Search results for: electric differential

Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

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An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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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: Lina Wu, Ye Li

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An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.

Keywords: closed forms, co-closed forms, harmonic forms, L^q spaces, p-balanced growth, simple differential k-forms

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Authors: Mustafa Bayram Gücen, Coşkun Yakar

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In this study, we have investigated the strict stability of fuzzy differential systems and we compare the classical notion of strict stability criteria of ordinary differential equations and the notion of strict stability of fuzzy differential systems. In addition that, we present definitions of stability and strict stability of fuzzy differential equations and also we have some theorems and comparison results. Strict Stability is a different stability definition and this stability type can give us an information about the rate of decay of the solutions. Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of fuzzy differential systems along with a comparison result that allows the prediction of behavior of a fuzzy differential system when the behavior of the null solution of a fuzzy comparison system is known. This method is a usefull for investigating strict stability of fuzzy systems. First of all, we present definitions and necessary background material. Secondly, we discuss and compare the differences between the classical notion of stability and the recent notion of strict stability. And then, we have a comparison result in which the stability properties of the null solution of the comparison system imply the corresponding stability properties of the fuzzy differential system. Consequently, we give the strict stability results and a comparison theorem. We have used Lyapunov second method and we have proved a comparison result with scalar differential equations.

Keywords: fuzzy systems, fuzzy differential equations, fuzzy stability, strict stability

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Authors: H. N. Agiza, M. A. Sohaly, M. A. Elfouly

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Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs.

Keywords: Parkinson's disease, step method, delay differential equation, two delays

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Authors: Brandon To, Yong S. Park

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As more battery electric vehicles are being introduced into the automobile industry, continuous advancements are constantly made in the electric vehicle space. Improvements in lithium-ion battery technology allow electric vehicles to be capable of traveling long distances. The batteries are capable of being charged faster, allowing for a sufficient range in shorter amounts of time. With increased reliance on battery technology and the changes in vehicle power trains, new challenges arise from this. Resulting electric vehicle fires caused by collisions are potentially more dangerous than those of the typical internal combustion engine. To further reduce the battery failures involved with side collisions, this project intends to reinforce an existing battery pack of an electric vehicle with honeycomb structures such that intrusion into the batteries can be minimized with weight restrictions in place. Honeycomb structures of hexagonal geometry are implemented into the side extrusions of the battery pack. With the use of explicit dynamics simulations performed in ANSYS, quantitative results such as deformation, strain, and stress are used to compare the performance of the battery pack with and without the implemented honeycomb structures.

Keywords: battery pack, electric vehicle, honeycomb, side impact

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Authors: Sadeep Sasidharan, T. B. Isha

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Modern day electric vehicles require compact high torque/power density motors for electric propulsion. This necessitates proper thermal management of the electric motors. The main focus of this paper is to compare the steady state thermal analysis of a conventional 20 kW 8/6 Switched Reluctance Motor (SRM) with that of an Induction Motor and Inset Permanent Magnet (IPM) motor of the same rating. The goal is to develop a proper thermal model of the three types of models for Finite Element Thermal Analysis. JMAG software is used for the development and simulation of the thermal models. The results show that the induction motor is subjected to more heating when used for electric vehicle application constantly, compared to the SRM and IPM.

Keywords: electric vehicles, induction motor, inset permanent magnet motor, loss models, switched reluctance motor, thermal analysis

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Authors: Farid Nouioua, Abdelouaheb Ardjouni

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In this article, we study the existence of positive periodic solutions of certain delay differential equations. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ Krasnoselskii's fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the differential equation. The obtained results improve and extend the results in the literature. Finally, an example is given to illustrate our results.

Keywords: delay differential equations, positive periodic solutions, integral equations, Krasnoselskii fixed point theorem

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Authors: Mehtap Lafcı

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In recent years, oscillation, periodicity and convergence of solutions of linear differential equations with piecewise constant arguments have been significantly considered but there are only a few papers for impulsive differential equations with piecewise constant arguments. In this paper, a first order nonlinear impulsive differential equation with piecewise constant arguments is studied and the existence of solutions and periodic solutions of this equation are investigated by using Carvalho’s method. Finally, an example is given to illustrate these results.

