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Publications related to Ordinary Differential Equations (4,980)
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We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximati...
Article
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We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximati...
Chapter
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En este capítulo se presenta un estilo para la enseñanza de las matemáticas a través de la modelación, que pueden utilizar los profesores que orientan cursos de Ecuaciones Diferenciales, y en el que se utilizan varios conceptos y presaberes. La modelación matemática en este contexto se puede ver como un instrumento para hacer o, en su defecto, apli...
Preprint
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This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with...
Preprint
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Deep residual networks (ResNets) have shown state-of-the-art performance in various real-world applications. Recently, the ResNets model was reparameterized and interpreted as solutions to a continuous ordinary differential equation or Neural-ODE model. In this study, we propose a neural generalized ordinary differential equation (Neural-GODE) mode...
Preprint
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The idea of neural Ordinary Differential Equations (ODE) is to approximate the derivative of a function (data model) instead of the function itself. In residual networks, instead of having a discrete sequence of hidden layers, the derivative of the continuous dynamics of hidden state can be parameterized by an ODE. It has been shown that this type...
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State estimation is an important aspect in many robotics applications. In this work, we consider the task of obtaining accurate state estimates for robotic systems by enhancing the dynamics model used in state estimation algorithms. Existing frameworks such as moving horizon estimation (MHE) and the unscented Kalman filter (UKF) provide the flexibi...
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This document, as the title stated, is meant to provide a vectorized implementation of adjoint dynamics calculation for Graph Convolutional Neural Ordinary Differential Equations (GCDE). The adjoint sensitivity method is the gradient approximation method for neural ODEs that replaces the back propagation. When implemented on libraries such as PyTor...
Preprint
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In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of first degree divided by the same...
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Neurological studies show that injured neurons can regain their functionality with therapeutics such as Chondroitinase ABC (ChABC). These therapeutics promote axon elongation by manipulating the injured neuron and its intercellular space to modify tubulin protein concentration. This fundamental protein is the source of axon elongation, and its spat...
Article
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This study addresses CuO-TiO2/CMC-water hybridnano-liquid in the influence of mixed convection flow and thermal radiative flow past a stretchable vertical surface. Cross nanofluid containing Titanium dioxide (TiO2), and Copper Oxide (CuO) are scattered in a base fluid of kind CMC water. In addition, theirreversibility analysis is also examined in t...
Conference Paper
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Recent research in deep learning has shown that neural networks can learn differential equations governing dynamical systems. In this paper, we adapt this concept to Virtual Analog (VA) modeling to learn the ordinary differential equations (ODEs) governing the first-order and the second-order diode clipper. The proposed models achieve performance c...
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Chemical kinetics mechanisms are essential for understanding, analyzing, and simulating complex combustion phenomena. In this study, a Neural Ordinary Differential Equation (Neural ODE) framework is employed to optimize kinetics parameters of reaction mechanisms. Given experimental or high-cost simulated observations as training data, the proposed...
Article
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We study the existence of solutions to a self-adjoint coupled system of three nonlinear second-order ordinary differential equations equipped with cyclic boundary conditions. We apply the tools of fixed point theory to obtain criteria ensuring the existence and uniqueness of solutions to the problem at hand. Examples are presented for the illustrat...
Article
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This study employed Lyapunov function method to investigate the stability of nonlinear ordinary differential equations. Using Lyapunov direct method, we constructed Lyapunov function to investigate the stability of sixth order nonlinear ordinary differential equations. We find V (x), a quadratic form, positive definite and U(x) which is also positi...
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This paper proposes a novel inverse kinematics (IK) solver of articulated robotic systems for path planning. IK is a traditional but essential problem for robot manipulation. Recently, data-driven methods have been proposed to quickly solve the IK for path planning. These methods can handle a large amount of IK requests at once with the advantage o...
Conference Paper
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This paper proposes a data-driven modeling approach for complex Anaerobic Digestion (AD) systems. This method is called Dynamic Mode Decomposition with Control (DMDc), which is an emerging equation-free technique for deducing global linear state-space input-output models with actuation for complex systems. DMDc is applied to a set of data generated...
