Boundary value problems for third-order nonlinear ordinary differential equations

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here.

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

The first D-pi-A octupolar cryptand molecule to exhibit bulk non-linearity

The cryptand derivative has H-bond mediated trigonal network structure that leads to octupolar bulk nonlinearity.

Non-linearity in piezo-fiber reinforced composites: an asymptotic approach

An asymptotically correct analysis is developed for Macro Fiber Composite unit cell using Variational Asymptotic Method (VAM). VAM splits the 3D nonlinear problem into two parts: A 1D nonlinear problem along the length of the fiber and a linear 2D cross-sectional problem. Closed form solutions are obtained for the 2D problem which are in terms of 1D parameters.

Singularity structure of third-order dynamical systems .2

The singularity structure of the solutions of a general third-order system, with polynomial right-hand sides of degree less than or equal to two, is studied about a movable singular point, An algorithm for transforming the given third-order system to a third-order Briot-Bouquet system is presented, The dominant behavior of a solution of the given system near a movable singularity is used to construct a transformation that changes the given system directly to a third-order Briot-Bouquet system. The results of Horn for the third-order Briot-Bouquet system are exploited to give the complete form of the series solutions of the given third-order system; convergence of these series in a deleted neighborhood of the singularity is ensured, This algorithm is used to study the singularity structure of the solutions of the Lorenz system, the Rikitake system, the three-wave interaction problem, the Rabinovich system, the Lotka-Volterra system, and the May-Leonard system for different sets of parameter values. The proposed approach goes far beyond the ARS algorithm.

Weakly non-linear stability of a hydrodynamic accretion disc

When the cold accretion disc coupling between neutral gas and a magnetic field is so weak that the magnetorotational instability is less effective or even stops working, it is of prime interest to investigate the pure hydrodynamic origin of turbulence and transport phenomena. As the Reynolds number increases, the relative importance of the non-linear term in the hydrodynamic equation increases. In an accretion disc where the molecular viscosity is too small, the Reynolds number is large enough for the non-linear term to have new effects. We investigate the scenario of the `weakly non-linear' evolution of the amplitude of the linear mode when the flow is bounded by two parallel walls. The unperturbed flow is similar to the plane Couette flow, but with the Coriolis force included in the hydrodynamic equation. Although there is no exponentially growing eigenmode, because of the self-interaction, the least stable eigenmode will grow in an intermediate phase. Later, this will lead to higher-order non-linearity and plausible turbulence. Although the non-linear term in the hydrodynamic equation is energy-conserving, within the weakly non-linear analysis it is possible to define a lower bound of the energy (alpha A(c)(2), where A(c) is the threshold amplitude) needed for the flow to transform to the turbulent phase. Such an unstable phase is possible only if the Reynolds number >= 10(3-4). The numerical difficulties in obtaining such a large Reynolds number might be the reason for the negative result of numerical simulations on a pure hydrodynamic Keplerian accretion disc.

Third order trace formula

In J. Funct. Anal. 257 (2009) 1092-1132, Dykema and Skripka showed the existence of higher order spectral shift functions when the unperturbed self-adjoint operator is bounded and the perturbation is Hilbert-Schmidt. In this article, we give a different proof for the existence of spectral shift function for the third order when the unperturbed operator is self-adjoint (bounded or unbounded, but bounded below).

Highly stable Ag nanoparticles in agar-agar matrix as inorganic-organic hybrid

A novel synthesis of inorganic-organic hybrid films containing well dispersed and almost uniform size Ag nanoparticles in agar-agar matrix has been reported. The films are found to be highly stable for more than a year. The colloidal particles of Ag can be obtained in large quantities in the form of a film or in the gel form when dispersed in agar-agar or by dissolving in a suitable solvent as solution. Characterization has been done by UV-visible spectroscopy and TEM. The hybrid may be of interest to study third-order non-linear susceptibility.

Structural and optical properties of (100-x)Li2B4O7 center dot x(Ba5Li2Ti2Nb8O30) glasses and glass nanocrystal composites

Optically clear glasses of various compositions in the system (100-x)Li2B4O7 center dot x(Ba5Li2Ti2Nb8O30) (5 <= x <= 20, in molar ratio) were fabricated by splat quenching technique. Controlled heat-treatment of the as-quenched glasses at 500 degrees C for 8 h yielded nanocrystallites embedded in the glass matrix. High Resolution Transmission Electron Microscopy (HRTEM) of these samples established the composition of the nano-crystallites to be that of Ba5Li2Ti2Nb8O30. B-11 NMR studies revealed the transformation of BO4 structural units into BO3 units owing to the increase in TiO6 and NbO6 structural units as the composition of Ba5Li2Ti2Nb8O30 increased in the glass. This, in turn, resulted in an increase in the density of the glasses. The influence of the nominal composition of the glasses and glass nanocrystal composites on optical band gap (E-opt), Urbach energy (Delta E), refractive index (n), molar refraction (R-m), optical polarizability (alpha(m)) and third order non-linear optical susceptibility (chi(3)) were studied.

Non-linear systems with quadratic and cubic damping - an analytical approach

The possible equivalence of second-order non-linear systems having quadratic and cubic damping with third-order linear systems is studied in this paper. It is shown that this equivalence can be established through transformation techniques under certain constraints on the form of the non-linearity of the given system.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

Equivalence conditions for classes of linear and non-linear distributed parameter systems

Equivalence of certain classes of second-order non-linear distributed parameter systems and corresponding linear third-order systems is established through a differential transformation technique. As linear systems are amenable to analysis through existing techniques, this study is expected to offer a method of tackling certain classes of non-linear problems which may otherwise prove to be formidable in nature.

Non-linear dynamic analysis of rotors by finite element method

Non-linear natural vibration characteristics and the dynamic response of hingeless and fully articulated rotors of rectangular cross-section are studied by using the finite element method. In the formulation of response problems, the global variables are augmented with appropriate additional variables, facilitating direct determination of sub-harmonic response. Numerical results are given showing the effect of the geometric non-linearity on the first three natural frequencies. Response analysis of typical rotors indicates a possibility of substantial sub-harmonic response especially in the fully articulated rotors widely adopted in helicopters.

Optical Properties of Nanocrystalline Potassium Lithium Niobate in the Glass System (100-x) TeO2-x(1.5K(2)O-Li2O-2.5Nb(2)O(5))

Optically clear glasses of various compositions in the system (100-x) TeO2-x(1.5K(2)O-Li2O-2.5Nb(2)O(5)) (2 <= x <= 12, in molar ratio) were prepared by the melt-quenching technique. The glassy nature of the as-quenched samples was established via differential scanning calorimetry (DSC). The amorphous and the crystalline nature of the as-quenched and heat-treated samples were confirmed by the X-ray powder diffraction and transmission electron microscopic (TEM) studies. Transparent glasses comprising potassium lithium niobate (K3Li2Nb5O15) microcrystallites on the surface and nanocrystallites within the glass were obtained by controlled heat-treatment of the as-quenched glasses just above the glass transition temperature (T-g). The optical transmission spectra of these glasses and glass-crystal composites of various compositions were recorded in the 200-2500 nm wavelength range. Various optical parameters such as optical band gap, Urbach energy, refractive index were determined. Second order optical non-linearity was established in the heat-treated samples by employing the Maker-Fringe method.