Search results for: Nonlinear effects
Commenced in January 2007
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Edition: International
Paper Count: 3860

Search results for: Nonlinear effects

3140 Fast Accurate Detection of Frequency Jumps Using Kalman Filter with Non Linear Improvements

Authors: Mahmoud E. Mohamed, Ahmed F. Shalash, Hanan A. Kamal

Abstract:

In communication systems, frequency jump is a serious problem caused by the oscillators used. Kalman filters are used to detect that jump, despite the tradeoff between the noise level and the speed of the detection. In this paper, an improvement is introduced in the Kalman filter, through a nonlinear change in the bandwidth of the filter. Simulation results show a considerable improvement in the filter speed with a very low noise level. Additionally, the effect on the response to false alarms is also presented and false alarm rate show improvement.

Keywords: Kalman Filter, Innovation, False Detection.

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3139 Adaptation of Iterative Methods to Solve Fuzzy Mathematical Programming Problems

Authors: Ricardo C. Silva, Luiza A. P. Cantao, Akebo Yamakami

Abstract:

Based on the fuzzy set theory this work develops two adaptations of iterative methods that solve mathematical programming problems with uncertainties in the objective function and in the set of constraints. The first one uses the approach proposed by Zimmermann to fuzzy linear programming problems as a basis and the second one obtains cut levels and later maximizes the membership function of fuzzy decision making using the bound search method. We outline similarities between the two iterative methods studied. Selected examples from the literature are presented to validate the efficiency of the methods addressed.

Keywords: Fuzzy Theory, Nonlinear Optimization, Fuzzy Mathematics Programming.

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3138 An Assessment of the Effects of Microbial Products on the Specific Oxygen Uptake in Submerged Membrane Bioreactor

Authors: M. F. R. Zuthi, H. H. Ngo, W. S. Guo, S. S. Chen, N. C. Nguyen, L. J. Deng, T. D. C. Tran

Abstract:

Sustaining a desired rate of oxygen transfer for microbial activity is a matter of major concern for biological wastewater treatment (MBR). The study reported in the paper was aimed at assessing the effects of microbial products on the specific oxygen uptake rate (SOUR) in a conventional membrane bioreactor (CMBR) and that in a sponge submerged MBR (SSMBR). The production and progressive accumulation of soluble microbial products (SMP) and bound-extracellular polymeric substances (bEPS) were affecting the SOUR of the microorganisms which varied at different stages of operation of the MBR systems depending on the variable concentrations of the SMP/bEPS. The effect of bEPS on the SOUR was stronger in the SSMBR compared to that of the SMP, while relative high concentrations of SMP had adverse effects on the SOUR of the CMBR system. Of the different mathematical correlations analyzed in the study, logarithmic mathematical correlations could be established between SOUR and bEPS in SSMBR, and similar correlations could also be found between SOUR and SMP concentrations in the CMBR.

Keywords: Microbial products, Microbial activity, Specific oxygen uptake rate, Membrane bioreactor.

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3137 Traffic Flow on Road Junctions

Authors: Wah Wah Aung, Cho Cho San

Abstract:

The paper deals with a mathematical model for fluid dynamic flows on road networks which is based on conservation laws. This nonlinear framework is based on the conservation of cars. We focus on traffic circle, which is a finite number of roads that meet at some junctions. The traffic circle with junctions having either one incoming and two outgoing or two incoming and one outgoing roads. We describe the numerical schemes with the particular boundary conditions used to produce approximated solutions of the problem.

Keywords: boundary conditions, conservation laws, finite difference Schemes, traffic flow.

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3136 Dual-Task – Immersion in the Interactions of Simultaneously Performed Tasks

Authors: M. Liebherr, P. Schubert, S. Kersten, C. Dietz, L. Franz, C. T. Haas

Abstract:

With a long history, dual-task has become one of the most intriguing research fields regarding human brain functioning and cognition. However, findings considering effects of taskinterrelations are limited (especially, in combined motor and cognitive tasks). Therefore, we aimed at developing a measurement system in order to analyse interrelation effects of cognitive and motor tasks. On the one hand, the present study demonstrates the applicability of the measurement system and on the other hand first results regarding a systematisation of different task combinations are shown. Future investigations should combine imagine technologies and this developed measurement system.

Keywords: Dual-task, interference, cognition, measurement.

