Search results for: fractional order differentiator

Authors: Seok Hong Min, Tae Kwon Ha

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With the increasing interest on Ti alloys, the extraction process of Ti from its typical ore, TiO₂, has long been and will be important issue. As an intermediate product for the production of pigment or titanium metal sponge, tetrachloride (TiCl₄) is produced by fluidized bed using high TiO₂ feedstock. The purity of TiCl₄ after chlorination is subjected to the quality of the titanium feedstock. Since the impurities in the TiCl₄ product are reported to final products, the purification process of the crude TiCl₄ is required. The purification process includes fractional distillation and chemical treatment, which depends on the nature of the impurities present and the required quality of the final product. In this study, thermodynamic analysis on the impurity effect in the chlorination process, which is the first step of extraction of Ti from TiO₂, has been conducted. All thermodynamic calculations were performed using the FactSage thermodynamical software.

Keywords: rutile, titanium, chlorination process, impurities, thermodynamic calculation, FactSage

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Authors: Henri Champliaud, Zhengkun Feng, Ngan Van Lê, Javad Gholipour

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In this article, a method is presented to effectively estimate the deformed shape of a thick plate due to line heating. The method uses a fifth order spline interpolation, with up to C3 continuity at specific points to compute the shape of the deformed geometry. First and second order derivatives over a surface are the resulting parameters of a given heating line on a plate. These parameters are determined through experiments and/or finite element simulations. Very accurate kriging models are fitted to real or virtual surfaces to build-up a database of maps. Maps of first and second order derivatives are then applied on numerical plate models to evaluate their evolving shapes through a sequence of heating lines. Adding an optimization process to this approach would allow determining the trajectories of heating lines needed to shape complex geometries, such as Francis turbine blades.

Keywords: deformation, kriging, fifth order spline interpolation, first, second and third order derivatives, C3 continuity, line heating, plate forming, thermal forming

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Authors: Wei Yao

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A steerable and flexible drill is needed in orthopaedic surgery. For example, osteoarthritis is a common condition affecting millions of people for which joint replacement is an effective treatment which improves the quality and duration of life in elderly sufferers. Conventional surgery is not very accurate. Computer navigation and robotics can help increase the accuracy. For example, In Total Hip Arthroplasty (THA), robotic surgery is currently practiced mainly on acetabular side helping cup positioning and orientation. However, femoral stem positioning mostly uses hand-rasping method rather than robots for accurate positioning. The other case for using a flexible drill in surgery is Anterior Cruciate Ligament (ACL) Reconstruction. The majority of ACL Reconstruction failures are primarily caused by technical mistakes and surgical errors resulting from drilling the anatomical bone tunnels required to accommodate the ligament graft. The proposed new steerable drill system will perform orthopedic surgery through curved tunneling leading to better accuracy and patient outcomes. It may reduce intra-operative fractures, dislocations, early failure and leg length discrepancy by making possible a new level of precision. This technology is based on a robotically assisted, steerable, hand-held flexible drill, with a drill-tip tracking device and a multi-modality navigation system. The critical differentiator is that this robotically assisted surgical technology now allows the surgeon to prepare 'patient specific' and more anatomically correct 'curved' bone tunnels during orthopedic surgery rather than drilling straight holes as occurs currently with existing surgical tools. The flexible and steerable drill and its navigation system for femoral milling in total hip arthroplasty had been tested on sawbones to evaluate the accuracy of the positioning and orientation of femoral stem relative to the pre-operative plan. The data show the accuracy of the navigation system is better than traditional hand-rasping method.

Keywords: navigation, robotic orthopedic surgery, steerable drill, tracking

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Authors: A. M. Sagir

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This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y^'''= f(x,y,y^',y^'' ), y(α)=y_0,〖y〗^' (α)=β,y^('' ) (α)=μ with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non-stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

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

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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: Mohamed Suleiman, Zarina Bibi Ibrahim, Nor Ain Azeany, Khairil Iskandar Othman

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In this paper, a class of variable order fully implicit multistep Block Backward Differentiation Formulas (VOBBDF) using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs) is developed. The code will combine three multistep block methods of order four, five and six. The order selection is based on approximation of the local errors with specific tolerance. These methods are constructed to produce two approximate solutions simultaneously at each iteration in order to further increase the efficiency. The proposed VOBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with single order Block Backward Differentiation Formula (BBDF). Numerical results shows the advantage of using VOBBDF for solving ODEs.

