Analysis Of Differential Equations Research Articles

Local models and curves of tangencies of foliation pairs

In this work we give a complete description of germs of foliation pairs –non dicritical and dicritical– given by generic analytic differential equations with degenerated non dicritical and dicritical singularities, by means of local analytic representatives (local models) and analytic normalizing transformations. We give a realization theorem of local models, prove the existence and uniqueness of them, and obtain the analytic invariants for non dicritical and dicritical foliation pairs.

Numerical simulations of equilibrium, kinetics, and chain length distribution in linear polycondensation proceeding in nano-scale small volumes: An apparent violation of the law of mass action

A simple polycondensation process Mi + Mj ⇄ Mi+j + Z, proceeding in dispersion of nano-scale droplets was described by analytical solutions and differential equations giving insight into equilibrium and kinetics of the process. Stochastic and deterministic simulations show that the statistical nature of polycondensation, both irreversible and reversible one, affects the way the macromolecules of different lengths are formed and distributed among droplets. The droplet distributions in respect to the number of reacting chains and the chain length distributions depend, for the given conversion, on the polycondensation equilibrium constant Kcond and the initial conditions: monomer concentration and the number of its molecules in a droplet. The apparent equilibrium constants of condensation K¯ij=[Mi+j]¯e[Z]¯e/[Mi]¯e[Mj]¯e depend on oligomer/polymer sizes as well as on the initial number of monomer molecules in a droplet. This apparent violation of the law of mass action was explained by stochasticity of involved processes. Moreover, when the polycondensation byproduct Z is being removed reversibly from the system, K¯ij depends also on average equilibrium concentration [Z]¯e. The indicated effects are observed not only for ideally dispersed systems (all droplets contain initially the same number of monomer molecules), but also when the initial numbers of monomer molecules conform the Poisson distribution. However, the chain length distribution as well as kinetics and K¯ij differ here from those observed for systems with uniform distribution.

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Numerical simulation of an external ballistic problem using analytical approach and atmosphere flow visualization by finite element method

Purpose. Analysis of the dynamic characteristics of the cargo in the presence of the speed of its carrier, taking into account the influence of the external environment and the determination of factors that affect the operational characteristics and parameters for the dynamic system control. Methodology. An approximate analytical approach to solving the nonlinear problem of external ballistics of a system with time-dependent parameters, which is implemented using the asymptotic perturbation method, and a numerical algorithm for modeling a dynamic process using a 3D software complex and the finite element method, which allows to visualize the nature of the flow around the object under wind load conditions. Findings. An analytical approach to solving the nonlinear problem of external ballistics of a system with time-varying parameters is proposed, as well as a numerical model, solution and visualization of a dynamic process that can be applied in problems of mathematical physics and engineering calculations. Originality. The use of an approximate analytical approach to solving the nonlinear problem of external ballistics, which includes a system of time-varying parameters, is a significant innovative step. For the first time, a three-dimensional distribution of the free fall parameters under atmospheric conditions was obtained depending on the time of free fall under the condition of an initial velocity different from zero. Obtaining the characteristics of the distribution, taking into account the aerodynamic quality of the object, made it possible to adapt the methods of analytical mechanics and differential equations with variable coefficients to the solution of the applied problem. The obtained characteristic three-dimensional surface allows to perform a study of the aerodynamic quality of the object under study using numerical methods and to visualize dynamic processes in three-dimensional space in order to obtain a qualitative picture of the perturbation of the object by the air flow. This integrated approach to the study of ballistic characteristics of cargo forms the scientific principle to perform engineering calculations to solve the problem of controlling dynamic systems. Practical value. An approximate analytical approach and a calculation model of the dynamic process of a mechatronic unmanned system make it possible to increase its efficiency in the presence of an initial speed and ensure the accuracy of the application of ballistic cargo delivery systems in real operating conditions.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines

In this paper, we analyze the problem of finding the minimum dimension n such that an analytic map/ordinary differential equation over R n can simulate a Turing machine in a way that is robust to perturbations. We show that one-dimensional analytic maps are sufficient to robustly simulate Turing machines; but the minimum dimension for the analytic ordinary differential equations to robustly simulate Turing machines is two, under some reasonable assumptions. We also show that any Turing machine can be simulated by a two-dimensional C ∞ ordinary differential equation on the compact sphere S 2 .