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3135 An Improved Particle Swarm Optimization Technique for Combined Economic and Environmental Power Dispatch Including Valve Point Loading Effects

Authors: Badr M. Alshammari, T. Guesmi

Abstract:

In recent years, the combined economic and emission power dispatch is one of the main problems of electrical power system. It aims to schedule the power generation of generators in order to minimize cost production and emission of harmful gases caused by fossil-fueled thermal units such as CO, CO2, NOx, and SO2. To solve this complicated multi-objective problem, an improved version of the particle swarm optimization technique that includes non-dominated sorting concept has been proposed. Valve point loading effects and system losses have been considered. The three-unit and ten-unit benchmark systems have been used to show the effectiveness of the suggested optimization technique for solving this kind of nonconvex problem. The simulation results have been compared with those obtained using genetic algorithm based method. Comparison results show that the proposed approach can provide a higher quality solution with better performance.

Keywords: Power dispatch, valve point loading effects, multiobjective optimization, Pareto solutions.

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3134 Positive Solutions of Initial Value Problem for the Systems of Second Order Integro-Differential Equations in Banach Space

Authors: Lv Yuhua

Abstract:

In this paper, by establishing a new comparison result, we investigate the existence of positive solutions for initial value problems of nonlinear systems of second order integro-differential equations in Banach space.We improve and generalize some results  (see[5,6]), and the results is new even in finite dimensional spaces.

Keywords: Systems of integro-differential equations, monotone iterative method, comparison result, cone.

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3133 Advanced Gronwall-Bellman-Type Integral Inequalities and Their Applications

Authors: Zixin Liu, Shu Lü, Shouming Zhong, Mao Ye

Abstract:

In this paper, some new nonlinear generalized Gronwall-Bellman-Type integral inequalities with mixed time delays are established. These inequalities can be used as handy tools to research stability problems of delayed differential and integral dynamic systems. As applications, based on these new established inequalities, some p-stable results of a integro-differential equation are also given. Two numerical examples are presented to illustrate the validity of the main results.

Keywords: Gronwall-Bellman-Type integral inequalities, integrodifferential equation, p-exponentially stable, mixed delays.

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3132 Nonlinear Modelling of Sloshing Waves and Solitary Waves in Shallow Basins

Authors: Mohammad R. Jalali, Mohammad M. Jalali

Abstract:

The earliest theories of sloshing waves and solitary waves based on potential theory idealisations and irrotational flow have been extended to be applicable to more realistic domains. To this end, the computational fluid dynamics (CFD) methods are widely used. Three-dimensional CFD methods such as Navier-Stokes solvers with volume of fluid treatment of the free surface and Navier-Stokes solvers with mappings of the free surface inherently impose high computational expense; therefore, considerable effort has gone into developing depth-averaged approaches. Examples of such approaches include Green–Naghdi (GN) equations. In Cartesian system, GN velocity profile depends on horizontal directions, x-direction and y-direction. The effect of vertical direction (z-direction) is also taken into consideration by applying weighting function in approximation. GN theory considers the effect of vertical acceleration and the consequent non-hydrostatic pressure. Moreover, in GN theory, the flow is rotational. The present study illustrates the application of GN equations to propagation of sloshing waves and solitary waves. For this purpose, GN equations solver is verified for the benchmark tests of Gaussian hump sloshing and solitary wave propagation in shallow basins. Analysis of the free surface sloshing of even harmonic components of an initial Gaussian hump demonstrates that the GN model gives predictions in satisfactory agreement with the linear analytical solutions. Discrepancies between the GN predictions and the linear analytical solutions arise from the effect of wave nonlinearities arising from the wave amplitude itself and wave-wave interactions. Numerically predicted solitary wave propagation indicates that the GN model produces simulations in good agreement with the analytical solution of the linearised wave theory. Comparison between the GN model numerical prediction and the result from perturbation analysis confirms that nonlinear interaction between solitary wave and a solid wall is satisfactorilly modelled. Moreover, solitary wave propagation at an angle to the x-axis and the interaction of solitary waves with each other are conducted to validate the developed model.

Keywords: Even harmonic components of sloshing waves, Green–Naghdi equations, nonlinearity, solitary waves.

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3131 The Symmetric Solutions for Boundary Value Problems of Second-Order Singular Differential Equation

Authors: Li Xiguang

Abstract:

In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper.

