Search results for: differential algebraic systems

Authors: Davide Forcellini, James Marshal Kelly

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Seismic isolation using multi-layer elastomeric isolators has been used in the United States for more than 20 years. Although isolation bearings normally have a large factor of safety against buckling due to low shear stiffness, this phenomenon has been widely studied. In particular, the linearly elastic theory adopted to study this phenomenon is relatively accurate and adequate for most design purposes. Unfortunately it cannot consider the large deformation response of a bearing when buckling occurs and the unresolved behaviour of the stability of the post-buckled state. The study conducted in this paper may be viewed as a development of the linear theory of multi-layered elastomeric bearing, simply replacing the differential equations by algebraic equations, showing how it is possible to evaluate the post-buckling behaviour and the interactions at large deformations.

Keywords: multi-layer elastomeric isolators, large deformation, compressive load, tensile load, post-buckling behaviour

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Authors: Ayub Khan, Net Ram Garg, Geeta Jain

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

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

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Authors: Alexander N. Savostyanov, Tatiana A. Dolgorukova, Elena A. Esipenko, Mikhail S. Zaleshin, Margherita Malanchini, Anna V. Budakova, Alexander E. Saprygin, Tatiana A. Golovko, Yulia V. Kovas

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EEG correlates of mathematical and trait anxiety level were studied in 52 healthy Russian-speakers during execution of error-recognition tasks with lexical, arithmetic and algebraic conditions. Event-related spectral perturbations were used as a measure of brain activity. The ERSP plots revealed alpha/beta desynchronizations within a 500-3000 ms interval after task onset and slow-wave synchronization within an interval of 150-350 ms. Amplitudes of these intervals reflected the accuracy of error recognition, and were differently associated with the three conditions. The correlates of anxiety were found in theta (4-8 Hz) and beta2 (16-20 Hz) frequency bands. In theta band the effects of mathematical anxiety were stronger expressed in lexical, than in arithmetic and algebraic condition. The mathematical anxiety effects in theta band were associated with differences between anterior and posterior cortical areas, whereas the effects of trait anxiety were associated with inter-hemispherical differences. In beta1 and beta2 bands effects of trait and mathematical anxiety were directed oppositely. The trait anxiety was associated with increase of amplitude of desynchronization, whereas the mathematical anxiety was associated with decrease of this amplitude. The effect of mathematical anxiety in beta2 band was insignificant for lexical condition but was the strongest in algebraic condition. EEG correlates of anxiety in theta band could be interpreted as indexes of task emotionality, whereas the reaction in beta2 band is related to tension of intellectual resources.

Keywords: EEG, brain activity, lexical and numerical error-recognition tasks, mathematical and trait anxiety

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Authors: Mahmood Otadi, Maryam Mosleh

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The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example.

Keywords: fuzzy number, fuzzy ODE, HAM, approximate method

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Authors: A. M. Sagir

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

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

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Authors: J. J. Cetina-Denis, R. M. López-Gutiérrez, R. Ramírez-Ramírez, C. Cruz-Hernández

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This work addresses the problem of designing an algorithm capable of generating chaotic trajectories for mobile robots. Particularly, the chaotic behavior is induced in the linear and angular velocities of a Khepera III differential mobile robot by infusing them with the states of the H´enon chaotic map. A possible application, using the properties of chaotic systems, is patrolling a work area. In this work, numerical and experimental results are reported and analyzed. In addition, two quantitative numerical tests are applied in order to measure how chaotic the generated trajectories really are.

Keywords: chaos, chaotic trajectories, differential mobile robot, Henon map, Khepera III robot, patrolling applications

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Authors: Martin K. Steiger, Lukas Heisler, Hans-Georg Brachtendorf

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Deep neural networks and their variants form the backbone of many AI applications. Based on the so-called residual networks, a continuous formulation of such models as ordinary differential equations (ODEs) has proven advantageous since different techniques may be applied that significantly increase the learning speed and enable controlled trade-offs with the resulting error at the same time. For the evaluation of such models, high-performance numerical differential equation solvers are used, which also provide the gradients required for training. However, whether classical gradient-based methods are even applicable or which one yields the best results has not been discussed yet. This paper aims to redeem this situation by providing empirical results for different applications.

