Rayleigh-type Equation Research Articles

Periodic Boundary-Value Problem for a Rayleigh-Type Equation Unsolved with Respect to the Derivative

Periodic Boundary-Value Problem for a Rayleigh-Type Equation Unsolved with Respect to the Derivative

Least-squares method in the theory of nonlinear boundary-value problems unsolved with respect to the derivative

UDC 517.9 We establish constructive necessary and sufficient conditions of solvability and a scheme for the construction of solutions for a nonlinear boundary-value problem unsolved with respect to the derivative. We also suggest convergent iterative schemes for finding approximate solutions of this problem. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems for a Rayleigh-type equation unsolved with respect to the derivative, in particular, in the case of a periodic problem for the equation used to describe the motion of satellites on elliptic orbits.

New results on the existence of periodic solutions for Rayleigh equations with state-dependent delay

Abstract We consider a Rayleigh-type equation with state–dependent delay ℘ ″ ( ϑ ) + 𝒱 1 ( ϑ , ℘ ( ϑ ) ) + A ( ℘ ( ϑ − 𝒱 2 ( ϑ , ℘ ( ϑ ) ) ) ) = B ( ϑ ) . \wp ''\left( \vartheta \right) + \mathcal{V}{_1}\left( {\vartheta ,\wp \left( \vartheta \right)} \right) + A\left( {\wp \left( {\vartheta - \mathcal{V}{_2}\left( {\vartheta ,\wp \left( \vartheta \right)} \right)} \right)} \right) = B\left( \vartheta \right). We establish a set of new su˚cient conditions on the existence of at least one positive periodic solution by using the continuation theorem of coincidence degree theory. Our results not only provide a new approach but also generalize previous results. An example with graphical representations are presented to illustrate the results.

Analytical and experimental study on Van der Pol-type and Rayleigh-type equations for modeling nonlinear aeroelastic instabilities

An empirical modeling of nonlinear aerodynamic force during aeroelastic instabilities, that is, vortex-induced vibration (VIV), galloping and flutter, is necessary in the estimation of vibration responses. Previous works on single-degree-of-freedom (SDOF) models suggest that nonlinear forms (Van der Pol or Rayleigh types) differ from section to section, which causes difficulty in practical application. Analytical evidences in this study have clarified that Van der Pol-type and Rayleigh-type models are equivalent in the amplitude-dependent aerodynamic damping; their difference lies in the higher-order harmonic responses. An identification algorithm of aerodynamic parameters is proposed to improve the robustness of aerodynamic parameters and guarantee the equivalence of both model types. Wind-tunnel tests of typical aeroelastic instabilities indicate that higher-order harmonic responses are small for VIV, galloping, and early-stage flutter instability when compared with the fundamental components due to weak nonlinearity. Van der Pol-type and Rayleigh-type models are both applicable until the flutter amplitude grows excessively large. It is clear that both model types are suitable for any section shape when use the proposed method of aerodynamic identification, and thus can be treated as a universal model for estimating the vibration amplitudes of nonlinear aeroelastic instabilities.

Nonlinear Boundary-Value Problems Unsolved with Respect to the Derivative

We establish constructive necessary and sufficient conditions of solvability and a scheme of construction of the solutions for a nonlinear boundary-value problem unsolved with respect to the derivative. We also suggest convergent iterative schemes for finding approximate solutions of this problem. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems for a Rayleigh-type equation unsolved with respect to the derivative, in particular, in the case of periodic problem for the equation used to describe the motion of satellites on elliptic orbits.

Нелінійні крайові задачі, не розв'язані відносно похідної

УДК 517.9 Знайдено конструктивні необхідні та достатні умови розв'язності, а також схему побудови розв'язків нелінійної крайової задачі, не розв'язаної відносно похідної. Побудовано збіжні ітераційні схеми для знаходження наближень до розв'язків зазначеної задачі. Як приклад застосування побудованої ітераційної схеми знайдено наближення до розв'язків періодичних крайових задач для рівняння типу Релея, не розв'язаного відносно похідної, зокрема, у випадку періодичної задачі для рівняння, яке визначає рух супутника на еліптичній орбіті.