Keywords: Banach space, cone, fixed point index, singular differential equation, p-Laplace operator, symmetric solutions.

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3130 A Study of the Change of Damping Coefficient Regarding Minimum Displacement

Authors: Tawiwat V., Narongkorn D., Auttapoom L.

Abstract:

This research proposes the change of damping coefficient regarding minimum displacement. From the mass with external forced and damper problem, when is the constant external forced transmitted to the understructure in the difference angle between 30 and 60 degrees. This force generates the vibration as general known; however, the objective of this problem is to have minimum displacement. As the angle is changed and the goal is the same; therefore, the damper of the system must be varied while keeping constant spring stiffness. The problem is solved by using nonlinear programming and the suitable changing of the damping coefficient is provided.

Keywords: Damping coefficient, Optimal control, Minimum Displacement and Vibration

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3129 Impact of Two Herbal Seeds Supplementation on Growth Performance and Some Biochemical Blood and Tissue Parameters of Broiler Chickens

Authors: Hamada A. Ahmed, Kadry M. Sadek, Ayman E. Taha

Abstract:

The effects of basil and/or chamomile seed supplementation on the growth of Hubbard broiler chicks were evaluated. The antioxidant effects of these supplements were also assessed. 120 1-day-old broiler chicks were randomly divided into four equal groups. The control group (group 1) was fed a basal diet (BD) without supplementation. Groups 2, 3, and 4 were fed the BD supplemented with 10g basil, 10g chamomile, and 5g basil plus 5g chamomile per kg of food, respectively. Basil supplementation alone or in combination with chamomile non-significantly (P≥0.05) increased final body weight (3.2% and 0.3%, respectively) and weight gain (3.5% and 3.6%, respectively) over the experimental period. Chamomile supplementation alone non-significantly (P≥0.05) reduced final body weight and weight gain over the experimental period by 1.7% and 1.7%, respectively. In comparison to the control group, herbal seed supplementation reduced feed intake and improved the feed conversion and protein efficiency ratios. In general, basil seed supplementation stimulated chicken growth and improved the feed efficiency more effectively than chamomile seed supplementation. The antioxidant activities of basil and/or chamomile supplementation were examined in the thymus, bursa, and spleen. In chickens that received supplements, the level of malondialdehyde was significantly decreased, whereas the activities of glutathione, superoxide dismutase, and catalase were significantly increased (P<0.05). Supplementation of basil and/or chamomile did not affect blood protein levels, but had lipid-lowering effects as evidenced by reduced serum levels of total lipids, triglycerides, and cholesterol. In conclusion, supplementation of basil and/or chamomile improved growth parameters in broiler chicks and had antioxidant and blood lipid-lowering effects. These beneficial effects of basil and/or chamomile supplementation resulted in economically viable production of high-quality white meat containing no harmful residues.

Keywords: Herbal additives, basil, chamomile, broiler, growth performance, antioxidant.

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3128 Performance Comparison of Cooperative Banks in the EU, USA and Canada

Authors: Matěj Kuc

Abstract:

This paper compares different types of profitability measures of cooperative banks from two developed regions: the European Union and the United States of America together with Canada. We created balanced dataset of more than 200 cooperative banks covering 2011-2016 period. We made series of tests and run Random Effects estimation on panel data. We found that American and Canadian cooperatives are more profitable in terms of return on assets (ROA) and return on equity (ROE). There is no significant difference in net interest margin (NIM). Our results show that the North American cooperative banks accommodated better to the current market environment.

Keywords: Cooperative banking, panel data, profitability measures, random effects.

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3127 Comparison of Noise Emissions in the Interior of Passenger Cars

Authors: Martin Kendra, Tomas Skrucany, Jaroslav Masek

Abstract:

The noise is one of the negative elements which affects the human health. This article presents the measurement of emitted noise by road vehicle and its parts during the operation. Measurement was done in the interior of common passenger cars with a digital sound meter. The results compare the noise value in different cars with different body shape, which influences the driver’s health. Transport has considerable ecological effects; many of them are detrimental to environmental sustainability. Roads and traffic exert a variety of direct and mostly detrimental effects on nature.

Keywords: Driver, noise measurement, passenger road vehicle, road transport.