Keywords: deep neural networks, gradient-based learning, image processing, ordinary differential equation networks

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Authors: Chao-Qing Dai

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In modern nonlinear science and textile engineering, nonlinear differential-difference equations are often used to describe some nonlinear phenomena. In this paper, we extend the variable coefficient Jacobian elliptic function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, we derive two series of Jacobian elliptic function solutions of the discrete sine-Gordon equation.

Keywords: discrete sine-Gordon equation, variable coefficient Jacobian elliptic function method, exact solutions, equation

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Authors: Kamel Al-Khaled

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A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving the Kuramoto-Sivashinsky equation is given. Both approaches depend on using Sinc basis functions. Firstly, a numerical scheme using Sinc-Galerkin method is developed to approximate the solution of Kuramoto-Sivashinsky equation. Sinc approximations to both derivatives and indefinite integrals reduces the solution to an explicit system of algebraic equations. The error in the solution is shown to converge to the exact solution at an exponential. The convergence proof of the solution for the discrete system is given using fixed-point iteration. Secondly, a combination of a Crank-Nicolson formula in the time direction, with the Sinc-collocation in the space direction is presented, where the derivatives in the space variable are replaced by the necessary matrices to produce a system of algebraic equations. The methods are tested on two examples. The demonstrated results show that both of the presented methods more or less have the same accuracy.

Keywords: Sinc-Collocation, nonlinear PDEs, numerical methods, fixed-point

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Authors: Abdorreza Rabiee, Shahla Mohammad Hoseini Mirzaei

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The advantages of distributed generations (DGs) based on renewable energy sources (RESs) leads to high penetration level of DGs in distribution network. With incorporation of DGs in distribution systems, the system reliability and security, as well as voltage profile, is improved. However, the protection of such systems is still challenging. In this paper, at first, the related papers are reviewed and then a practical scheme is proposed for coordination of OCRs in distribution system with DGs. The coordination problem is formulated as a nonlinear programming (NLP) optimization problem with the object function of minimizing total operating time of OCRs. The proposed method is studied based on a simple test system. The optimization problem is solved by General Algebraic Modeling System (GAMS) to calculate the optimal time dial setting (TDS) and also pickup current setting of OCRs. The results show the effectiveness of the proposed method and its applicability.

Keywords: distributed generation, DG, distribution network, over current relay, OCR, protection coordination, pickup current, time dial setting, TDS

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Authors: Mary Chriselda A

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This paper deals with the fact that the Hubble's parameter is not constant and tends to vary stochastically with time. This premise has been proven by converting it to a stochastic differential equation using the Ornstein-Uhlenbeck process. The formulated stochastic differential equation is further solved analytically using the Euler and the Kolmogorov Forward equations, thereby obtaining the probability density function using the Fourier transformation, thereby proving that the Hubble's parameter varies stochastically. This is further corroborated by simulating the observations using Python and R-software for validation of the premise postulated. We can further draw conclusion that the randomness in forces affecting the white noise can eventually affect the Hubble’s Parameter leading to scale invariance and thereby causing stochastic fluctuations in the density and the rate of expansion of the Universe.