Keywords: Carvalho's method, impulsive differential equation, periodic solution, piecewise constant arguments

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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: Saad Chahba, Rabia Sehab, Ahmad Akrad, Cristina Morel

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Nowadays, the electric machines used in urban electric vehicles are, in most cases, three-phase electric machines with or without a magnet in the rotor. Permanent Magnet Synchronous Machine (PMSM) and Induction Machine (IM) are the main components of drive trains of electric and hybrid vehicles. These machines have very good performance in healthy operation mode, but they are not redundant to ensure safety in faulty operation mode. Faced with the continued growth in the demand for electric vehicles in the automotive market, improving the reliability of electric vehicles is necessary over the lifecycle of the electric vehicle. Multiphase electric machines respond well to this constraint because, on the one hand, they have better robustness in the event of a breakdown (opening of a phase, opening of an arm of the power stage, intern-turn short circuit) and, on the other hand, better power density. In this work, a diagnosis approach using a neural network for an open circuit fault or more of a five-phase induction machine is developed. Validation on the simulator of the vehicle drivetrain, at reduced power, is carried out, creating one and more open circuit stator phases showing the efficiency and the reliability of the new approach to detect and to locate on-line one or more open phases of a five-induction machine.

Keywords: electric vehicle drivetrain, multiphase drives, induction machine, control, open circuit (OC) fault diagnosis, artificial neural network

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Authors: I. Sagaama, A. Kechiche, W. Trojet, F. Kamoun

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Replacing combustion engine vehicles by electric vehicles (EVs) is a major step in recent years due to their potential benefits. Battery autonomy and charging processes are still a big issue for that kind of vehicles. Therefore, reducing the energy consumption of electric vehicles becomes a necessity. Many researches target introducing recent information and communication technologies in EVs in order to propose reducing energy consumption services. Evaluation of realistic scenarios is a big challenge nowadays. In this paper, we will elaborate a state of the art of different proposed energy consumption models in the literature, then we will present a comparative study of these models, finally, we will extend previous works in order to propose an accurate and realistic energy model for calculating instantaneous power consumption of electric vehicles.

Keywords: electric vehicle, vehicular networks, energy models, traffic simulation

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Authors: Fakhreddin Abedi, Wah June Leong

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Exponential stability of stochastic differential equations with non trivial solutions is provided in terms of Lyapunov functions. The main result of this paper establishes that, under certain hypotheses for the dynamics f(.) and g(.), practical exponential stability in probability at the small neighborhood of the origin is equivalent to the existence of an appropriate Lyapunov function. Indeed, we establish exponential stability of stochastic differential equation when almost all the state trajectories are bounded and approach a sufficiently small neighborhood of the origin. We derive sufficient conditions for exponential stability of stochastic differential equations. Finally, we give a numerical example illustrating our results.

Keywords: exponential stability in probability, stochastic differential equations, Lyapunov technique, Ito's formula

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Authors: Kaan Karaoglu, Raif Bayir

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In this study, a cartesian robot is developed to improve the performance and efficiency of wireless charging of electric vehicles. The cartesian robot has three axes, each of which moves linearly. Magnetic positioning is used to align the cartesian robot transmitter charging pad. There are two different wireless charging methods, static and dynamic, for charging electric vehicles. The current state of charge information (SOC State of Charge) and location information are received wirelessly from the electric vehicle. Based on this information, the power to be transmitted is determined, and the transmitter and receiver charging pads are aligned for maximum efficiency. With this study, a fully automated cartesian robot structure will be used to charge electric vehicles with the highest possible efficiency. With the wireless communication established between the electric vehicle and the charging station, the charging status will be monitored in real-time. The cartesian robot developed in this study is a fully automatic system that can be easily used in static wireless charging systems with vehicle-machine communication.