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3126 Social and Economic Effects of Mining Industry Restructuring in Romania -Case Studies

Authors: Andra Costache, Gica Pehoiu

Abstract:

As in other countries from Central and Eastern Europe, the economic restructuring occurred in the last decade of the twentieth century affected the mining industry in Romania, an oversize and heavily subsidized sector before 1989. After more than a decade since the beginning of mining restructuring, an evaluation of current social implications of the process it is required, together with an efficiency analysis of the adaptation mechanisms developed at governmental level. This article aims to provide an insight into these issues through case studies conducted in the most important coal basin of Romania, Petroşani Depression.

Keywords: case studies, government programs, miningrestructuring, social effects.

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3125 The Sustainability of Public Debt in Taiwan

Authors: Chiung-Ju Huang

Abstract:

This study examines whether the Taiwan’s public debt is sustainable utilizing an unrestricted two-regime threshold autoregressive (TAR) model with an autoregressive unit root. The empirical results show that Taiwan’s public debt appears as a nonlinear series and is stationary in regime 1 but not in regime 2. This result implies that while Taiwan’s public debt was mostly sustainable over the 1996 to 2013 period examined in the study, it may no longer be sustainable in the most recent two years as the public debt ratio has increased cumulatively to 3.618%.

Keywords: Nonlinearity, public debt, sustainability, threshold autoregressive model.

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3124 Analytical Solution for the Zakharov-Kuznetsov Equations by Differential Transform Method

Authors: Saeideh Hesam, Alireza Nazemi, Ahmad Haghbin

Abstract:

This paper presents the approximate analytical solution of a Zakharov-Kuznetsov ZK(m, n, k) equation with the help of the differential transform method (DTM). The DTM method is a powerful and efficient technique for finding solutions of nonlinear equations without the need of a linearization process. In this approach the solution is found in the form of a rapidly convergent series with easily computed components. The two special cases, ZK(2,2,2) and ZK(3,3,3), are chosen to illustrate the concrete scheme of the DTM method in ZK(m, n, k) equations. The results demonstrate reliability and efficiency of the proposed method.

Keywords: Zakharov-Kuznetsov equation, differential transform method, closed form solution.

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3123 The Effects of Tissue Optical Parameters and Interface Reflectivity on Light Diffusion in Biological Tissues

Authors: MA. Ansari

Abstract:

In cancer progress, the optical properties of tissues like absorption and scattering coefficient change, so by these changes, we can trace the progress of cancer, even it can be applied for pre-detection of cancer. In this paper, we investigate the effects of changes of optical properties on light penetrated into tissues. The diffusion equation is widely used to simulate light propagation into biological tissues. In this study, the boundary integral method (BIM) is used to solve the diffusion equation. We illustrate that the changes of optical properties can modified the reflectance or penetrating light.

Keywords: Diffusion equation, boundary element method, refractive index

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3122 Forecasting the Istanbul Stock Exchange National 100 Index Using an Artificial Neural Network

Authors: Birol Yildiz, Abdullah Yalama, Metin Coskun

Abstract:

Many studies have shown that Artificial Neural Networks (ANN) have been widely used for forecasting financial markets, because of many financial and economic variables are nonlinear, and an ANN can model flexible linear or non-linear relationship among variables. The purpose of the study was to employ an ANN models to predict the direction of the Istanbul Stock Exchange National 100 Indices (ISE National-100). As a result of this study, the model forecast the direction of the ISE National-100 to an accuracy of 74, 51%.

Keywords: Artificial Neural Networks, Istanbul StockExchange, Non-linear Modeling.

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3121 Multi-fidelity Fluid-Structure Interaction Analysis of a Membrane Wing

Authors: M. Saeedi, R. Wuchner, K.-U. Bletzinger

Abstract:

In order to study the aerodynamic performance of a semi-flexible membrane wing, Fluid-Structure Interaction simulations have been performed. The fluid problem has been modeled using two different approaches which are the vortex panel method and the numerical solution of the Navier-Stokes equations. Nonlinear analysis of the structural problem is performed using the Finite Element Method. Comparison between the two fluid solvers has been made. Aerodynamic performance of the wing is discussed regarding its lift and drag coefficients and they are compared with those of the equivalent rigid wing.

Keywords: CFD, FSI, Membrane wing, Vortex panel method.

