat obscure anomalies in physical field. The stability versus computation complexity and accuracy can be regulated by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation. The presented methodology is relatively simple to understand and implement, which makes it potentially powerful tool for engineering simulations. Besides simplicity and straightforward implementation, there are many opportunities to fully exploit modern computer architectures through different parallel computing strategies. The performance of the method is presented on the lid driven cavity problem, backward facing step problem, de Vahl Davis natural convection test, extended also to low Prandtl fluid and Darcy porous flow. Results are presented in terms of velocity profiles, convergence plots, and stability analyses. Results of all cases are also compared against published data.

Keywords: fluid flow, meshless, low Pr problem, natural convection

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Authors: Ofoegbu Ositadinma Edward

Abstract:

This research paper discusses a web based monitoring system that enables effective monitoring of fuel pump output and sales volume from distributed fuel stations under the domain of a single company/organization. The traditional method of operation by these organizations in Nigeria is non-automated and accounting for dispensed product is usually approximated and manual as there is little or no technology implemented to presently provide information relating to the state of affairs in the station both to on-ground staff and to supervisory staff that are not physically present in the station. This results in unaccountable losses in product and revenue as well as slow decision making. Remote monitoring technology as a vast research field with numerous application areas incorporating various data collation techniques and sensor networks can be applied to provide information relating to fuel pump status in distributed fuel stations reliably. Thus, the proposed system relies upon a microcontroller, keypad and pump to demonstrate the traditional fuel dispenser. A web-enabled PC with an accompanying graphic user interface (GUI) was designed using virtual basic which is connected to the microcontroller via the serial port which is to provide the web implementation.

Keywords: fuel pump, microcontroller, GUI, web

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Authors: Marek Dlapa

Abstract:

The algebraic approach is applied to the control of the HiMAT (Highly Maneuverable Aircraft Technology). The objective is to find a robust controller which guarantees robust stability and decoupled control of longitudinal model of a scaled remotely controlled vehicle version of the advanced fighter HiMAT. Control design is performed by decoupling the nominal MIMO (multi-input multi-output) system into two identical SISO (single-input single-output) plants which are approximated by a 4th order transfer function. The algebraic approach is then used for pole placement design, and the nominal closed-loop poles are tuned so that the peak of the µ-function is minimal. As an optimization tool, evolutionary algorithm Differential Migration is used in order to overcome the multimodality of the cost function yielding simple controller with decoupling for nominal plant which is compared with the D-K iteration through simulations of standard longitudinal manoeuvres documenting decoupled control obtained from algebraic approach for nominal plant as well as worst case perturbation.

Keywords: algebraic approach, evolutionary computation, genetic algorithms, HiMAT, robust control, structured singular value

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Authors: Muhammad Jawwad, Riffat Iqbal

Abstract:

The Socratic method, also known as the dialectical method and elenctic method, has significant relevance in the contemporary educational system. It can be incorporated into modern-day educational systems theoretically as well as practically. Being interactive and dialogue-based in nature, this teaching approach is followed by critical thinking and innovation. The pragmatic value of the Dialectical Method has been discussed in this article, and the limitations of the Socratic method have also been highlighted. The interactive Method of Socrates can be used in many subjects for students of different grades. The Limitations and delimitations of the Method have also been discussed for its proper implementation. This article has attempted to elaborate and analyze the teaching method of Socrates with all its pre-suppositions and Epistemological character.

Keywords: Socratic method, dialectical method, knowledge, teaching, virtue

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Authors: Y. N. Reddy

Abstract:

The class of differential-difference equations which have characteristics of both classes, i.e., delay/advance and singularly perturbed behaviour is known as singularly perturbed differential-difference equations. The expression ‘positive shift’ and ‘negative shift’ are also used for ‘advance’ and ‘delay’ respectively. In general, an ordinary differential equation in which the highest order derivative is multiplied by a small positive parameter and containing at least one delay/advance is known as singularly perturbed differential-difference equation. Singularly perturbed differential-difference equations arise in the modelling of various practical phenomena in bioscience, engineering, control theory, specifically in variational problems, in describing the human pupil-light reflex, in a variety of models for physiological processes or diseases and first exit time problems in the modelling of the determination of expected time for the generation of action potential in nerve cells by random synaptic inputs in dendrites. In this paper, we envisage the use of Liouville Green Transformation to find the solution of singularly perturbed differential difference equations. First, using Taylor series, the given singularly perturbed differential difference equation is approximated by an asymptotically equivalent singularly perturbation problem. Then the Liouville Green Transformation is applied to get the solution. Several model examples are solved, and the results are compared with other methods. It is observed that the present method gives better approximate solutions.

Keywords: difference equations, differential equations, singular perturbations, boundary layer

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