Preprint
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A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this lim...
Preprint
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Despite the tremendous progress made by deep learning models in image semantic segmentation, they typically require large annotated examples, and increasing attention is being diverted to problem settings like Few-Shot Learning (FSL) where only a small amount of annotation is needed for generalisation to novel classes. This is especially seen in me...
Preprint
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Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages...
Preprint
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The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar cubic vector field with an i...
Research Proposal
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My main research area is Analysis of Ordinary Differential Equations. In particular, my research interest so far has been centered around the formal differential expression Ju ′ + qu = wf (1) where J is a constant skew-hermitian n × n matrix and q, w are n × n-matrices of distributions of order 0, i.e. (local) measures.
Article
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Individual organizations, such as hospitals, pharmaceutical companies, and health insurance providers, are currently limited in their ability to collect data that are fully representative of a disease population. This can, in turn, negatively impact the generalization ability of statistical models and scientific insights. However, sharing data acro...
Article
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The purpose of the current study is to investigate the non-Newtonian unsteady Williamson fluid on a stretching/shrinking surface along with thermophoresis and Brownian effects. Basically, the model consists of a time-dependent magnetic field. The fluid is considered to be electrically conducting due to the effect of the external magnetic field. The...
Article
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In the present paper, we investigate the influence of the choice of continuous linear operator for obtaining the approximate periodic solutions of ordinary second-order differential equations. In most of these problems, the periods are unknown, and the determination of these periods and periodic solutions is a difficult issue. So, a new computation...
Preprint
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The library QIBSH++ is a C++ object oriented library for the solution of Quasi Interpolation problems. The library is based on a Hermite Quasi Interpolating operator, which was derived as continuous extensions of linear multistep methods applied for the numerical solution of Boundary Value Problems for Ordinary Differential Equations. The library i...
Article
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In order to solve general seventh-order ordinary differential equations (ODEs), this study will develop an implicit block method with three points of the form y(7)(x) = f (x, y(x), y0(x), y00(x), y000(x), y(4)(x), y(5)(x), y(6)(x)) directly. The general implicit block method with Hermite interpolation in three points (GIBM3P) has been derived to so...
Article
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This paper deals with the asymptotic stabilization of a class of port-Hamiltonian (pH) 1-D Partial Differential Equations (PDE) with spatial varying parameters, interconnected with a class of linear Ordinary Differential Equations (ODE), with control input on the ODE. The class of considered ODE contains the effect of a proportional term, that can...
Preprint
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Fundamental diagrams describe the relationship between speed, flow, and density for some roadway (or set of roadway) configuration(s). These diagrams typically do not reflect, however, information on how speed-flow relationships change as a function of exogenous variables such as curb configuration, weather or other exogenous, contextual informatio...
Article
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The derivation of the Embedded pair Diagonally Implicit Type Runge-Kutta Method (EDITRKM) for solving 3rd special order ordinary differential equations (ODEs) is introduced in the current study. The EDITRKM techniques are the name of the approach. This approach in the present study has two types: EDITRKM 4(3) for orders 4 and 3 of the first pair an...
Conference Paper
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Electrocardiogram (ECG) signals provide rich information on individuals' potential cardiovascular conditions and disease, ranging from coronary artery disease to the risk of a heart attack. While health providers store and share these information for medical and research purposes, such data is highly vulnerable to privacy concerns, similar to many...
Preprint
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Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods which exploit "parallelism within the method" so that arbitrary user ODEs can be parallelized. We...
Preprint
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The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g. civil or mechanical structures), which are typically high-dimensional in nature. In the scope of physics-informed machine learn...
Article
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Based on the theory of ordinary differential equations, this paper proposes a stable and discriminative method for piano tone tuning. We perform discriminative training on the hidden Markov tone model according to ordinary differential equations' feature extraction parameters and model parameters. The model can improve the recognition rate of piano...
