The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System

For many nonlinear systems in our life, the chaos phenomenon generated under certain conditions in special cases will split the system and result in a crash-down of the system. This paper discusses the stable control of one class of chaotic systems and a control method based on the accurate exponential solution of a differential equation is used. Compared with other methods, the advantages are: this method determines that the system can exponentially converge at the origin and the convergence rate can be easily regulated. The chaotic system with unknown parameters is also deduced and validated by using this method. In practical application, it is found that the ship’s electric system also has the same model, so it has certain practical significance.

THE LOCALIZED RESONANCE EXCITATIONS ARE DESCRIBED BY EXACT SOLUTIONS OF THE MODIFIED KORTEWEG-DE VRIES EQUATION

The exact soliton solutions of modified Korteweg-de Vries equation are obtained by procedure based on Hirota method. It has shown that thesе solutions described the bound state of soliton-antisoliton pairs which are formed in result resonance interaction of two solitons. Keywords: exact solution, rational-exponential solution, Hirota method

On Liouvillian Solutions of Third Order Homogeneous Linear Differential Equations

In this article we will consider third order homogeneous differential equations:  L(y)=y'''+a_1y'+a_0y(a_0,a_1 ∈k) whose Galois group G（L） is imprimitive. This case is characterised by the fact that the third symmetric power equation L ^ⓢ3(y)=0 has an exponential solution whose square is rational (Singer & Ulmer 1993). If L(y)=0 has a Liouvillian solution z whose logarithmic derivative u=z'/z  is algebraic over a differential field (k,') ,we will give an algorithm to find the relation between a_0, a_1 , the semi-invariant S=Y_1Y_2Y_3 which is unique up to multiplication by a constant, the coefficients C_0, C_1 of the minimal polynomial P(u) of u  and their derivatives. The aim of this work is to diminutize the number of constants C_m  stated in the algorithm of Singer & Ulmer (Singer & Ulmer 1993 Algorithm p. 31) whose determination is not easy to do, and we will achieve this by using Groebner Basis.

A Molecular Dynamics-Based Model for Knudsen Number and Slip Velocity

Much research has been devoted to incestigating the relationship between Knudsen number and slip velocity using different lattice Boltzmann methods. However, these models are complex to implement for simulations in continuum regime, and have shown to diverge when compared with Direct Simulation Monte Carlo (DSMC) simulations at high Knudsen numbers. In this study, a molecular dynamics (MD)-based Knudsen number is presented, and the relationship between Knudsen number and slip velocity is presented. The proposed slip model directly correlates the Knudsen number with the slip velocity. The model is implemented on a shear-driven MD simulation of a Couette flow, and curve fitting is used to get an exponential solution for the slip velocity. The solution obtained from the proposed model as well as the solutions from the literature are compared with a DSMC simulation. The results show that the proposed exponential solution agrees well with DSMC simulations in comparison with the models from the literature. The exponential solution can serve as boundary conditions for simulating flows at different Knudsen numbers in continuum regime.

Klein-Gordon-Fock equation from Einstein general relativity

A time-space symmetry based cylindrical model of geometrical dynamics was proposed. Accordingly, the solution of Einstein gravitational equation in vacuum has a duality: an exponential solution and a wave-like one. The former leads to a "microscopic" cosmological model with Hubble expansion. Due to interaction of a Higgs-like cosmological potential, the original time-space symmetry is spontaneously broken, inducing a strong time-like curvature and a weak space-like deviation curve. In the result, the wave-like solution leads to Klein-Gordon-Fock equation which would serve an explicit approach to the problem of consistency between quantum mechanics and general relativity.

Estimating Subsurface Velocities from Surface Fields with Idealized Stratification

A previously published method by Wang et al. for predicting subsurface velocities and density from sea surface buoyancy and surface height is extended by incorporating analytical solutions to make the vertical projection. One solution employs exponential stratification and the second has a weakly stratified surface layer, approximating a mixed layer. The results are evaluated using fields from a numerical simulation of the North Atlantic. The simple exponential solution yields realistic subsurface density and vorticity fields to nearly 1000 m in depth. Including a mixed layer improves the response in the mixed layer itself and at high latitudes where the mixed layer is deeper. It is in the mixed layer that the surface quasigeostrophic approximation is most applicable. Below that the first baroclinic mode dominates, and that mode is well approximated by the analytical solution with exponential stratification.

Analytic Model and FEM Characterization of Two Piezoresistive Microphone Membrane

In this paper we will present the analytical modeling of two different microphone membranes (one square and one circular) fabricated using the MultiMEMS process at Sensonor in Norway. In particular we built up a mathematical model of the pressurebehavior inside the cavity, when an external pressure is applied. The time constant of the exponential solution, gave us the possibility to estimate the minimum working frequencies of these devices. We compared the results that we found with both the mechanical resonance estimates from FEM models and the initial measurements of the devices presented in [.