Keywords: block backward differentiation formulas, uniform step size, ordinary differential equations

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Authors: Parshanth Maroju, Ramandeep Behl, Sandile S. Motsa

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In this study, we proposed a simple and interesting family of fourth-order multi-point methods without memory for obtaining simple roots. This family requires only three functional evaluations (viz. two of functions f(xn), f(yn) and third one of its first-order derivative f'(xn)) per iteration. Moreover, the accuracy and validity of new schemes is tested by a number of numerical examples are also proposed to illustrate their accuracy by comparing them with the new existing optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations. Further, the dynamic study of these methods also supports the theoretical aspect.

Keywords: basins of attraction, nonlinear equations, simple roots, Newton's method

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Authors: Sandeep Singh

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In this article, two classes of iterative schemes are proposed for approximating solutions of nonlinear systems of equations whose orders of convergence are six and eight respectively. Sixth order scheme requires the evaluation of two vector-functions, two first Fr'echet derivatives and three matrices inversion per iteration. This three-step sixth-order method is further extended to eighth-order method which requires one more step and the evaluation of one extra vector-function. Moreover, computational efficiency is compared with some other recently published methods in which we found, our methods are more efficient than existing numerical methods for higher and medium size nonlinear system of equations. Numerical tests are performed to validate the proposed schemes.

Keywords: Nonlinear systems, Computational complexity, order of convergence, Jarratt-type scheme

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Authors: Gubhinder Kundhi, Paul Rilstone

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Refined procedures for higher-order asymptotic theory for non-linear models are developed. These include a new method for deriving stochastic expansions of arbitrary order, new methods for evaluating the moments of polynomials of sample averages, a new method for deriving the approximate moments of the stochastic expansions; an application of these techniques to gather improved inferences with the weak instruments problem is considered. It is well established that Instrumental Variable (IV) estimators in the presence of weak instruments can be poorly behaved, in particular, be quite biased in finite samples. In our application, finite sample approximations to the distributions of these estimators are obtained using Edgeworth and Saddlepoint expansions. Departures from normality of the distributions of these estimators are analyzed using higher order analytical corrections in these expansions. In a Monte-Carlo experiment, the performance of these expansions is compared to the first order approximation and other methods commonly used in finite samples such as the bootstrap.

Keywords: edgeworth expansions, higher order asymptotics, saddlepoint expansions, weak instruments

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Authors: Ould Bouamama, M. Bressel, D. Hissel, M. Hilairet

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Fuel cell (FC) is one of the best alternatives of fossil energy. Recently, the research community of fuel cell has shown a considerable interest for diagnosis in view to ensure safety, security, and availability when faults occur in the process. The problematic for model based FC diagnosis consists in that the model is complex because of coupling of several kind of energies and the numerical values of parameters are not always known or are uncertain. The present paper deals with use of one tool: the Linear Fractional Transformation bond graph tool not only for uncertain modelling but also for monitorability (ability to detect and isolate faults) analysis and formal generation of robust fault indicators with respect to parameter uncertainties.The developed theory applied to a nonlinear FC system has proved its efficiency.

Keywords: bond graph, fuel cell, fault detection and isolation (FDI), robust diagnosis, structural analysis

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Authors: Ying Yi Wu

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The study of finite groups is a central topic in algebraic structures, and one of the most fundamental questions in this field is the classification of finite groups up to isomorphism. In this paper, we investigate the cyclic property of groups of prime order, which is a crucial result in the classification of finite abelian groups. We prove the following statement: If p is a prime, then every group G of order p is cyclic. Our proof utilizes the properties of group actions and the class equation, which provide a powerful tool for studying the structure of finite groups. In particular, we first show that any non-identity element of G generates a cyclic subgroup of G. Then, we establish the existence of an element of order p, which implies that G is generated by a single element. Finally, we demonstrate that any two generators of G are conjugate, which shows that G is a cyclic group. Our result has significant implications in the classification of finite groups, as it implies that any group of prime order is isomorphic to the cyclic group of the same order. Moreover, it provides a useful tool for understanding the structure of more complicated finite groups, as any finite abelian group can be decomposed into a direct product of cyclic groups. Our proof technique can also be extended to other areas of group theory, such as the classification of finite p-groups, where p is a prime. Therefore, our work has implications beyond the specific result we prove and can contribute to further research in algebraic structures.

Keywords: group theory, finite groups, cyclic groups, prime order, classification.

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Authors: Moh'd Alodat

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In this paper, we discuss the power function distribution and derive the maximum likelihood estimator of its parameter as well as the reliability parameter. We derive the large sample properties of the estimators based on the selective order statistic scheme. We conduct simulation studies to investigate the significance of the selective order statistic scheme in our setup and to compare the efficiency of the new proposed estimators.