Keywords: Chapman Kolmogorov forward differential equations, fourier transformation, hubble's parameter, ornstein-uhlenbeck process , stochastic differential equations

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Authors: Raquel C. A. G. Mota, Livia R. Menezes, Emerson O. da Silva

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Both silver nanoparticles and titanium dioxide have been extensively used in tissue engineering since they’ve been approved by the Food and Drug Administration (FDA), and present a bactericide effect when added to a polymeric matrix. In this work, the focus is on fabricating binary systems with both nanoparticles so that the synergistic effect can be investigated. The systems were tested by Nuclear Magnetic Resonance (NMR), Thermogravimetric Analysis (TGA), Fourier-Transformed Infrared (FTIR), and Differential Scanning Calorimetry (DSC), and X-ray Diffraction (XRD), and had both their bioactivity and bactericide effect tested. The binary systems presented different properties than the individual systems, enhancing both the thermal and biological properties as was to be expected. The crystallinity was also affected, as indicated by the finding of the DSC and XDR techniques, and the NMR showed a good dispersion of both nanoparticles in the polymer matrix. These findings indicate the potential of combining TiO₂ and silver nanoparticles in biomedicine.

Keywords: metallic nanoparticles, nanotechnology, polymer nanocomposites, polymer science

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Authors: T. Vamsee Kiran, A. Gopi

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The undesired torque and flux ripple may occur in conventional direct torque control (DTC) induction motor drive. DTC can improve the system performance at low speeds by continuously tuning the regulator by adjusting the Kp, Ki values. In this differential evolution (DE) is proposed to adjust the parameters (Kp, Ki) of the speed controller in order to minimize torque ripple, flux ripple, and stator current distortion.The DE based PI controller has resulted is maintaining a constant speed of the motor irrespective of the load torque fluctuations.

Keywords: differential evolution, direct torque control, PI controller

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Authors: Abbes Lounis, Kahina Louadj, Mohamed Aidene

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This paper presents an optimal control problem applied to a robot; the problem is to determine a command which makes it possible to reach a final state from a given initial state in record time. The approach followed to solve this optimization problem with constraints on the control starts by presenting the equations of motion of the dynamic system then by applying Pontryagin's maximum principle (PMP) to determine the optimal control, and Picard's successive approximation method combined with the shooting method to solve the resulting differential system.

Keywords: robotics, Pontryagin's Maximum Principle, PMP, Picard's method, shooting method, non-linear differential systems

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Authors: Ritu Rani

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In this paper, we apply minimalistically upheld linear semi-orthogonal B-spline wavelets, exceptionally developed for the limited interim to rough the obscure function present in the integral equations. Semi-orthogonal wavelets utilizing B-spline uniquely developed for the limited interim and these wavelets can be spoken to in a shut frame. This gives a minimized help. Semi-orthogonal wavelets frame the premise in the space L²(R). Utilizing this premise, an arbitrary function in L²(R) can be communicated as the wavelet arrangement. For the limited interim, the wavelet arrangement cannot be totally introduced by utilizing this premise. This is on the grounds that backings of some premise are truncated at the left or right end purposes of the interim. Subsequently, an uncommon premise must be brought into the wavelet development on the limited interim. These functions are alluded to as the limit scaling functions and limit wavelet functions. B-spline wavelet method has been connected to fathom linear and nonlinear integral equations and their systems. The above method diminishes the integral equations to systems of algebraic equations and afterward these systems can be illuminated by any standard numerical methods. Here, we have connected Newton's method with suitable starting speculation for solving these systems.

Keywords: semi-orthogonal, wavelet arrangement, integral equations, wavelet development

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Authors: Jiya Mohammed, Ibrahim Ismail Giwa

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Unsteady squeezing flow of a viscous fluid between parallel plates is considered. The two plates are considered to be approaching each other symmetrically, causing the squeezing flow. Two-dimensional rectangular Cartesian coordinate is considered. The Navier-Stokes equation was reduced using similarity transformation to a single fourth order non-linear ordinary differential equation. The energy equation was transformed to a second order coupled differential equation. We obtained solution to the resulting ordinary differential equations via Homotopy Perturbation Method (HPM). HPM deforms a differential problem into a set of problem that are easier to solve and it produces analytic approximate expression in the form of an infinite power series by using only sixth and fifth terms for the velocity and temperature respectively. The results reveal that the proposed method is very effective and simple. Comparisons among present and existing solutions were provided and it is shown that the proposed method is in good agreement with Variation of Parameter Method (VPM). The effects of appropriate dimensionless parameters on the velocity profiles and temperature field are demonstrated with the aid of comprehensive graphs and tables.