Thermal circuit model of prismatic lithium cell considering dynamic non-uniform characteristics during charging-discharging in energy storage

Current analytical and simulation models for lithium battery thermal behaviour encounter efficiency or accuracy challenges in energy storage applications. In this paper, an analytical thermal analysis approach for prismatic lithium cells considering dynamic non-uniform characteristics is proposed to calculate the dynamic temperature distribution. A thermal circuit model and analytical differential equation group were formulated to capture the dynamic non-uniform thermal characteristics during charging/discharging. A solving algorithm was provided to balance accuracy and efficiency by establishing an iterative mechanism over the modelling parameters. The proposed model and algorithm were applied to three commercial lithium cells. The experiments under various charge-discharge conditions verified the millisecond-level computational cost and less than 3 % calculation accuracy. The parameter calibration, iteration mechanism, and reproducible codes enhanced the engineering applicability. The performance of the approach suggested its excellent potential in battery thermal management for energy storage.

An efficient multi-state evaluation approach for lithium-ion pouch cells under dynamic conditions in pressure/current/temperature

The current analytical and simulation models for lithium-ion pouch cells encounter challenges in accuracy and efficiency for battery management applications. This paper proposes an analytical model and decoupling algorithm for lithium-ion pouch cells. It can evaluate the multi-state distribution and evolution in gas/heat generation, temperature, pressure, and deformation. Analytic differential equations are formulated, including mechanical, thermal circuit and reaction sub-models. The analysis framework considers dynamic conditions (e.g., mechanical pressure, charging/discharging, heat dissipation) and uniform characteristics (e.g., local heating, anisotropic heat conduction, shell deformation). A sequential decoupling algorithm from the global to the local is proposed to address the convergence issue from the high-dimensional and strongly nonlinear equations. Iterative mechanisms are established to determine the solving parameters for eliminating confusion in engineering applications. The numerical and experimental results on three commercial pouch cells verified the approach performances, including stable convergence, millisecond efficiency, and 2% temperature error. The performances suggest its application prospects in energy storage and vehicle power systems.

Three Examples in the Dynamical Systems Theory

We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms $R$, $S$ of a closed two-dimensional annulus that possess the intersection property but their composition $RS$ does not ($R$ being just the rotation by $\pi/2$). The second example is that of a non-Lagrangian $n$-torus $L_0$ in the cotangent bundle $T^\ast{\mathbb T}^n$ of ${\mathbb T}^n$ ($n\geq 2$) such that $L_0$ intersects neither its images under almost all the rotations of $T^\ast{\mathbb T}^n$ nor the zero section of $T^\ast{\mathbb T}^n$. The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form $\dot{x}=f(x,y)$, $\dot{y}=\mu g(x,y)$ in the closed upper half-plane $\{y\geq 0\}$ such that for each family, the corresponding phase portraits for $01$ are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.

INNOVATIVE METHODS OF TEACHING THE DISCIPLINE HIGHER MATHEMATICS TO STUDENTS STUDYING CHEMICAL TECHNOLOGY AND ENGINEERING

The article discusses some innovative methods that can be used in practical classes in higher mathematics, teaching students of chemical specialties. The possibilities for the development of competencies in complex interuniversity projects are closely related to the issues of classifying all types of interrelationships of disciplines within the framework of courses according to curricula, as well as the choice of additional universal competencies. Mathematics for chemical process engineers is, first of all, a useful tool for solving many chemical and technological problems and tasks. The typical curriculum takes into account the modern needs of related and special disciplines in the mathematical education of students, and consists of four main sections: the foundations of algebra and analytical geometry, mathematical analysis, differential equations, probability theory and mathematical statistics. When writing the article, many years of experience in teaching students of chemical specialties by the department 'Higher Mathematics' of the National Technical University «KhPI» were used. The purpose of the scientific research of teachers and students presented in the article is to increase the competitiveness of Ukrainian technical education in the world market by developing and implementing innovative models and methods. When writing this article, the authors pursued three goals. First, to give the general course of mathematics for students of chemical and related specialties an appropriate professional orientation; secondly, to form in students of the first years of study ideas about the mathematical apparatus, information technologies and the mathematical modeling of modern chemistry and, thirdly, to instill in students the primary skills of building mathematical models of the simplest physical and mathematical processes when studying a mathematics course.