Formation of cylindrical shock wave in two-phase liquid layer with free surface

Numerical model for formation of 1D cylindrical shock-wave (SW) in a liquid layer with a free surface is considered. The liquid states are pure water and distilled water containing free micro-bubbles (1.5 I¼m, 106 cm-3). The SW initiation is performed on the axis by giving the pulse of mass velocity as the exponent for maximal amplitudes from 60 to 20 m/s. A two-phase mathematical model is the system of equation describing average pressure, velocity and density (including the Rayleigh-type equation). For pure water, the distributions of maximal amplitudes both from the axis along the radius for SW and from free surface up to the axis of symmetry for rarefaction wave (RW) were calculated. The distribution of maximal amplitudes of positive SW and negative RW along the radius appears to be completely symmetric. It was shown, the SW amplitude decreases proportionally to r -0.45 (within 3 cm from axis) and then asymptotics (r -0.72) is registered. The increase of RW amplitude during propagation to the axis is a cumulative effect. In two-phase model of distilled water the cavitation begins behind the RW front. Beginning from free surface, the volumetric gas concentration increases in 300 initial values (maximum velocity 60 m/s).Numerical model for formation of 1D cylindrical shock-wave (SW) in a liquid layer with a free surface is considered. The liquid states are pure water and distilled water containing free micro-bubbles (1.5 I¼m, 106 cm-3). The SW initiation is performed on the axis by giving the pulse of mass velocity as the exponent for maximal amplitudes from 60 to 20 m/s. A two-phase mathematical model is the system of equation describing average pressure, velocity and density (including the Rayleigh-type equation). For pure water, the distributions of maximal amplitudes both from the axis along the radius for SW and from free surface up to the axis of symmetry for rarefaction wave (RW) were calculated. The distribution of maximal amplitudes of positive SW and negative RW along the radius appears to be completely symmetric. It was shown, the SW amplitude decreases proportionally to r -0.45 (within 3 cm from axis) and then asymptotics (r -0.72) is registered. The increase of RW amplitude during propagation to the axis is ...

Roles of hydroxyl and sulfate radicals in degradation of trichloroethene by persulfate activated with Fe2+ and zero-valent iron: Insights from carbon isotope fractionation

Active species including hydroxyl (HO) and sulfate radicals (SO4−) play important roles in contaminant degradation during the persulfate based in-situ chemical oxidation (ISCO) process. The generation and contribution of active species are critical and can potentially be evaluated using compound specific isotope analysis (CSIA). However, the evaluation of stable isotope fractionation (or isotope enrichment factor ε values) for contaminants degraded by individual active species of concern is required but lacking. This study firstly determined the carbon isotope fractionation of trichloroethene (TCE) degradation by SO4·− with chemical probe methods to obtain ε values from −6.4±0.7 to −6.9±0.5‰. The ε values were significantly different from those reported for TCE degradation by HO, which could be used to identify the competing TCE degradation by HO and SO4−. Relying on the observed ε values and the extended Rayleigh-type equation, the contributions of SO4− and HO to TCE degradation were evaluated in persulfate activated by Fe0 or Fe(II). This study provides an illuminating idea to determine stable isotope fractionation for contaminant degradation by individual active species, which is crucial for the application of CSIA in relevant environments.

On the zeros of the Pearcey integral and a Rayleigh-type equation

In this work we find a sequence of functions at which the Pearcey function is identically zero. The sequence of functions can be expressed in terms of a second order non-linear ODE, which happens to be the Rayleigh-type. As a byproduct of these facts, we develop a methodology to find a class of functions which solve the moving boundary problem of the heat equation. To this end, we make use of generalized Airy functions, which in some particular cases fall within the category of functions with infinitely many real zeros, studied by P\'olya.

Existence of the stationary solution of a Rayleigh-type equation

A fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (“lower deck”) and a classical Prandtl boundary layer (“upper deck”). The aim of this paper is to prove the existence and uniqueness of the stationary solution of a Rayleigh-type equation, which describes oscillations of the vertical velocity component in the classical boundary layer.

