Second-order Linear Ordinary Differential Equation Research Articles

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

Explicit Solutions of Singular Differential Equation by Means of Fractional Calculus Operators

Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

The Quantum Arnold Transformation for the damped harmonic oscillator: from the Caldirola-Kanai model toward the Bateman model

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations (LSODE), including systems with friction linear in velocity such as the damped harmonic oscillator, can be related to the quantum free-particle dynamical system. This implies that symmetries and simple computations in the free particle can be exported to the LSODE-system. The quantum Arnold transformation is given explicitly for the damped harmonic oscillator, and an algebraic connection between the Caldirola-Kanai model for the damped harmonic oscillator and the Bateman system will be sketched out.

An improved method for determining the solution of Riccati equations

In this paper, we present an improved method for solving Riccati equations, which possesses a fast convergence rate and high accuracy. This modification is based on a previous approach used for solving Riccati equations (Tang and Li in Appl Math Comput 194:431–440, 2007). The Riccati equation is first transformed to a second-order linear ordinary differential equation, and then to a Volterra integro-differential equation. The Taylor expansion for the unknown function and integration method are employed to reduce the resulting integral equations to a system of linear equations for the unknown and its derivatives. Also, we improve the accuracy of the obtained approximate solutions through the application of the Pade approximant. Some numerical illustrations are given to show the effectiveness of the proposed modification.

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.

Coupled Higgs field equation and Hamiltonian amplitude equation: Lie classical approach and (G′/G)-expansion method

In this paper, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian amplitude equation using (G′/G)-expansion method, where G = G(ξ) satisfies a second-order linear ordinary differential equation (ODE). The travelling wave solutions expressed by hyperbolic, trigonometric and the rational functions are obtained.

Novel interacting phenomena in (2+1) dimensional AKNS system

By means of an new auxiliary equation method based on mapping method and the second-order linear ordinary differential equation, some types of variable separation solutions with two arbitrary lower dimensional functions of the 2+1 dimensional Ablowitz–Kaup–Newell–Segur system are derived. Based on the derived variable separation excitation, some new special types of non-localized solutions such as doubly periodic wave and line periodic wave are constructed by choosing appropriate functions. In addition, we also investigate the annihilation phenomenon and non-elastic interaction graphically.

Propagator for the general time-dependent harmonic oscillator with application to an ion trap

We present the simplest possible formula for the propagator of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second-order ordinary differential equation corresponding to the classical quadratic Hamiltonian. We give an example for the case of the Paul trap.

Analytical Solution of Transversely Isotropic Elastic Subgrade in Rectangular Coordinates

In this paper, basing on Hu haichang’s theory and using Fourier transformation, the problem is changed into solving fourth-order and second-order linear ordinary differential equations. The eigenvalue method is used for solving these equations, and then both displacement and stress with undetermined coefficients can be received. Considering the boundary conditions, the exact displacement and stress of the transversely isotropic foundation can also be gotten. Especially on the transversely isotropic elastic half space, with the aid of computer software, a more simple vertical displacement is presented. When the foundation is degenerated into isotropic elastic half space, the displacement is also degenerated into elastic half space displacement in a rectangular coordinate system which is given in the reference. And this exactly proves the result which is received from this paper is right. On this basis, we can analyze a settlement of transversely isotropic foundation and the static characteristics of rectangular plate on the transversely isotropic foundation.

A procedure for solving some second-order linear ordinary differential equations

The purpose of this work is to introduce the concept of pseudo-exactness for second-order linear ordinary differential equations (ODEs), and then to try to solve some specific ODEs.

On two comparison tests for linear second-order ordinary differential equations

We prove two comparison tests for second-order linear differential equations. The well-known Sturm comparison theorem is a straightforward corollary of the first of them. The second (integral) test permits one to use specific integral relations between the coefficients of two equations to prove the oscillation of one of the equations assuming the oscillation of the other.

Exact travelling solutions for some nonlinear physical models by ( G′/ G)-expansion method

In this paper, we establish exact solutions for some special nonlinear partial differential equations. The (G′/G)-expansion method is used to construct travelling wave solutions of the two-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrodinger–KdV equations, which appear in many fields such as, solid-state physics, nonlinear optics, fluid dynamics, fluid flow, quantum field theory, electromagnetic waves and so on. In this method we take the advantage of general solutions of second-order linear ordinary differential equation (LODE) to solve many nonlinear evolution equations effectively. The (G′/G)-expansion method is direct, concise and elementary and can be used with a wider applicability for handling many nonlinear wave equations.

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin–Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin–Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.

Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

In this paper, a new difference scheme based on quartic splines is derived for solving linear and nonlinear second-order ordinary differential equations subject to Neumann-type boundary conditions. The scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary, which is superior to the well-known Numerov’s scheme with the accuracy being fourth order. Convergence analysis of the present method for linear cases is discussed. Finally, numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method.

The quantum Arnold transformation

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg–Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.

Comment on “Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients”

The present paper corrects the way of using Jordan canonical forms for studying the symmetry structures of systems of linear second-order ordinary differential equations with constant coefficients applied in [1]. The approach is demonstrated for a system consisting of two equations.

Reformulating the Schrödinger equation as a Shabat–Zakharov system

We reformulate the second-order Schrödinger equation as a set of two coupled first-order differential equations, a so-called “Shabat–Zakharov system” (sometimes called a “Zakharov–Shabat” system). There is considerable flexibility in this approach, and we emphasize the utility of introducing an “auxiliary condition” or “gauge condition” that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schrödinger equation. The general solution depends on three arbitrarily chosen functions, and a path-ordered exponential matrix. If one considers path ordering to be an “elementary” process, then this represents complete quadrature, albeit formal, of the second-order linear ordinary differential equation.

Oscillation for Certain Nonlinear Neutral Partial Differential Equations

We present some new oscillation criteria for second‐order neutral partial functional differential equations of the form , (x, t) ∈ Ω × R+ ≡ G, where Ω is a bounded domain in the Euclidean N‐space RN with a piecewise smooth boundary ∂Ω and Δ is the Laplacian in RN. Our results improve some known results and show that the oscillation of some second‐order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced courses in mathematics. In this study, we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery-based approach where students employ their existing skills as a framework for constructing the solutions of first and second-order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first-year undergraduate class. Finally, we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

Modelling of cortical and thalamic 600 Hz activity by means of oscillatory networks

The purpose of this study is to investigate information processing in the primary somatosensory system with the help of oscillatory network modelling. Specifically, we consider interactions in the oscillatory 600 Hz activity between the thalamus and the cortical Brodmann areas 3b and 1. This type of cortical activity occurs after electrical stimulation of peripheral nerves such as the median nerve. Our measurements consist of simultaneous 31-channel MEG and 32-channel EEG recordings and individual 3D MRI data. We perform source localization by means of a multi-dipole model. The dipole activation time courses are then modelled by a set of coupled oscillators, described by linear second-order ordinary delay differential equations (DDEs). In particular, a new model for the thalamic activity is included in the oscillatory network. The parameters of the DDE system are successfully fitted to the data by a nonlinear evolutionary optimization method. To activate the oscillatory network, an individual input function is used, based on measurements of the propagated stimulation signal at the biceps. A significant feedback from the cortex to the thalamus could be detected by comparing the network modelling with and without feedback connections. Our finding in humans is supported by earlier animal studies. We conclude that this type of rhythmic brain activity can be modelled by oscillatory networks in order to disentangle feed forward and feedback information transfer.