Keywords: coupled differential equation, Homotopy Perturbation Method, plates, squeezing flow

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Authors: Maryam Mosleh, Mahmood Otadi

Abstract:

The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example.

Keywords: Fuzzy number, parametric form of a fuzzy number, fuzzy integrodifferential equation, homotopy analysis method

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Authors: Jay-R M. Mendoza

Abstract:

In almost all mathematics classrooms, students demonstrated discrepancies in their knowledge, skills, and understanding. OECD reports predicted that this continued to aggravate as not all teachers were sufficiently trained to handle this concentration. In response, the paper explored the potential of reSolve’s professional learning module 3 (PLM3) as an affordable and accessible professional development (PD) resource. Participants’ hands-on experience and exposure to PLM3 were audio recorded. After it was transcribed and examined and their work samples were analysed, there were four issues emerged: (1) criticality of conducting preliminary data collections and increasing the validity of inferences about what students can and cannot do by addressing the probabilistic nature of their performance; (2) criticality of the conclusion: a > b and/or (a-b) ∈ Z⁺ among students’ algebraic reasoning; (3) enabling and extending prompts provided by reSolve were found useful; and (4) dynamic adaptation of reSolve PLM3 through developing transferable skills and collaboration among teachers. PLM3 provided valuable insights on assessment, teaching, and planning to include all students in mathematics experiences.

Keywords: algebraic reasoning, building teacher capacity, including all students in mathematics experiences, professional development

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Authors: Anandhi Santhosh

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In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations has been used to describe the system model. In this article, the problem of approximate controllability for nonlinear impulsive integrodifferential equations with state-dependent delay is investigated. We study the approximate controllability for nonlinear impulsive integrodifferential system under the assumption that the corresponding linear control system is approximately controllable. Using methods of functional analysis and semigroup theory, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory.

Keywords: approximate controllability, impulsive differential system, fixed point theorem, state-dependent delay

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Authors: Joon-Hoon Park

Abstract:

The fundamental concept of observability is important in both theoretical and practical points of modern control systems. In modern control theory, a control system has criteria for determining the design solution exists for the system parameters and design objectives. The idea of observability relates to the condition of observing or estimating the state variables from the output variables that is generally measurable. To design closed-loop control system, the practical problems of implementing the feedback of the state variables must be considered and implementing state feedback control problem has been existed in this case. All the state variables are not available, so it is requisite to design and implement an observer that will estimate the state variables form the output parameters. However sometimes unknown inputs are presented in control systems as practical cases. This paper presents a design method and algorithm for observer of control system with unknown input parameters based on Rationalized Haar transform. The proposed method is more advantageous than the other numerical method.

Keywords: orthogonal functions, rationalized Haar transforms, control system observer, algebraic method

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Authors: Yingxu Wang, Jianhua Lu, Jun Peng, Jiawei Zhang

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The fundamental demand in contemporary intelligent science towards Autonomous AI (AI*) is the creation of unprecedented formal means of Intelligent Mathematics (IM). It is discovered that natural intelligence is inductively created rather than exhaustively trained. Therefore, IM is a family of algebraic and denotational mathematics encompassing Inference Algebra, Real-Time Process Algebra, Concept Algebra, Semantic Algebra, Visual Frame Algebra, etc., developed in our labs. IM plays indispensable roles in training-free AI* theories and systems beyond traditional empirical data-driven technologies. A set of applications of IM-driven AI* systems will be demonstrated in contemporary intelligence science, AI*, and cognitive computers.