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3120 Analytical solution of Gas Flow Through a Micro-Nano Porous Media by Homotopy Perturbation method

Authors: Jamal Amani Rad, Kourosh Parand

Abstract:

In this paper, we have applied the homotopy perturbation method (HPM) for obtaining the analytical solution of unsteady flow of gas through a porous medium and we have also compared the findings of this research with some other analytical results. Results showed a very good agreement between results of HPM and the numerical solutions of the problem rather than other analytical solutions which have previously been applied. The results of homotopy perturbation method are of high accuracy and the method is very effective and succinct.

Keywords: Unsteady gas equation, Homotopy perturbation method(HPM), Porous medium, Nonlinear ODE

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3119 Surface and Guided Waves in Composites with Nematic Coatings

Authors: Dmitry D. Zakharov

Abstract:

The theoretical prediction of the acoustical polarization effects in the heterogeneous composites, made of thick elastic solids with thin nematic films, is presented. The numericalanalytical solution to the problem of the different wave propagation exhibits some new physical effects in the low frequency domain: the appearance of the critical frequency and the existence of the narrow transition zone where the wave rapidly changes its speed. The associated wave attenuation is highly perturbed in this zone. We also show the possible appearance of the critical frequencies where the attenuation changes the sign. The numerical results of parametrical analysis are presented and discussed.

Keywords: Surface wave, guided wave, heterogeneous composite, nematic coating.

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3118 Effects of Level Densities and Those of a-Parameter in the Framework of Preequilibrium Model for 63,65Cu(n,xp) Reactions in Neutrons at 9 to 15 MeV

Authors: L. Yettou

Abstract:

In this study, the calculations of proton emission spectra produced by 63Cu(n,xp) and 65Cu(n,xp) reactions are used in the framework of preequilibrium models using the EMPIRE code and TALYS code. Exciton Model predidtions combined with the Kalbach angular distribution systematics and the Hybrid Monte Carlo Simulation (HMS) were used. The effects of levels densities and those of a-parameter have been investigated for our calculations. The comparison with experimental data shows clear improvement over the Exciton Model and HMS calculations.

Keywords: Preequilibrium models, level density, level density a-parameter, 63Cu(n, xp) and 65Cu(n, xp) reactions.

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3117 Probability of Globality

Authors: Eva Eggeling, Dieter W. Fellner, Torsten Ullrich

Abstract:

The objective of global optimization is to find the globally best solution of a model. Nonlinear models are ubiquitous in many applications and their solution often requires a global search approach; i.e. for a function f from a set A ⊂ Rn to the real numbers, an element x0 ∈ A is sought-after, such that ∀ x ∈ A : f(x0) ≤ f(x). Depending on the field of application, the question whether a found solution x0 is not only a local minimum but a global one is very important. This article presents a probabilistic approach to determine the probability of a solution being a global minimum. The approach is independent of the used global search method and only requires a limited, convex parameter domain A as well as a Lipschitz continuous function f whose Lipschitz constant is not needed to be known.

Keywords: global optimization, probability theory, probability of globality

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3116 Spectral Analysis of Speech: A New Technique

Authors: Neeta Awasthy, J.P.Saini, D.S.Chauhan

Abstract:

ICA which is generally used for blind source separation problem has been tested for feature extraction in Speech recognition system to replace the phoneme based approach of MFCC. Applying the Cepstral coefficients generated to ICA as preprocessing has developed a new signal processing approach. This gives much better results against MFCC and ICA separately, both for word and speaker recognition. The mixing matrix A is different before and after MFCC as expected. As Mel is a nonlinear scale. However, cepstrals generated from Linear Predictive Coefficient being independent prove to be the right candidate for ICA. Matlab is the tool used for all comparisons. The database used is samples of ISOLET.

Keywords: Cepstral Coefficient, Distance measures, Independent Component Analysis, Linear Predictive Coefficients.