Online inference of lane changing events for connected and automated vehicle applications with analytical logistic diffusion stochastic differential equation

Inferences of the lane changing behaviors (including lane changing initiation moment LCIM, cross-lane-mark moment CLMM, and lane changing duration LCD) of the surrounding vehicles and taking actions to avoid collision are essential for the safe operation of connected and autonomous vehicles (CAVs). The majority of current models rely on data-driven methods that need to be trained before deployment. Besides, an analytical and stochastic formulation of the lateral model which can generate CLMM and LCD distribution and adjust the parameters online is still lacking. To the best of our knowledge, currently, no method can provide an analytically stochastic lateral movement that explicitly considers the system noise and simultaneously outputs the information of lane changing initiation moment, cross-lane-mark moment, and lane changing duration in a real-time manner. To fill such a gap, a stochastic lateral trajectory framework with parsimonious parameters is established and an online simultaneous inference of LCIM, CLMM, and LCD is developed. The proposed method is tested with the highD and NGSIM datasets. The results show that 1) computational efficiency-wise, the algorithm takes milliseconds to run, which suggests promising prospects for field deployment; 2) the false negative and false positive errors of the LCIM inference are as low as 1.5%, indicating that the method is robust to the stochastic noise; 3) LCIM inference is insensitive to the lane width, which means the model is error tolerant when such information is lacking; and 4) the error of CLMM inference is within 2 sec, while most of the error of LCD locates within 4 sec, suggesting satisfactory performance.

Formula omitted]: Over- and Under-approximating Reachable Sets for analytic time-invariant differential equations

We present OURS, a precision-oriented MATLAB tool for computing Over- and Under- approximations of Reachable Sets for analytic time-invariant differential equations. The main theoretical framework behind OURS is introduced, including the concept of evolution function, whose zero sub-level sets are used to describe reachable sets, and a series representation of evolution function. Especially, using the partial sums of this series, OURS finds over- and under-approximations of evolution function at time-instants: it consecutively estimates each remainder of the corresponding partial sum of the series with interval arithmetics until one remainder satisfies the designated precision, and then builds over- and under-approximations with this remainder. The structure of OURS is also presented, such as the forms of inputs and outputs, and technical implementations of the crucial steps inside. Moreover, we compare OURS with two other existing methods on some benchmarks. Finally, OURS is additionally extended to deal with a class of time-invariant differential systems with disturbances described by uncertain parameters. The performance of OURS in dealing with parametric uncertainties is also shown by two examples with comparisons and discussions.

Learning Dynamic Systems Using Gaussian Process Regression with Analytic Ordinary Differential Equations as Prior Information

Recently the data-driven learning of dynamic systems has become a promising approach because no physical knowledge is needed. Pure machine learning approaches such as Gaussian process regression (GPR) learns a dynamic model from data, with all physical knowledge about the system discarded. This goes from one extreme, namely methods based on optimizing parametric physical models derived from physical laws, to the other. GPR has high flexibility and is able to model any dynamics as long as they are locally smooth, but can not generalize well to unexplored areas with little or no training data. The analytic physical model derived under assumptions is an abstract approximation of the true system, but has global generalization ability. Hence the optimal learning strategy is to combine GPR with the analytic physical model. This paper proposes a method to learn dynamic systems using GPR with analytic ordinary differential equations (ODEs) as prior information. The one-time-step integration of analytic ODEs is used as the mean function of the Gaussian process prior. The total parameters to be trained include physical parameters of analytic ODEs and parameters of GPR. A novel method is proposed to simultaneously learn all parameters, which is realized by the fully Bayesian GPR and more promising to learn an optimal model. The standard Gaussian process regression, the ODE method and the existing method in the literature are chosen as baselines to verify the benefit of the proposed method. The predictive performance is evaluated by both one-time-step prediction and long-term prediction. By simulation of the cart-pole system, it is demonstrated that the proposed method has better predictive performances.