Keywords: electric vehicle, wireless charging systems, energy efficiency, cartesian robot, location detection, trajectory planning

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Authors: Hongda Guo, Wenxia Sima

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Transformer oil is widely used in power systems because of its excellent insulation properties. The accurate measurement of electric field and space charge distribution in transformer oil under high voltage impulse has important theoretical and practical significance, but still remains challenging to date because of its low Kerr constant. In this study, the continuous electric field and space charge distribution over time between parallel-plate electrodes in high-voltage pulsed transformer oil based on the Kerr effect is directly measured using a linear array photoelectrical detector. Experimental results demonstrate the applicability and reliability of this method. This study provides a feasible approach to further study the space charge effects and breakdown mechanisms in transformer oil.

Keywords: electric field, Kerr, space charge, transformer oil

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Authors: Armin Ardekani, Mohammad Akbari

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We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields.

Keywords: computational mathematics, differential equations, engineering, series

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Authors: Zhan Chen, Wenquan Bi

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This paper focuses on examining the strength of Midori against impossible differential attack. The Midori family of light weight block cipher orienting to energy-efficiency is proposed in ASIACRYPT2015. Using a 6-round property, the authors implement an 11-round impossible differential attack on Midori64 by extending two rounds on the top and three rounds on the bottom. There is enough key space to consider pre-whitening keys in this attack. An impossible differential path that minimises the key bits involved is used to reduce computational complexity. Several additional observations such as partial abort technique are used to further reduce data and time complexities. This attack has data complexity of 2 ⁶⁹·² chosen plaintexts, requires 2 ¹⁴·⁵⁸ blocks of memory and 2 ⁹⁴·⁷ 11- round Midori64 encryptions.

Keywords: cryptanalysis, impossible differential, light weight block cipher, Midori

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Authors: M. Jamil Amir, Rabia Iqbal, M. Yaseen

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In this paper, we solve some differential equations analytically by using differential transform method. For this purpose, we consider four models of Laplace equation with two Dirichlet and two Neumann boundary conditions and K(2,2) equation and obtain the corresponding exact solutions. The obtained results show the simplicity of the method and massive reduction in calculations when one compares it with other iterative methods, available in literature. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

Keywords: differential transform method, laplace equation, Dirichlet boundary conditions, Neumann boundary conditions

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Authors: Y. N. Reddy

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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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Authors: Radia Chemlal, Salim Mehenni, Dahbia Leila Anes-boulahbal, Mohamed Kherat, Nabil Mameri

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The application of the electric field is considered to be a very promising method in cancer therapy. Indeed, cancer cells are very sensitive to the electric field, although the cellular response is not entirely clear. The tests carried out consisted in subjecting the RD cell line under the effect of the continuous electric field while varying certain parameters (voltage, exposure time, and cell concentration). The response surface methodology (RSM) was used to assess the effect of the chosen parameters, as well as the existence of interactions between them. The results obtained showed that the voltage, the cell concentration as well as the interaction between voltage and exposure time have an influence on the mortality rate of the RD cell line.

Keywords: continuous electric field, RD cancer cell line, RSM, voltage

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Authors: Chelsea-Anne Edwards, Paul Parker

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Flight school members are facing a major disruption in the technologies available for them to fly as electric planes enter the aviation industry. The year 2020 marked a new era in aviation with the first type certification of an electric plane. The Pipistrel Velis Electro is a two-seat electric aircraft (e-plane) designed for flight training. Electric flight training has the potential to deeply reduce emissions, noise, and cost of pilot training. Though these are all attractive features, understanding must be developed on the perceptions of the essential actor of the technology, the pilot. This study asks student pilots, flight instructors, flight center managers, and other members of flight schools about their perceptions of e-planes. The questions were divided into three categories: safety and trust of the technology, expected costs in comparison to conventional planes, and interest in the technology, including their desire to fly electric planes. Participants were recruited from flight schools using a protocol approved by the Office of Research Ethics. None of these flight schools have an e-plane in their fleet so these views are based on perceptions rather than direct experience. The results revealed perceptions that were strongly positive with many qualitative comments indicating great excitement about the potential of the new electric aviation technology. Some concerns were raised regarding battery endurance limits. Overall, the flight school community is clearly in favor of introducing electric propulsion technology and reducing the environmental impacts of their industry.

