Second-order Homogeneous Differential Equations Research Articles

On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order

In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and L2[0,∞)-solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) [1]. An example is given to illustrate the obtained results.

Analytical study of optical pumping for the D1 line of 85Rb atoms

We present analytical solutions for the populations of the D1 transition line of 85Rb atoms. The time evolution of the populations of the magnetic sublevels in the excited state at the weak intensity limit exhibits several kinds of homogeneous or inhomogeneous second-order differential equations. General solutions of the differential equations result in an analytical form of the population in each magnetic sublevel as a function of the elapsed time, laser intensity, and laser frequency detuning.

FINSLER METRIZABLE ISOTROPIC SPRAYS AND HILBERT’S FOURTH PROBLEM

AbstractIt is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main result we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. We call these conditions the scalar flag curvature (SFC) test. The proof of the main result provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. Hilbert’s fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert’s fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the constant flag curvature (CFC) test, which we developed in our previous paper, Bucataru and Muzsnay [‘Sprays metrizable by Finsler functions of constant flag curvature’, Differential Geom. Appl.31 (3)(2013), 405–415] as well as the SFC test to decide whether or not the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the CFC and SFC tests to construct solutions to Hilbert’s fourth problem.

Solution and show thermodynamic problem by using mechanical and electrical analogies

In this paper, for the case of gas oscillator two characteristic thermodynamic change of state, and for them defined functional dependence of the force from moving the piston in a nonlinear form were considered. In this paper small oscillations are observed and the linearization is performed by neglecting nonlinear term as the small size at isothermal change and with development obtained function in Taylor series for adiabatic change. Based on the analysis of analog oscillator of mechanical system, or the equivalent scheme of problems, characteristics oscillation parameters and the motion law of the observed thermodynamic system are defined. With this procedure greatly simplifying for problem solution was achieved with respect to classical procedure. At the end of the paper, an analysis of the obtained results were done and appropriate electrical analogy was setting. This is used for the presentation of the problem through appropriate electric circuits and analogy mathematical relations, which involve voltage and electric currents as electric sizes. Also, it was shown that the electrical analogy can be used for all mechanical and thermodynamic processes which describes by homogeneous second-order differential equations which are the most common in technical practice.

Periodic solutions of second-order differential equations with discontinuous nonlinearity

We study periodic solutions of homogeneous second-order differential equations with discontinuous nonlinearity. For equations that have chaotic solutions, we determine the value of the spatial entropy with respect to periodic solutions.

Construction of Exactly Solvable Ring-Shaped Potentials

We propose a method for construction of exactly solvable ring-shaped potentials where the linear homogeneous second-order differential equation satisfied by special function is subjected to the extended transformation comprising a coordinate transformation and a functional transformation to retrieve the standard Schr?dinger polar angle equation form in non-relativistic quantum mechanics. By invoking plausible ansatze, exactly solvable ring-shaped potentials and corresponding angular wave functions are constructed. The method is illustrated using Jacobi and hypergeometric polynomials and the wave functions for the constructed ring-shaped potentials are normalized.

Study of Solutions to Some Functional Differential Equations with Piecewise Constant Arguments

We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second‐order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.

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.

Generalized perturbation equations in bouncing cosmologies

We consider linear perturbation equations for long-wavelength scalar metric perturbations in generalized gravity, applicable to nonsingular cosmological models including a bounce from collapse to expansion in the very early universe. We present the general form for the perturbation equations, which follows from requiring that the inhomogeneous universe on large scales obeys the same local equations as the homogeneous Friedmann-Robertson-Walker background cosmology (the separate universes approach). In a pseudolongitudinal gauge this becomes a homogeneous second-order differential equation for adiabatic perturbations, which reduces to the usual equation for the longitudinal gauge metric perturbation in general relativity with vanishing anisotropic stress. As an application we show that the scale-invariant spectrum of perturbations in the longitudinal gauge generated during an ekpyrotic collapse are not transferred to the growing mode of adiabatic density perturbations in the expanding phase in a simple bounce model.

Geometry of nonlinear connections

We show that locally diffeomorphic exponential maps can be defined for any second-order differential equation, and give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. We introduce vertically homogeneous connections as the natural correspondents of homogeneous second-order differential equations. We provide significant support for the prospect of studying nonlinear connections via certain, closely associated second-order differential equations. One of the most important is our generalized Ambrose-Palais-Singer correspondence.

Exact solution of a model of three bound oscillators with Stark nonlinearity

A model of three bound boson oscillators with Stark nonlinearity is introduced and solved by the quantum inverse scattering method. For the trilinear oscillator, the eigenvalue problem is reduced to the spectral problem for the second-order homogeneous differential equation. Bibliography: 11 titles.

Confluence of fuchsian second-order differential equations

Ordinary linear homogeneous second-order differential equations with polynomial coefficients including one in front of the second derivative are studied. Fundamental definitions for these equations: of s-rank of the singularity (different from Poincare rank), of s-multisymbol of the equation, and of s-homotopic transformations are proposed. The generalization of Fuchs' theorem for confluent Fuchsian equations is proved. The tree structure of types of equations is shown, and the generalized confluence theorem is proved.