Keywords: intelligence mathematics, foundations of intelligent science, autonomous AI, cognitive computers, inference algebra, real-time process algebra, concept algebra, semantic algebra, applications

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Authors: Eusebio Jr. Lina, Ederlina Nocon

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Cyclic codes are a fundamental subclass of linear codes that enjoy a very interesting algebraic structure. The class of skew cyclic codes (or θ-cyclic codes) is a generalization of the notion of cyclic codes. This a very large class of linear codes which can be used to systematically search for codes with good properties. A linear code with complementary dual (LCD code) is a linear code C satisfying C ∩ C^⊥ = {0}. This subclass of linear codes provides an optimum linear coding solution for a two-user binary adder channel and plays an important role in countermeasures to passive and active side-channel analyses on embedded cryptosystems. This paper aims to identify LCD codes from the class of skew cyclic codes. Let F_q be a finite field of order q, and θ be an automorphism of F_q. Some conditions for a skew cyclic code to be LCD were given. To this end, the properties of a noncommutative skew polynomial ring F_q[x, θ] of automorphism type were revisited, and the algebraic structure of skew cyclic code using its skew polynomial representation was examined. Using the result that skew cyclic codes are left ideals of the ring F_q[x, θ]/〈x^n-1〉, a characterization of a skew cyclic LCD code of length n was derived. A necessary condition for a skew cyclic code to be LCD was also given.

Keywords: LCD cyclic codes, skew cyclic LCD codes, skew cyclic complementary dual codes, theta-cyclic codes with complementary duals

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Authors: Somveer Singh, Vineet Kumar Singh

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We present a new model based on viscoelasticity for the Non-Newtonian fluids.We use a matrix formulated algorithm to approximate solutions of a class of partial integro-differential equations with the given initial and boundary conditions. Some numerical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis, error estimation and numerical stability of the method are also investigated. Finally, some test examples are given to demonstrate accuracy and efficiency of the proposed method.

Keywords: Legendre Wavelets, operational matrices, partial integro-differential equation, viscoelasticity

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Authors: Muhammad Luqman, Yusuf Kurniawan

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S-Boxes is one of the linear parts of the cryptographic algorithm. The existence of S-Box in the cryptographic algorithm is needed to maintain non-linearity of the algorithm. Nowadays, modern cryptographic algorithms use an S-Box as a part of algorithm process. Despite the fact that several cryptographic algorithms today reuse theoretically secure and carefully constructed S-Boxes, there is an evaluation characteristic that can measure security properties of S-Boxes and hence the corresponding primitives. Analysis of an S-Box usually is done using manual mathematics calculation. Several S-Boxes are presented as a Truth Table without any mathematical background algorithm. Then, it’s rather difficult to determine the strength of Truth Table S-Box without a mathematical algorithm. A comprehensive analysis should be applied to the Truth Table S-Box to determine the characteristic. Several important characteristics should be owned by the S-Boxes, they are Nonlinearity, Balancedness, Algebraic degree, LAT, DAT, differential delta uniformity, correlation immunity and global avalanche criterion. Then, a comprehensive tool will be present to automatically calculate the characteristics of S-Boxes and determine the strength of S-Box. Comprehensive analysis is done on a deterministic process to produce a sequence of S-Boxes characteristic and give advice for a better S-Box construction.

Keywords: cryptographic properties, Truth Table S-Boxes, S-Boxes characteristic, deterministic process

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Authors: Abdullah M. Almeshal, Mohammad R. Alenezi, Muhammad Moaz

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In this paper, we discuss the performance of applying hybrid spiral dynamic bacterial chemotaxis (HSDBC) optimisation algorithm on an intelligent controller for a differential drive robot. A unicycle class of differential drive robot is utilised to serve as a basis application to evaluate the performance of the HSDBC algorithm. A hybrid fuzzy logic controller is developed and implemented for the unicycle robot to follow a predefined trajectory. Trajectories of various frictional profiles and levels were simulated to evaluate the performance of the robot at different operating conditions. Controller gains and scaling factors were optimised using HSDBC and the performance is evaluated in comparison to previously adopted optimisation algorithms. The HSDBC has proven its feasibility in achieving a faster convergence toward the optimal gains and resulted in a superior performance.