Keywords: electric planes, flight training, green aircraft, student pilots, sustainable aviation

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Authors: M. Mourad, K. Mahmoud

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Recently hybrid vehicles have become a major concern as one alternative vehicles. This type of hybrid vehicle contributes greatly to reducing pollution. Therefore, this work studies the influence of electrification phase of hybrid electric vehicle on emission of vehicle at different road conditions. To accomplish this investigation, a simulation model was used to evaluate the external characteristics of the hybrid electric vehicle according to variant conditions of road resistances. Therefore, this paper reports a methodology to decrease the vehicle emission especially greenhouse gas emission inside cities. The results show the effect of electrification on vehicle performance characteristics. The results show that CO₂ emission of vehicle decreases up to 50.6% according to an urban driving cycle due to applying the electrification strategy for hybrid electric vehicle.

Keywords: electrification strategy, hybrid electric vehicle, driving cycle, CO2 emission

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Authors: O. Acan, Y. Keskin

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In this study, we will carry out a comparative study between the reduced differential transform method, the adomian decomposition method, the variational iteration method and the homotopy analysis method. These methods are used in many fields of engineering. This is been achieved by handling a kind of 2-Dimensional and forth-order partial differential equations called the Kuramoto–Sivashinsky equations. Three numerical examples have also been carried out to validate and demonstrate efficiency of the four methods. Furthermost, it is shown that the reduced differential transform method has advantage over other methods. This method is very effective and simple and could be applied for nonlinear problems which used in engineering.

Keywords: reduced differential transform method, adomian decomposition method, variational iteration method, homotopy analysis method

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Authors: Muhammad Asghar, Aamer Iqbal Bhatti, Qadeer Ahmed, Tahir Izhar

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Three wheeler auto Rickshaw, often termed as ‘auto rickshaw’ is very common in Pakistan and is considered as the most affordable means of transportation to the local people. Problems caused by the gasoline engine on the environment and people, the researchers and the automotive industry have turned to the hybrid electric vehicles and electrical powered vehicle. The research in this paper explains the design of energy efficient Electric auto Rickshaw. An electric auto rickshaw is being developed at Center for Energy Research and Development, (Lahore), which is running on the roads of Lahore city. Energy storage capacity of batteries is at least 25 times heavier than fossil fuel and having volume 10 times in comparison to fuel, resulting an increase of the Rickshaw weight. A set of specifications is derived according to the mobility requirements of the electric auto rickshaw. The design choices considering the power-train and component selection are explained in detail. It was concluded that electric auto rickshaw has many advantages and benefits over the conventional auto rickshaw. It is cleaner and much more energy efficient but limited to the distance it can travel before recharging of battery. In addition, a brief future view of the battery technology is given.

Keywords: conventional auto rickshaw, energy efficiency, electric auto rickshaw, internal combustion engine, environment

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Authors: Anne Y. Faxer, Ellen Olausson, Jens Hagman, Ana Magazinius, Jenny J. Stier, Tommy Fransson, Oscar Enerback

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While electric vehicles (both battery electric vehicles and plug-in hybrids) have been on the market for around 6 years, they are still far from mainstream and the knowledge of them is still low among the public. This is likely one of the reasons that Sweden, having one of the highest penetrations of electric vehicles in Europe, still has a long way to go in reaching a fossil free vehicle fleet. Car dealerships are an important medium that connects consumers to vehicles, but somehow, their role in introducing electric vehicles has not yet been thoroughly studied. Research from other domains shows that salespeople can affect customer decisions in their choice of products. The aim of this study is to explore the role of car dealerships when it comes to promoting electric vehicles. The long-term goal is to understand how they could be a key in the effort of achieving a mass introduction of electric vehicles in Sweden. By emulating the customer’s experience, this study investigates the interaction between car salespeople and customers, particularly examining whether they present electric vehicles as viable options. Covert participatory observations were conducted for data collection from four different brands at in total twelve car dealers. The observers worked in pairs and played the role of a customer with needs that could be matched by an electric vehicle. The data was summarized in observation protocols and analyzed using thematic coding. The result shows that only one of twelve salespeople offered an electric vehicle as the first option. When environmental factors were brought up by the observers, the salespeople followed up with lower fuel consumption internal combustion engine vehicles rather than suggesting an electric vehicle. All salespeople possessed at least basic knowledge about electric vehicles but their interest of selling them were low in most cases. One of the reasons could be that the price of electric vehicles is usually higher. This could be inferred from the finding that salespeople tend to have a strong focus on price and economy in their dialogues with customers, regardless which type of car they were selling. In conclusion, the study suggests that car salespeople have the potential to help the market to achieve mass introduction of electric vehicles; however, their potential needs to be exploited further. To encourage salespeople to prioritize electric vehicles in the sales process, right incentives need to be in place.