Stieltjes sums for zeros of orthogonal polynomials

Stieltjes considered sums of reciprocals of differences of zeros of a solution of a homogeneous second-order linear differential equation. Here we re-examine the derivation of these sums with a view to extending the class of differential equations to which the theory applies and including sums involving the zeros of the derivatives as well as those of the polynomials themselves.

The central two-point connection problem of Heun's class of differential equations

Boundary-value problems of ordinary, linear, homogeneous second-order differential equations belong to the most important and thus well-investigated problems in mathematical physics. This statement is true only as long asirregular singularities of the differential equation at hand are not involved. If singular points of irregular type enter the problem one will hardly find a systematic investigation of such a topic from a practical point of view. This paper is devoted to an outline of an approach to boundary-value problems of the class of Heun's differential equation when irregular singularities may be located at the endpoints of the relevant interval. We present an approach to the central two-point connection problem for all of these equations in a quite uniform manner. The essential point is an investigation of the Birkhoff sets of irregular difference equations, which, on the one hand, gives a detailed insight into the structure of the singularities of the underlying differential equation and, on the other hand, yields the basis of quite convenient algorithms for numerical investigations of the boundary values.

Density perturbations in Bianchi type-I universes.

The evolution of small perturbations in the matter density in a Bianchi type-I universe, which was highly anisotropic in its early history and virtually isotropic at the decoupling time, is studied. It is shown that the evolution of density perturbations in a dust-filled Bianchi type-I universe is in the general theory of relativity described by a linear second-order homogeneous differential equation whose growing and decaying solutions are gauge invariant. Moreover, we show that, in the study of density perturbations in Bianchi type-I universes, only axially symmetric universes need to be considered.

On the use of the characteristic method to solve linear homogeneous second-order differential equations with constant matrix coefficients for multicomponent reacting systems

A simple procedure to determine the solution of a second-order differential equations with constant matrices as coefficients is presented. This may be helpful in analysing simultaneous diffusion and convection problems in multicomponent systems

On the Development of Caustics in Shear Flows Over Rigid Walls

In this paper the scattered field produced by a source inside a shear layer of arbitrary profile, flowing above an infinite, rigid wall, is examined. The shear layer is topped by a uniform flow. A representation of the solution is obtained in terms of a pair of functions that satisfy a homogeneous second-order ordinary differential equation, with variable coefficients. The asymptotic form of these functions is obtained in the high-frequency limit. The representation yields, in this limit, a pair of parametric equations whose solutions describe the formation of infinite families of caustics inside the shear layer and downstream of the source. This is in agreement with previous results found by employing ray-theory techniques. Applications are given to the linear profile. The advantages of the present method and its application to other physical problems are discussed.

A search for shape-invariant solvable potentials

The author investigates a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions. It turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions, satisfying linear homogeneous second-order differential equations, can be solutions of the Schrodinger equation with potentials of the form V(x)=W2(x)-W'(x). The author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape-invariant potentials.

A Continuously Stratified Nonlinear Ventilated Thermocline

Three exact, closed-form analytical solutions for the subtropical gyre are presented for the ideal fluid thermocline equations. Specifically, the flow is exactly geostrophic, hydrostatic, and mass and buoyancy conserving. Ekman pumping and density can be chosen as fairly arbitrary functions at the surface. No flow is permitted through the ocean's eastern boundary, or through its bottom. The solutions are continuous extensions of existing layered models. The first solution, discovered simultaneously with Janowitz's solution, uses a deep resting isopycnal layer; the surface density may only be a function of latitude for this solution. A second nonunique solution requires velocities to tend to zero at great depth, giving an additional degree of freedom which permits surface density to be specified almost arbitrarily. This second solution is unphysical in the sense that depth-integrated mass fluxes and energies are infinite. However, a small change in the solution (which returns surface density to a function of latitude only) permits solutions with finite fluxes once more. A third solution requires partial homogenization of the potential vorticity of fluid layers which, while overlying a deep resting iopycnal layer, are not directly ventilated from the surface. Again, fairly arbitrary surface density and Ekman pumping are permitted. All the problems reduce to linear homogeneous second-order differential equations when density replaces depth as the vertical coordinate. The importance of the bottom boundary for closing the problem is stressed.

De Vogelaere's method for the automatic solution of systems of coupled homogeneous second-order differential equations

Systems of coupled second-order ordinary differential equations, arising in quantum mechanical scattering problems, are frequently solved over a fixed mesh by de Vogelaere's method (a fourth-order step-by-step method) and the reactance matrix subsequently calculated. The numerical integration stage of any such problem invariably consumes a large proportion of the computer time required to solve the problem in full. Initial tests with an automatic implementation of de Vogelaere's method to calculate the reactance matrix for electron-hydrogen scattering in the “strong-coupling approximation” appear very promising, with savings of over one-half of the total number of function evaluations required in the numerical integration stage.