Article
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The article uses ordinary differential to solve inhomogeneous equations by the constructivist learning concept. We use the equivalent equations to study the n-th order non-homogeneous linear ordinary differential equations with constant coefficients and get the method of solving this equation. Then we use the Filippov transformation and comparison...
Article
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Mango with the botanical name of Mangifera Indica (MI) (Linn.) is one of the traditionally used herb trees in the third-world country used for different curative and prophylaxis measure for different health challenges. This article presents updated information on its mathematical modeling using ordinary differential equations (S,E,I,P,R) for theore...
Preprint
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Continuous deep learning models, referred to as Neural Ordinary Differential Equations (Neural ODEs), have received considerable attention over the last several years. Despite their burgeoning impact, there is a lack of formal analysis techniques for these systems. In this paper, we consider a general class of neural ODEs with varying architectures...
Preprint
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Different from the charge properties of electrons, the spin properties of electrons have produced a revolution since their discovery. Spintronics have opened a window for researchers because of their low energy consumption properties. This study explores the application of spintronics in artificial intelligence and computer vision, and achieves hig...
Article
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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differe...
Article
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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differe...
Conference Paper
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En este trabajo se presenta una propuesta de diseño de tarea de aprendizaje matemático que ayude a los estudiantes en su proceso de modelación de una aplicación de una ecuación diferencial ordinaria (EDO), concretamente sobre la Ley de Enfriamiento de Newton en un contexto experimental y con el uso de herramientas digitales para la enseñanza.
Article
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This study proposes one-step block method with generalized three-hybrid points for solving initial value problems of third order ordinary differential equations directly using interpolation and collocation strategy. In deriving this method, the approximate power series function is interpolated at {x n , x n+r , x n+s } while its third and fourth de...
Article
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Floating solar park deployment increasing as the demand for clean energy increases around the world. These solar parks introduce a new avenue of investment for the clean energy producers, nevertheless increasing offshore floating solar parks affects the marine ecosystem. In this paper mathematical model related to the effect of solar irradiance on...
Article
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RESUMO O presente trabalho busca dar continuidade a pesquisa na área de equações diferenciais ordinárias e o estudo das simulações numéricas com a utilização do Software MatLab® (Math Works Inc.) no processo de modelagem matemática. Para a implementação desse estudo foi utilizado o clássico modelo epidemiológico SIR juntamente aos dados da COVID-19...
Article
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A influência da resistência do ar sobre um corpo deslocando-se sobre um cabo inclinado foi estudada por uma abordagem integrada englobando modelagem, solução analítica de equações diferenciais ordinárias, experimentação e estimativa de erros para os dados experimentais. O sistema utilizado permite realizar o experimento com três configurações difer...
Article
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Cancer with all its more than 200 variants continues to be a major health problem around the world with nearly 10 million deaths recorded in 2020, and leukemia accounted for more than 300,000 cases according to the Global Cancer Observatory. Although new treatment strategies are currently being developed in several ongoing clinical trials, the high...
Article
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Fire detection methods based on the Convolutional Neural Networks (CNN) have advantages of high accuracy, wide coverage and robustness, receiving significant attention from researchers. Among CNN‐based methods, ResNet has achieved better performance than other CNN frameworks in fire detection system, since it uses stacked residual blocks to enlarge...
Preprint
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Multi-fidelity modeling and learning are important in physical simulation-related applications. It can leverage both low-fidelity and high-fidelity examples for training so as to reduce the cost of data generation while still achieving good performance. While existing approaches only model finite, discrete fidelities, in practice, the fidelity choi...
Article
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Este estudo teve por objetivo explorar a Lei de Resfriamento proposta por Newton e realizar uma experimentação com materiais de fácil acesso, contemplando uma atividade experimental para o estudo, no Ensino Médio, da função exponencial, bem como algumas formas de caracterizá-la. No que tange ao desenvolvimento matemático da lei de resfriamento, fiz...
