On properties of a deformed Freud weight

59 - Exact Second Order Equations

Correction for ‘On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations’ by V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan (Proc. R. Soc. A 461, 2451–2476. (doi: 10.1098/rspa.2005.1465)). The sentence preceeding equation (4.48) is incorrect and should read as follows.

This article concerns the positive periodic solutions for a system of second-order nonlinear ordinary differential equations, in which the nonlinear term is sublinear in one equation and superlinear in the other equation.By using the fixed point theorem of cone expansion and compression we obtain the existence of positive periodic solutions.

Acoustic nonlinearities in a quasi 1-D duct with arbitrary mean properties and mean flow

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

The aim of this paper is to estimate an interval of the existence of positive solutions for a boundary value problem for the system of three nonlinear second-order ordinary differential equations. Krasnosel'skiĭ–Precup fixed point theorem is used to determine this interval theoretically. For Dirichlet boundary conditions theoretical result is compared with result obtained numerically.

In this paper, we propose the reduced model for the full dynamics of a bicycle and analyze its nonlinear behavior under a proportional control law for steering. Based on the Gibbs-Appell equations for the Whipple bicycle, we obtain a second-order nonlinear ordinary differential equation (ODE) that governs the bicycle’s controlled motion. Two types of equilibrium points for the governing equation are found, which correspond to the bicycle’s uniform straight forward and circular motions, respectively. By applying the Hurwitz criterion to the linearized equation, we find that the steer coefficient must be negative, consistent with the human’s intuition of turning toward a fall. Under this condition, a critical angular velocity of the rear wheel exists, above which the uniform straight forward motion is stable, and slightly below which a pair of symmetrical stable uniform circular motions will occur. These theoretical findings are verified by both numerical simulations and experiments performed on a powered autonomous bicycle.

We prove an existence theorem concerning the identifica tion of a nonlinearity term in an overspecified boundary value problem related to a (nonlinear) second-order ordinary differential equation containing a spectral parameter λ.

Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes–Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a “force controlled by the movements of the oscillator itself”, i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a “controlled nonlinear oscillator”. To that end, they introduced a change of variables and parameters to transform the “controlled nonlinear oscillator” that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes–Rand equation, Chua’s cubic circuit, Chen’s system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of “generalized controlled nonlinear oscillators”, which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Kármán equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where Ω1≅ω1 and Ω2≅ω2. From the frequency response equations, the stability of the system is investigated according to the Routh–Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using matlab 7.0 software. A comparison is made with the available published work.

Low Reynolds number flows in a microscopic and tapered tube with a permeability are theoretically investigated in steady conditions. A momentum equation for the main flow in the tube is derived from a set of incompressible Navier-Stokes equations for an axisymmetric flow. The second-order nonlinear ordinary differential equation on a pressure is derived using a velocity profile of the fourth-order polynomials for the main flow, in which there are nonlinear terms of both orders of Re2 and Re. Solutions obtained are classified into two types; one is a flow with a Reynolds number far less than 161, whilst another is a flow with a Reynolds number larger than the former case but smaller than 248. They are compared with an exact solution of the original nonlinear differential equation. Effects of the permeability of the tube wall and the Reynolds number of the main flow upon the pressure loss, the axial and radial components of the main flow velocity, and the resistance coefficient are clarified. Furthermore, a flow in a permeable and straight tube is investigated as a special case of the permeable and tapered tube, and it is demonstrated that an adverse pressure gradient takes place at a relatively large Reynolds number when the main flow is withdrawn through the permeable wall. As a consequence, the results obtained for the permeable and tapered tube contain flows in both the impermeable tapered tube and the impermeable straight tube, the latter of which is Hagen-Poiseuille flow, as special cases.

The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m², and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σ_rr = (-1.2973e-05)kg/m/s². We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.

The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations