Second-order Linear Differential Equations Research Articles

Multi-phase iterative learning control for high-order systems with arbitrary initial shifts

Aiming at the second-order tracking system with arbitrary initial shifts, this paper presents a multi-phase iterative learning control strategy. Firstly, utilizing the form of solution of the second-order non-homogeneous linear differential equation with constant coefficients and the initial shifts, we can select the appropriate control gain to ensure that the second-order systems are stable and reach the stable output after a fixed time. Secondly, on the premise that the second-order systems have reached the fixed output, two methods are proposed for rectifying the fixed shift, namely, shifts rectifying control and varied trajectory control. Theoretical analysis shows that the multi-stage iterative learning control strategy proposed in this paper can ensure that the second-order systems achieve complete tracking in the specified interval. Finally, the simulation examples affirm the validation of the designed algorithms.

Fast Generalized Sliding Sinusoidal Transforms

Discrete cosine and sine transforms closely approximate the Karhunen–Loeve transform for first-order Markov stationary signals with high and low correlation coefficients, respectively. Discrete sinusoidal transforms can be used in data compression, digital filtering, spectral analysis and pattern recognition. Short-time transforms based on discrete sinusoidal transforms are suitable for the adaptive processing and time–frequency analysis of quasi-stationary data. The generalized sliding discrete transform is a type of short-time transform, that is, a fixed-length windowed transform that slides over a signal with an arbitrary integer step. In this paper, eight fast algorithms for calculating various sliding sinusoidal transforms based on a generalized solution of a second-order linear nonhomogeneous difference equation and pruned discrete sine transforms are proposed. The performances of the algorithms in terms of computational complexity and execution time were compared with those of recursive sliding and fast discrete sinusoidal algorithms. The low complexity of the proposed algorithms resulted in significant time savings.

Some Approaches for Solving Multiplicative Second-Order Linear Differential Equations with Variable Exponentials and Multiplicative Airy’s Equation

This paper offers several approaches for solving multiplicative second-order linear differential equations with variable exponentials, such as normalization and reduction to Riccati equations. In addition, in this paper, the multiplicative version of the Airy equation, which emerges in fluid mechanics, geophysics, and atomic physics, is solved using the multiplicative power series solution method.

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

Gradient elasticity solutions of 2D nano-beams

In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed.

Nested invariant tori foliating a vector field and its curl: Toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.

Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets

ABSTRACT In this paper, we present an effective method under Taylor wavelets and collocation technique to find an approximate solution of linear and non-linear second-order singular value differential equations. Application of this method on various problems confirms the efficiency, easy applicability, and rapid computation. The obtained solution is compared with some other existing numerical solutions. For many problems, this method gives a solution same as the exact solution of the problem which confirms the effectiveness and accuracy of this method. Additionally, we include graphs and figures to demonstrate that the Taylor wavelet method offers better accuracy for a number of problems. MATLAB software is used to process relative data and execute calculations.

Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations

Abstract This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.

Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

Separating Long-Term Equilibrium Adaptation from Short-Term Self-Regulation Dynamics Using Latent Differential Equations

Self-regulating systems change along different timescales. Within a given week, a depressed person’s affect might oscillate around a low equilibrium point. However, when the timeframe is expanded to capture the year during which they onboarded antidepressant medication, their equilibrium and oscillatory patterns might reorganize around a higher affective point. To simultaneously account for the meaningful change processes that happen at different time scales in complex self-regulatory systems, we propose a single model that combines a second-order linear differential equation for short timescale regulation and a first-order linear differential equation for long timescale adaptation of equilibrium. This model allows for individual-level moderation of short-timescale model parameters. The model is tested in a simulation study which shows that, surprisingly, the short and long timescales can fully overlap and the model still converges to the reasonable estimates. Finally, an application of this model to self-regulation of emotional well-being in recent widows is presented and discussed.

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

General solutions’ laws of linear partial differential equations II

This paper uses Z transformations to obtain the general solutions of a large number of second-order, third-order and fourth-order linear partial differential equations for the first time, including one-dimensional inhomogeneous wave equation. Using general solutions, we have obtained plenty of exact solutions of the typical definite solution problems, which proves the important value of general solutions. We propose the Z4 transformation for the first time and initially use it to solve a specific case. We successfully obtain the Fourier series solution using the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famous unresolved debate in the history of mathematics.

Stability Analysis of Second-Order Linear PDEs on Time Scales

This paper presents the stability analysis of a class of second-order linear partial differential equations (PDEs) on time scales with diffusion operator and first-order partial derivative. According to the time scale theory, the Lyapunov functional method, and some inequality techniques, sufficient conditions for exponential stability are strictly obtained, and the results are generalized for that where both the discrete-time and continuous-time cases are considered jointly. In addition, the theoretical results are applied to exponential synchronization of reaction-diffusion neural networks (RDNNs). Simulation examples are given to verify the feasibility of our results.

Некоторые аспекты качественного анализа модели высокочастотной геоакустической эмиссии

Геоакустическая эмиссия является индикатором напряженно-деформированного состояния геосферы, поэтому она играет важную роль в разработке методики прогнозирования сильных землетрясений в сейсмоактивных регионах, таких как Камчатка. В работе исследуются некоторые аспекты качественного анализа математической модели высокочастотной геоакустической эмиссии. Математическая модель высокочастотной геоакустической эмиссии представляет собой цепочку из двух связанных осцилляторов, которая описывается системой из двух линейных обыкновенных дифференциальных уравнений второго порядка с непостоянными коэффициентами. Непостоянные коэффициенты обладают свойством непрерывного затухания при больших временах. Каждое дифференциальное уравнение описывает импульс высокочастотной геоакустической эмиссии со своими характеристиками, а взаимодействие между импульсами – обмен энергией осуществляется с помощью коэффициента линейной связи. Для математической модели были исследованы вопросы существования и единственности решения, доказана соответствующая теорема на основе принципа сжимающих отображений из функционального анализа. Исследована устойчивость нулевого решения математической модели геоакустической эмиссии, результаты были сформулированы в виде теоремы, а также исследована устойчивость при больших временах с помощью критерия Рауса-Гурвица. Проведено исследование на жесткость, показано, какие параметры в модели могут влиять на жесткость исследуемой системы дифференциальных уравнений, приведена визуализация исследований зависимости жесткости от времени. С помощью численного метода Розенброка, реализуемого в среде компьютерной математики Maple были построены осциллограммы и фазовые траектории при различных условиях: наличия жесткости, неустойчивости и т.д. Проведена интерпретация результатов исследования и даны направления дальнейшего исследования математической модели высокочастотной геоакустической эмиссии. Geoacoustic emission is an indicator of the stress-strain state of the geosphere, so it plays an important role in the development of methods for predicting strong earthquakes in seismically active regions such as Kamchatka. The paper investigates some aspects of the qualitative analysis of the mathematical model of high-frequency geoacoustic emission. The mathematical model of high-frequency geoacoustic emission is a chain of two coupled oscillators, which is described by a system of two second-order linear ordinary differential equations with non-constant coefficients. Non-constant coefficients have the property of continuous damping at large times. Each differential equation describes a pulse of high-frequency geoacoustic emission with its own characteristics, and the interaction between pulses – energy exchange is carried out using a linear coupling coefficient. For a mathematical model, the existence and uniqueness of a solution were investigated, and the corresponding theorem was proved based on the contraction mapping principle from functional analysis. The stability of the zero solution of the mathematical model of geoacoustic emission was studied, the results were formulated in the form of a theorem, and the stability at large times was also studied using the Routh-Hurwitz criterion. A study of stiffness was carried out, it was shown which parameters in the model can affect the stiffness of the system of differential equations under study, and visualization of studies of the dependence of stiffness on time is given. Using the Rosenbrock numerical method implemented in the Maple computer mathematics environment, oscillograms and phase trajectories were constructed under various conditions: the presence of rigidity, instability, etc. The results of the study are interpreted and directions for further research of the mathematical model of high-frequency geoacoustic emission are given.

Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations

We extend the invariant subspace method (ISM) to a class of Hamilton–Jacobi equations (HJEs) and a family of third-order time-fractional dispersive PDEs with the Caputo fractional derivative in this letter. More precisely, the complete classification is presented for such HJEs that admit invariant subspaces governed by solutions of the second-order and third-order linear ordinary differential equations (ODEs). Meanwhile, some concrete equations are derived for the construction of new exact solutions u(x,t)=∑i=1nCi(t)fi(x). Then a set of invariant subspaces of the considered third-order time-fractional non-linear dispersive equations are obtained. Based on the Laplace transform method (LTM) and applying several properties of the well known Mitta-Leffer (ML) function, the different types of explicit solutions of a family of third-order time-fractional dispersive PDEs are finally derived.

A localized reduced basis approach for unfitted domain methods on parameterized geometries

This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on (i) extension of snapshots on the background mesh and (ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.

Mechanical Analysis and Optimal Design of Wave Energy Device

At present, the global energy problem is serious. Wave energy is the kinetic energy and potential energy of waves on the ocean surface. Because of its high energy density, wide distribution, and long continuous operation of the device, it has become the focus of academic research. The key problem to be solved urgently for the large-scale application of wave energy is how to optimize the energy conversion efficiency of wave energy devices. This paper starts from the Newtonian dynamic equation, analyzes the motion state of the wave energy device in different wave environments, and studies the optimization and control of its power output. This paper first adopts the analysis method of first whole and then isolation, and establishes a one-dimensional coordinate system with the zero point of sea level. Then, the force analysis of the system is carried out, the dynamic differential equation is established, and the initial draft of the system is solved through the initial balance equation. Since the dynamic equation is a second-order linear ordinary differential inhomogeneous linear equation, the fourth-order Runge is used. The Kutta method is used to solve the problem, and the numerical solution for the position and velocity of the float is obtained. Secondly, the movable coordinate system is established with the float as the reference system, and the force analysis of the oscillator is carried out to obtain its relative dynamic equation. Then, the initial relative position of the oscillator is solved through the initial balance equation, and the relative dynamic differential equation is obtained by using the fourth-order Runge-Kutta method to obtain the numerical solution of the relative position and relative velocity of the oscillator with respect to time. Finally, the numerical solution of the absolute position and absolute velocity of the oscillator with respect to time is obtained through coordinate transformation.

The indefinite integrals of Meijer's G-functions from integrating factors

Conway [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26:812–824] introduced a new and simple method named the ‘Lagrangian method’ for deriving indefinite integrals of both elementary and special functions, provided the function satisfies the second-order linear differential equation. In this paper, different Meijer's G-functions have been used for deriving indefinite integrals, which satisfy second-order differential equations. We have derived recurrence relations and the integration of those recurrence relations by the Lagrangian method. We have discussed the integration of Euler identity, which is given in terms of Meijer's G-function. Different additional relations of Meijer's G-functions have also been discussed.

Solution processes for second-order linear fractional differential equations with random inhomogeneous parts

The novelty of this paper is to derive a mean square solution of a second-order (Caputo) fractional linear differential equation in which the coefficients and initial conditions are random variables, and the forcing term is a second order stochastic process. Using the so-called mean square calculus and assuming mild conditions on the random variables of the equation together with an exponential growth condition on the forcing term, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and correlation obtained from the truncated power series solution are convergent as well. The theory is illustrated with several examples in which different kind of distributions on the input parameters are assumed.