Article
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In this paper, a mathematical model for COVID-19 disease incorporating clinical management based on a system of Ordinary Differential Equations is developed. The existence of the steady states of the model are determined and the effective reproduction number derived using the next generation matrix approach. Stability analysis of the model is carri...
Preprint
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A novel way of using neural networks to learn the dynamics of time delay systems from sequential data is proposed. A neural network with trainable delays is used to approximate the right hand side of a delay differential equation. We relate the delay differential equation to an ordinary differential equation by discretizing the time history and tra...
Article
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Continuous-depth neural networks, such as the Neural Ordinary Differential Equations (ODEs), have aroused a great deal of interest from the communities of machine learning and data science in recent years, which bridge the connection between deep neural networks and dynamical systems. In this article, we introduce a new sort of continuous-depth neu...
Article
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As long as the field of Engineering, Science and Technology exists, the place of Mathematical modelling that involves stiff systems cannot be overemphasized. Models involving stiff system may result in ordinary differential equations (ODEs) or sometimes as system of ordinary differential equations which must be solved by experts working in that fie...
Article
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Abstract---This paper introduces a new complex integral transformation obtained by inserting a complex parameter into the well-known Rangaig integral transform kernel function. The new integral transform is denoted by the acronym SEL and is called the Complex (Serifenur-Emad-Luay) integral transform. The proposed SEL integral transform feature...
Article
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In the paper, we determine the period of an n-dimensional nonlinear dynamical system by using a derived formula in an (n + 1)-dimensional augmented space. To form a periodic motion, the periodic conditions in the state space and nonlinear first-order differential equations constitute a special periodic problem within a time interval with an unknown...
Article
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Organism network systems provide a biological data with high complex level. Besides, these data reflect the complex activities in organisms that identifies nonlinear behavior as well. Hence, mathematical modelling methods such as Ordinary Differential Equations model (ODE's) are becoming significant tools to predict, and expose implied knowledge an...
Preprint
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The combination of ordinary differential equations and neural networks, i.e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles. However, deciphering the numerical integration in Neural ODE is still an open challenge, as many researches demonstrated that numerical integration significantly affects the...
Article
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In this paper, a robust modification of the variational iteration method that gives a numerical solution for a system of linear/nonlinear differential equations of fractional order was proposed. This technique does not need the perturbation theory or linearization. The conformable fractional derivative initiated by the authors Khalil et al. is cons...
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Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the mini...
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A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints.
Preprint
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Neural Ordinary Differential Equations model dynamical systems with \textit{ODE}s learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in engineering and biological systems. Broader classes of differential equations (DE) have been proposed as remedi...
Article
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A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real-life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well-known and popular concept in mathematics because of its massive applicati...
Article
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A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real-life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well-known and popular concept in mathematics because of its massive applicati...
Article
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This work presents a novel method to calculate the distributed water depth and flow velocity of heavy rainfall-induced stormwater runoff. Although the shallow water equations or other numerical models are often employed to deal with the calculation of distributed water depth and flow velocity of runoff, it still suffers from low computational effic...
Article
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This paper is a contribution to the spectral theory associated with the differential equation Ju′+qu=wf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ju'+qu=wf$$\end{d...
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The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the It\^o process and is of great importance to the literature of statistical physics and machine learning. The FPE can be regarded as a continuity equation where the change of the density is completely determined by a time varying velocity f...
Article
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In this paper, we study the existence of dominated weak solutions u of the boundary-value problem (-g(t)(u(t)′)γ)′=f(t,u(t)),in(0,∞),u(0)=0,u(∞)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin...
Article
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Many problems in mathematical physics are modelled by second order ordinary differential equations (ODES). Consequently, numerically solving second order ODES has attracted much attention of many mathematicians and physicists. Most of the existing methods reduce second order ODES to a system of first order ODES. In the present study, we are motivat...
Article
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We consider a nonlinear ordinary differential equation of arbitrary order with coefficients in the form of power series that converge in a neighborhood of the origin. The methods created in power geometry in recent years make it possible to compute formal solutions to that equation in the form of Dulac series. We describe the corresponding algorith...