Keywords: car dealerships, covert participatory observation, customer perspective , electric vehicle, market penetration

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Authors: Anurag Savarnya

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The reliability of the various parts of electric mining shovel has been assessed through the application of Weibull Model. The study was initiated to find reliability of components of electric mining shovel. The paper aims to optimize the reliability of components and increase the life cycle of component. A multilevel decomposition of the electric mining shovel was done and maintenance records were used to evaluate the failure data and appropriate system characterization was done to model the system in terms of reasonable number of components. The approach used develops a mathematical model to assess the reliability of the electric mining shovel components. The model can be used to predict reliability of components of the hydraulic mining shovel and system performance. Reliability is an inherent attribute to a system. When the life-cycle costs of a system are being analyzed, reliability plays an important role as a major driver of these costs and has considerable influence on system performance. It is an iterative process that begins with specification of reliability goals consistent with cost and performance objectives. The data were collected from an Indian open cast coal mine and the reliability of various components of the electric mining shovel has been assessed by following a Weibull Model.

Keywords: reliability, Weibull model, electric mining shovel

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Authors: Mohamed A. Alsharif, Peter A. Wallace, Donald M. Hepburn, Chengke Zhou

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Worldwide, most PILC MV underground cables in use are approaching the end of their design life; hence, failures are likely to increase. This paper studies the electric field and potential distributions within the PILC insulted cable containing common void-defect. The finite element model of the performance of the belted PILC MV underground cable is presented. The variation of the electric field stress within the cable using the Finite Element Method (FEM) is concentrated. The effects of the void-defect within the insulation are given. Outcomes will lead to deeper understanding of the modeling of Paper Insulated Lead Covered (PILC) and electric field response of belted PILC insulted cable containing void defect.

Keywords: MV PILC cables, finite element model/COMSOL multiphysics, electric field stress, partial discharge degradation

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Authors: R. Mkahl, A. Nait-Sidi-Moh, M. Wack

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Batteries of electric vehicles (BEV) are becoming more attractive with the advancement of new battery technologies and promotion of electric vehicles. BEV batteries are recharged on board vehicles using either the grid (G2V for Grid to Vehicle) or renewable energies in a stand-alone application (H2V for Home to Vehicle). This paper deals with the modeling, sizing and control of a photo voltaic stand-alone application that can charge the BEV at home. The modeling approach and developed mathematical models describing the system components are detailed. Simulation and experimental results are presented and commented.

Keywords: electric vehicles, photovoltaic energy, lead-acid batteries, charging process, modeling, simulation, experimental tests

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Authors: Katerina Kormusheva

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Pricing plays a pivotal role in the marketing discipline as it directly influences consumer perceptions, purchase decisions, and overall market positioning of a product or service. This paper seeks to expand current knowledge in the area of discriminatory and differential pricing, a main area of marketing research. The methodology includes developing a framework and a model for determining how many price points to implement in differential pricing. We focus on choosing the levels of differentiation, derive a function form of the model framework proposed, and lastly, test it empirically with data from a large-scale marketing pricing experiment of services in telecommunications.

Keywords: marketing, differential pricing, price points, optimization

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Authors: Khosrow Maleknejad, Yaser Rostami

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In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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