Fractional ModelArchitecturefora crude-gasturbulent flow

Currently, petroleum is one of the primary economic resources for Mexico and other developing countries. The legal production begins at the wells and storage systems; once required, it is transported to the refinery or the ship exportation. The oil industry uses various mathematical models to know the pressure drop in a pipeline or reproduce different variables included; however, they require lengthy analysis times that can be reduced with solutions of analytical differential equations. In the present work, a model is developed to know the pressure drops in the pipeline but from the fractional calculation, reducing the analysis steps and developing predictions quickly. The obtained model allows the reproductionof the velocity profiles for the various Reynold values that characterize the exposed flow.

A local Hopf lemma and unique continuation for elliptic equations

We prove results on unique continuation at the boundary for the solutions of real analytic elliptic partial differential equations of the formΔu+∑i=1nbi(x)∂u∂xi+c(x)u=0. The work is motivated by and generalizes the main results of X. Huang et al. in [15] and [16], and M.S. Baouendi and L.P. Rothschild in [5].

Field loss calculation of a wind‐powered axial flux alternator by analytical equations

AbstractVarious techniques have been investigated and proposed for core loss minimization in electrical machines. Nevertheless, many of such methods are mostly complicated and not suitable for consideration at a preliminary design stage. In this work, a simplified procedure which uses an analytical approach to minimizing the field's losses of an Axial Flux Permanent Magnet Alternator (AFPMA), is presented. First, the output equation of an AFPMA is referred, and then the minimization of the losses is investigated by analytical differential equations. The result of the derived‐specific magnetic loading is investigated using three different core materials, namely 35RM300, 50JN350, and 65JN800, and is found to reduce with increased frequencies. The 35RM300 core material gives the maximum specific magnetic loading and minimum power loss at investigated frequencies of 50 to 500 Hz. Although the 35RM300 core material gives the best performance, the optimal values are only determined as suitable by the manufacturer's design criteria. This study is a key indicator for a simple and efficient core material selection in the design of a Wind‐Powered AFPMA without the need for complicated analyses at the preliminary design stage.

Analysis of third order linear analytic differential equations with a regular singularity: Bessel and other classical equations

Analysis of third order linear analytic differential equations with a regular singularity: Bessel and other classical equations

Two-loop integrals for planar five-point one-mass processes

We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for two-jet-associated W-boson production at leading color in QCD. We provide a set of pure integrals together with differential equations in canonical form. We obtain analytic differential equations efficiently from numerical samples over finite fields, fitting an ansatz built from symbol letters. The symbol alphabet itself is constructed from cut differential equations and we find that it can be written in a remarkably compact form. We comment on the analytic properties of the integrals and confirm the extended Steinmann relations, which govern the double discontinuities of Feynman integrals, to all orders in ϵ. We solve the differential equations in terms of generalized power series on single-parameter contours in the space of Mandelstam invariants. This form of the solution trivializes the analytic continuation and the integrals can be evaluated in all kinematic regions with arbitrary numerical precision.

Development and assessment of adjoint sensitivity analysis method for transient two-phase flow simulations

The adjoint method is efficient for calculating sensitivities of a few responses to a large number of parameters. The cost of solving an adjoint equation is comparable to the cost of solving the forward governing equation. Once the adjoint solution is available, the sensitivities to any number of parameters can be calculated with little effort. There are two methods to develop the adjoint equations: continuous method and discrete method. In the continuous method, the control theory is applied to the forward governing equation and produces an analytical partial differential equation for solving the adjoint variable; in the discrete method, the control theory is applied to the discrete form of the forward governing equation and produces a linear system of equations for solving the adjoint variable. In this article, an adjoint sensitivity analysis framework is developed using both the continuous and discrete methods. These two methods are assessed with one steady-state and one transient test cases. Adjoint sensitivities from both methods are verified by sensitivities calculated with a perturbation method. Adjoint sensitivities from both methods are physically reasonable and match each other. Though continuous method is computationally more efficient than the discrete method, difficulties are observed in solving the continuous adjoint equation for cases where the adjoint equation contains sharp discontinuities in the source terms. In such cases, the continuous method is not as robust as the discrete adjoint method.

A hypersingular integro-differential equation of the Euler type

In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.