Periodicity of a time-delay system of fractional order joining n-deviating arguments

ABSTRACTIn this study, we generalize a time-delay system joining n-deviating arguments by utilizing the concept of the Riemann–Liouville fractional calculus. The class of equations is taken in view of the Rayleigh-type equation. Our tool is based on the fixed-point theorems. The existence of the outcomes is delivered under some certain conditions. The application is illustrated in the sequel.

New Results on the Existence of Periodic Solutions for Rayleigh Equation with State-Dependent Delay

We give certain sufficient conditions for the existence of periodic solutions to a Rayleigh-type equation with state-dependent delay. With this work we extend and improve some known results in the literature.

On the Existence of Periodic Solutions for a Class of Rayleigh-Type p-Laplacian Equations with Deviating Arguments

We propose new sufficient conditions for the existence of periodic solutions for a class of Rayleigh-type p-Laplacian equations with deviating arguments. The results obtained complement or improve the existing ones.

Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

The Rayleigh equation with two deviating argumentsx′′(t)+f(x'(t))+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=e(t)is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

Existence and uniqueness of periodic solution for forced Rayleigh type equations

By using the coincidence degree theory and some analysis skill, some new sufficient conditions for the existence and uniqueness of periodic solution for a kind of forced Rayleigh type equation are established, which are complement of previously known results.

Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method

In this paper, reproducing kernel theorem is employed to solve anti-periodic solutions for Rayleigh-type equations. A simple algorithm is given to obtain the approximate solutions of the equations. By comparing the approximate solution with the exact analytical solution, we find that the simple algorithm is of good accuracy and it can be also applied to some ordinary or partial differential equations with initial-boundary value conditions and nonlocal boundary value conditions.

EXISTENCE AND STABILITY OF PERIODIC SOLUTIONS OF A RAYLEIGH TYPE EQUATION

AbstractIn this work, we shall be concerned with the following forced Rayleigh type equation: Under certain assumptions, some criteria for guaranteeing the existence, uniqueness and asymptotic stability (in the Lyapunov sense) of periodic solutions of this equation are presented by applying the Manásevich–Mawhin continuation theorem, Floquet theory, Lyapunov stability theory and some analysis techniques. Moreover, an example is provided to demonstrate the applications of our results.

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form x ″ + f ( x ′ ( t ) ) + g ( t , x ( t ) ) = e ( t ) .

Existence of asymptotically stable periodic solutions of a Rayleigh type equation

In this work, we investigate the following Rayleigh type equation: x ″ ( t ) + f ( x ′ ( t ) ) + g ( t , x ( t ) ) = e ( t ) . Some criteria for guaranteeing the existence of asymptotically stable (in the sense of Lyapunov) periodic solutions of this equation is presented by using Mawhin’s continuation theorem, Floquet theory, Lyapunov stability theory and some analysis techniques. Moreover, an example is provided to illustrate the validity of our results.

Axisymmetric bubble collapse in a quiescent liquid pool. II. Experimental study

We present an experimental study of the detachment of a gas bubble growing quasistatically at constant flow rate conditions from a vertical nozzle placed at the bottom of a quiescent pool of water. In particular, we focus on the dynamics of the necking process and on its dependence on both the Bond and Weber numbers, respectively, defined as Bo=ρga2/σ, and WeQ=ρQ2/(π2a3σ). Here, a, ρ, σ, g, and Q are the inner radius of the nozzle, the liquid density, the gas-liquid surface tension, the gravitational acceleration, and the gas flow rate. Our experimental data indicate that the collapse process is not only driven by capillarity but also by the liquid hydrostatic pressure. Good agreement is achieved between the measurements of the collapse time and that given by the scaling proposed as tc=tσ/1+121/3Bo2/3 where tσ=(ρa3/σ)1/2 is the capillary time, valid in the limit WeQ→0. In addition, the details of the final instants previous to pinch-off have been analyzed by recording the time evolution of both the bubble neck radius, R0, and the axial curvature at the minimum radius, 2r1, using a high speed digital video camera and an appropriate set of microscopic lenses. We find that the dimensionless, asymptotic law, recently obtained for the inviscid pinch-off of a bubble, given by τ∝R02 exp[−ln(R02)], is never achieved down to about 20 μm. However, the experimental results are accurately reproduced by a pair of two-dimensional Rayleigh-type equations that include liquid inertia as well as surface tension effects.