Keywords: fisher information, maximum likelihood estimator, power function distribution, ranked set sampling, selective order statistics sampling

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Authors: A. A. James, A. O. Adesanya, M. R. Odekunle, D. G. Yakubu

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This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compete favorably with the existing methods.

Keywords: interpolation, approximate solution, collocation, differential system, half step, converges, block method, efficiency

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Authors: Huei Chu Weng

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This paper presents a study on the effect of second-order slip on forced convection through a long isoflux heated or cooled planar microchannel. The fully developed solutions of flow and thermal fields are analytically obtained on the basis of the second-order Maxwell-Burnett slip and local heat flux boundary conditions. Results reveal that when the average flow velocity increases or the wall heat flux amount decreases, the role of thermal creep becomes more insignificant, while the effect of second-order slip becomes larger. The second-order term in the Deissler slip boundary condition is found to contribute a positive velocity slip and then to lead to a lower pressure drop as well as a lower temperature rise for the heated-wall case or to a higher temperature rise for the cooled-wall case. These findings are contrary to predictions made by the Karniadakis slip model.

Keywords: microfluidics, forced convection, thermal creep, second-order boundary conditions

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Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

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Authors: A. Boateng, La Gil-Alana, M. Lesaoana; Hj. Siweya, A. Belete

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This study examines long memory or long-range dependence in the CPI inflation rates of Ghana and South Africa using Whittle methods and autoregressive fractionally integrated moving average (ARFIMA) models. Standard I(0)/I(1) methods such as Augmented Dickey-Fuller (ADF), Philips-Perron (PP) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests were also employed. Our findings indicate that long memory exists in the CPI inflation rates of both countries. After processing fractional differencing and determining the short memory components, the models were specified as ARFIMA (4,0.35,2) and ARFIMA (3,0.49,3) respectively for Ghana and South Africa. Consequently, the CPI inflation rates of both countries are fractionally integrated and mean reverting. The implication of this result will assist in policy formulation and identification of inflationary pressures in an economy.

Keywords: Consumer Price Index (CPI) inflation rates, Whittle method, long memory, ARFIMA model

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Authors: Muheddin Hamza, Entisar Etter

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This study is a part of investigating the possibility of blending two crude oils obtained from Libyan oil fields, namely crude oil (A) and crude oil (B) with different ratios, prior to blending the crude oils have to be compatible in order to avoid phase out and precipitation of asphaltene from the bulk of crude. The physical properties of both crudes such as density, viscosity, pour point and sulphur content were measured according to (ASTM) method. To examine the stability of both crudes and their blends, the oil compatibility model using microscopic, colloidal instability index (CII) using SARA analysis and asphaltene stabilization test using Turbiscan tests were conducted in the Libyan Petroleum Institute laboratories. Compatibility tests were carried out with both crude oils, the insolubility number (IN), and the solubility blending number (SBN), for both crude oils and their blends were calculated. The criteria for compatibility of any blend is that the volume average solubility blending number (SBN) is greater than the insolubility number (IN) of any component in the blend, the results indicated that both crudes were compatible. To support the results of compatibility tests the SARA analysis was done for the fractional determination of (saturates, aromatics, resins and asphaltenes) content. From this result, the colloidal Instability index (CII) and resin to asphaltenes ratio (R/A) were calculated for crudes and their blends. The results show that crude oil (B) which has higher (R/A) and lower (CII) is more stable than crude oil (A) and as the ratio of crude (B) increases in the blend the (CII) and (R/A) were improved, and the blends becomes more stable. Asphaltene stabilization test was also conducted for the crudes and their blends using Turbiscan MA200 according to the standard test method ASTM D7061-04, the Turbiscan shows that the crude (B) is more stable than crude (A) which shows a fair tendency. The (CII) and (R/A) were compared with the solubility number (SBN) for each crude and the blends along with Turbiscan results. The solubility blending number (SBN) of the crudes and their blends show that the crudes are compatible, also by comparing (R/A) and (SBN) values of the blends, it can be seen that they are complements of each other. All the experimental results show that the blends of both crudes are more stability.