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3115 UEFA Super Cup: Economic Effects on Georgian Economy

Authors: Giorgi Bregadze

Abstract:

Tourism is the most viable and sustainable economic development option for Georgia and one of the main sources of foreign exchange earnings. Events are considered as one of the most effective ways to attract foreign visitors to the country, and, recently, the government of Georgia has begun investing in this sector very actively. This article stresses the necessity of research based economic policy in the tourism sector. In this regard, it is of paramount importance to measure the economic effects of the events which are subsidized by taxpayers’ money. The economic effect of events can be analyzed from two perspectives; financial perspective of the government and perspective of economic effects of the tourism administration. The article emphasizes more realistic and all-inclusive focus of the economic effect analysis of the tourism administration as it concentrates on the income of residents and local businesses, part of which generate tax revenues for the government. The public would like to know what the economic returns to investment are. In this article, the methodology used to describe the economic effects of UEFA Super Cup held in Tbilisi, will help to answer this question. Methodology is based on three main principles and covers three stages. Using the suggested methodology article estimates the direct economic effect of UEFA Super cup on Georgian economy. Although the attempt to make an economic effect analysis of the event was successful in Georgia, some obstacles and insufficiencies were identified during the survey. The article offers several recommendations that will help to refine methodology and improve the accuracy of the data. Furthermore, it is very important to receive the correct standard of measurement of events in Georgia. In this caseü non-ethical acts of measurement which are widely utilized by different research companies will not trigger others to show overestimated effects. It is worth mentioning that to author’s best knowledge, this is the first attempt to measure the economic effect of an event held in Georgia.

Keywords: Biased economic effect analysis, expenditure of local citizens, time switchers and casuals, UEFA super cup.

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3114 Experimental Investigation on the Effects of Electroless Nickel Phosphorus Deposition, pH and Temperature with the Varying Coating Bath Parameters on Impact Energy by Taguchi Method

Authors: D. Kari Basavaraja, M. G. Skanda, C. Soumya, V. Ramesh

Abstract:

This paper discusses the effects of sodium hypophosphite concentration, pH, and temperature on deposition rate. This paper also discusses the evaluation of coating strength, surface, and subsurface by varying the bath parameters, percentage of phosphate, plating temperature, and pH of the plating solution. Taguchi technique has been used for the analysis. In the experiment, nickel chloride which is a source of nickel when mixed with sodium hypophosphite has been used as the reducing agent and the source of phosphate and sodium hydroxide has been used to vary the pH of the coating bath. The coated samples are tested for impact energy by conducting impact test. Finally, the effects of coating bath parameters on the impact energy absorbed have been plotted, and analysis has been carried out. Further, percentage contribution of coating bath parameters using Design of Experiments approach (DOE) has been analysed. Finally, it can be concluded that the bath parameters of the Ni-P coating will certainly influence on the strength of the specimen.

Keywords: Bath parameters, coatings, design of experiment, fracture toughness, impact strength.

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3113 The Symmetric Solutions for Three-Point Singular Boundary Value Problems of Differential Equation

Authors: Li Xiguang

Abstract:

In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper.

Keywords: Banach space, cone, fixed point index, singular differential equation, p-Laplace operator, symmetric solutions.

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3112 Ginzburg-Landau Model for Curved Two-Phase Shallow Mixing Layers

Authors: Irina Eglite, Andrei A. Kolyshkin

Abstract:

Method of multiple scales is used in the paper in order to derive an amplitude evolution equation for the most unstable mode from two-dimensional shallow water equations under the rigid-lid assumption. It is assumed that shallow mixing layer is slightly curved in the longitudinal direction and contains small particles. Dynamic interaction between carrier fluid and particles is neglected. It is shown that the evolution equation is the complex Ginzburg-Landau equation. Explicit formulas for the computation of the coefficients of the equation are obtained.

Keywords: Shallow water equations, mixing layer, weakly nonlinear analysis, Ginzburg-Landau equation

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3111 4D Flight Trajectory Optimization Based on Pseudospectral Methods

Authors: Kouamana Bousson, Paulo Machado

Abstract:

The optimization and control problem for 4D trajectories is a subject rarely addressed in literature. In the 4D navigation problem we define waypoints, for each mission, where the arrival time is specified in each of them. One way to design trajectories for achieving this kind of mission is to use the trajectory optimization concepts. To solve a trajectory optimization problem we can use the indirect or direct methods. The indirect methods are based on maximum principle of Pontryagin, on the other hand, in the direct methods it is necessary to transform into a nonlinear programming problem. We propose an approach based on direct methods with a pseudospectral integration scheme built on Chebyshev polynomials.

Keywords: Pseudospectral Methods, Trajectory Optimization, 4DTrajectories

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