Keywords: differential drive robot, hybrid fuzzy controller, optimization, path tracking, unicycle robot

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Authors: Ramdas Sonawane, Mahaveer Gadiya

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The control problem associated with nuclear dynamics is represented by nonlinear integro-differential equation with additive controls. To control chain reaction, certain amount of neutrons is added into (or withdrawn out of) chamber as and when required. It is not realistic. So, we can think of controlling the reactor dynamics by bilinear control, which enters the system as coefficient of state. In this paper, we study the approximate and exact controllability of parabolic integro-differential equation controlled by bilinear control with non-homogeneous boundary conditions in bounded domain. We prove the existence of control and propose an explicit control strategy.

Keywords: approximate control, exact control, bilinear control, nuclear dynamics, integro-differential equations

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Authors: Gilbert Makanda, Roelf Sypkens

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A mathematical model for knowledge acquisition in teaching and learning is proposed. In this study we adopt the mathematical model that is normally used for disease modelling into teaching and learning. We derive mathematical conditions which facilitate knowledge acquisition. This study compares the effects of dropping out of the course at early stages with later stages of learning. The study also investigates effect of individual interaction and learning from other sources to facilitate learning. The study fits actual data to a general mathematical model using Matlab ODE45 and lsqnonlin to obtain a unique mathematical model that can be used to predict knowledge acquisition. The data used in this study was obtained from the tutorial test results for mathematics 2 students from the Central University of Technology, Free State, South Africa in the department of Mathematical and Physical Sciences. The study confirms already known results that increasing dropout rates and forgetting taught concepts reduce the population of knowledgeable students. Increasing teaching contacts and access to other learning materials facilitate knowledge acquisition. The effect of increasing dropout rates is more enhanced in the later stages of learning than earlier stages. The study opens up a new direction in further investigations in teaching and learning using differential equations.

Keywords: differential equations, knowledge acquisition, least squares nonlinear, dynamical systems

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Authors: Iyai Davies, Olivier L. C. Haas

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In this paper, the problem of stability and stabilization for neutral delay-differential systems with infinite delay is investigated. Using Lyapunov method, new delay-independent sufficient condition for the stability of neutral systems with infinite delay is obtained in terms of linear matrix inequality (LMI). Memory-less state feedback controllers are then designed for the stabilization of the system using the feasible solution of the resulting LMI, which are easily solved using any optimization algorithms. Numerical examples are given to illustrate the results of the proposed methods.

Keywords: infinite delays, Lyapunov method, linear matrix inequality, neutral systems, stability

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Authors: Hassan Manouzi, Taous-Meriem Laleg-Kirati

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In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Ito chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm alternative. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: eigenvalue problem, Wick product, SPDEs, finite element, Wiener-Ito chaos expansion

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Authors: Qinghua Zhang, Yanhe Zhu, Xiang Zhao, Yeqin Yang, Hongwei Jing, Guoan Zhang, Jie Zhao

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This paper presents a wearable reconfigurable supernumerary robotic limb with differential actuated joints, which is lightweight, compact and comfortable for the wearers. Compared to the existing supernumerary robotic limbs which mostly adopted series structure with large movement space but poor carrying capacity, a prototype with the series-parallel configuration to better adapt to different task requirements has been developed in this design. To achieve a compact structure, two kinds of cable-driven mechanical structures based on guide pulleys and differential actuated joints were designed. Moreover, two different tension devices were also designed to ensure the reliability and accuracy of the cable-driven transmission. The proposed device also employed self-designed bearings which greatly simplified the structure and reduced the cost.

Keywords: cable-driven, differential actuated joints, reconfigurable, supernumerary robotic limb

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