Keywords: asphaltene, crude oil, compatibility, oil blends, resin, SARA

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Authors: Asma Ameur, Dhafer Malouche

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Nowadays, user comments on the Internet have an important impact on hotel bookings. This confirms that the e-reputation issue can influence the likelihood of customer loyalty to a hotel. In this way, e-reputation has become a real differentiator between hotels. For this reason, we have a unique opportunity in the opinion mining field to analyze the comments. In fact, this field provides the possibility of extracting information related to the polarity of user reviews. This sentimental study (Opinion Mining) represents a new line of research for analyzing the unstructured textual data. Knowing the score of e-reputation helps the hotelier to better manage his marketing strategy. The score we then obtain is translated into the image of hotels to differentiate between them. Therefore, this present research highlights the importance of hotel satisfaction ‘scoring. To calculate the satisfaction score, the sentimental analysis can be manipulated by several techniques of machine learning. In fact, this study treats the extracted textual data by using the Artificial Neural Networks Approach (ANNs). In this context, we adopt the aforementioned technique to extract information from the comments available in the ‘Trip Advisor’ website. This actual paper details the description and the modeling of the ANNs approach for the scoring of online hotel reviews. In summary, the validation of this used method provides a significant model for hotel sentiment analysis. So, it provides the possibility to determine precisely the polarity of the hotel users reviews. The empirical results show that the ANNs are an accurate approach for sentiment analysis. The obtained results show also that this proposed approach serves to the dimensionality reduction for textual data’ clustering. Thus, this study provides researchers with a useful exploration of this technique. Finally, we outline guidelines for future research in the hotel e-reputation field as comparing the ANNs with other technique.

Keywords: clustering, consumer behavior, data mining, e-reputation, machine learning, neural network, online hotel ‘reviews, opinion mining, scoring

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Authors: F. Majeed, D. V. Thiel, M. Shahpari

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An electrically small meander line antenna is designed for impedance matching with RF transceiver chip CC1101. The design provides the flexibility of tuning the reactance of the antenna over a wide range of values: highly capacitive to highly inductive. The antenna was printed with silver ink on FR4 substrate using the screen printing design process. The antenna impedance was perfectly matched to CC1101 at 433 MHz. The measured radiation efficiency of the antenna was 81.3% at resonance. The 3 dB and 10 dB fractional bandwidth of the antenna was 14.5% and 4.78%, respectively. The read range of the antenna was compared with a copper wire monopole antenna over a distance of five meters. The antenna, with a perfect impedance match with RF transceiver chip CC1101, shows improvement in the read range compared to a monopole antenna over the specified distance.

Keywords: meander line antenna, RFID, silver ink printing, impedance matching

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Authors: Brando Vagenende, Marie-Anne Guerry

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For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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Authors: Rafat Alshorman, Fadi Awawdeh

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By employing a simplified bilinear method, a class of generalized fifth-order KdV (gfKdV) equations which arise in nonlinear lattice, plasma physics and ocean dynamics are investigated. With the aid of symbolic computation, both solitary wave solutions and multiple-soliton solutions are obtained. These new exact solutions will extend previous results and help us explain the properties of nonlinear solitary waves in many physical models in shallow water. Parametric analysis is carried out in order to illustrate that the soliton amplitude, width and velocity are affected by the coefficient parameters in the equation.

Keywords: multiple soliton solutions, fifth-order evolution equations, Cole-Hopf transformation, Hirota bilinear method

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Authors: Virginie Hergault, Siham Chebbah, Bertrand Frere

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Following in-house objectives, Central laboratory of Paris police Prefecture conducted a general review on models and Computational Fluid Dynamics (CFD) codes used to simulate pollutant dispersion in the atmosphere. Starting from that review and considering main features of Large Eddy Simulation, Central Laboratory Of Paris Police Prefecture (LCPP) postulates that the Fire Dynamics Simulator (FDS) model, from National Institute of Standards and Technology (NIST), should be well suited for air pollutant dispersion modeling. This paper focuses on the implementation and the evaluation of FDS in the frame of the European COST ES1006 Action. This action aimed at quantifying the performance of modeling approaches. In this paper, the CUTE dataset carried out in the city of Hamburg, and its mock-up has been used. We have performed a comparison of FDS results with wind tunnel measurements from CUTE trials on the one hand, and, on the other, with the models results involved in the COST Action. The most time-consuming part of creating input data for simulations is the transfer of obstacle geometry information to the format required by SDS. Thus, we have developed Python codes to convert automatically building and topographic data to the FDS input file. In order to evaluate the predictions of FDS with observations, statistical performance measures have been used. These metrics include the fractional bias (FB), the normalized mean square error (NMSE) and the fraction of predictions within a factor of two of observations (FAC2). As well as the CFD models tested in the COST Action, FDS results demonstrate a good agreement with measured concentrations. Furthermore, the metrics assessment indicate that FB and NMSE meet the tolerance acceptable.

Keywords: numerical simulations, atmospheric dispersion, cost ES1006 action, CFD model, cute experiments, wind tunnel data, numerical results

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Authors: S. A. M. Yatim, Z. B. Ibrahim, K. I. Othman, M. Suleiman

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The method of solving second order stiff ordinary differential equation (ODEs) that is based on backward differentiation formula (BDF) is considered in this paper. We derived the method by increasing the order of the existing method using an improved strategy in choosing the step size. Numerical results are presented to compare the efficiency of the proposed method to the MATLAB’s suite of ODEs solvers namely ode15s and ode23s. The method was found to be efficient to solve second order ordinary differential equation.

Keywords: backward differentiation formulae, block backward differentiation formulae, stiff ordinary differential equation, variable step size

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Authors: Matthew C. Best

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Linear Multi-Input Multi-Output (MIMO) dynamic models can be identified, with no a priori knowledge of model structure or order, using a new Generalised Identifying Filter (GIF). Based on an Extended Kalman Filter, the new filter identifies the model iteratively, in a continuous modal canonical form, using only input and output time histories. The filter’s self-propagating state error covariance matrix allows easy determination of convergence and conditioning, and by progressively increasing model order, the best fitting reduced-order model can be identified. The method is shown to be resistant to noise and can easily be extended to identification of smoothly nonlinear systems.

Keywords: system identification, Kalman filter, linear model, MIMO, model order reduction

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Authors: A. M. Sagir

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Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

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

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Authors: F. Maass, P. Martin, J. Olivares

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The aim of this work is the application of the program GeoGebra in teaching the study of nonhomogeneous linear second order differential equations with constant coefficients. Different kind of functions or forces will be considered in the right hand side of the differential equations, in particular, the emphasis will be placed in the case of trigonometrical functions producing the resonance phenomena. In order to obtain this, the frequencies of the trigonometrical functions will be changed. Once the resonances appear, these have to be correlationated with the roots of the second order algebraic equation determined by the coefficients of the differential equation. In this way, the physics and engineering students will understand resonance effects and its consequences in the simplest way. A large variety of examples will be shown, using different kind of functions for the nonhomogeneous part of the differential equations.

Keywords: education, geogebra, ordinary differential equations, resonance

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Authors: Hussein Shaman, Faris Almansour

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This paper proposes a modified optimum bandstop filter with ultra-wideband stopband. The filter consists of three shunt open-circuited stubs and two non-redundant unit elements. The proposed bandstop filter is designed with unequal electrical lengths of the open-circuited stubs at the mid-stopband. Therefore, the filter can exhibit a quasi-elliptic function response that improves the selectivity and enhances the rejection bandwidth. The filter is designed to exhibit a fractional bandwidth of about 114% at a mid-stopband frequency of 3.0 GHz. The filter is successfully realized in theory, simulated, fabricated and measured. An excellent agreement is obtained between calculated, simulated and measured. The fabricated filter has a compact size with a low insertion loss in the passbands, high selectivity and good attenuation level inside the desired stopband

Keywords: microstrip filter, bandstop filter, UWB filter, transmission line filter

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Authors: Aurore Cauquis, Philippe Heinrich, Mario Ricchiuto, Audrey Gailler

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In this talk, we look for an accurate time scheme to model the propagation of waves. Several numerical schemes have been developed to solve the extended weakly nonlinear weakly dispersive Boussinesq Equations. The temporal schemes used are two Lax-Wendroff schemes, second or third order accurate, two Runge-Kutta schemes of second and third order and a simplified third order accurate Lax-Wendroff scheme. Spatial derivatives are evaluated with fourth order accuracy. The numerical model is applied to two monodimensional benchmarks on a flat bottom. It is also applied to the simulation of the Algerian tsunami generated by a Mw=6 seism on the 18th March 2021. The tsunami propagation was highly dispersive and propagated across the Mediterranean Sea. We study here the effects of the order of temporal discretization on the accuracy of the results and on the time of computation.

Keywords: numerical analysis, tsunami propagation, water wave, boussinesq equations

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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: Olusola Ezekiel Abolarin, Gift E. Noah

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This research work considered an innovative procedure to numerically approximate higher-order Initial value problems (IVP) of ordinary differential equations (ODE) using the Legendre polynomial as the basis function. The proposed method is a half-step, self-starting Block integrator employed to approximate fourth and fifth order IVPs without reduction to lower order. The method was developed through a collocation and interpolation approach. The basic properties of the method, such as convergence, consistency and stability, were well investigated. Several test problems were considered, and the results compared favorably with both exact solutions and other existing methods.

Keywords: initial value problem, ordinary differential equation, implicit off-grid block method, collocation, interpolation

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