Variable-coefficient System Research Articles

The similarities and differences of different plane solitons controlled by (3 + 1) – Dimensional coupled variable coefficient system

In this paper, a system with controllable parameters for describing the evolution of polarization modes in nonlinear fibers is studied. Using the Horita’s method, the coupled nonlinear Schrödinger equations are transformed into the bilinear equations, and the one- and two- bright soliton solutions of system(3) are obtained. Then, the influencing factors on velocity and intensity in the process of soliton transmission are analyzed. The fusion, splitting and deformation of the solitons caused by their interactions are discussed. Finally, a method for adjusting the inconsistencies of sine-wave soliton transmission is given. The conclusions of this paper may be helpful for the related research of wavelength division multiplexing systems.

Ion-acoustic shock in a collisional plasma

The paper is concerned with the propagation of ion-acoustic shock waves in a collision dominated plasma whose equations of motion are described by the one-dimensional isothermal Navier-Stokes-Poisson system for ions with the electron density determined by the Boltzmann relation. The main results include three parts: (a) We establish the existence and uniqueness of a small-amplitude smooth traveling wave by solving a 3-D ODE in terms of the center manifold theorem. (b) We study the shock structure in a specific asymptotic regime where the viscosity coefficient and the shock strength are proportional to ε and the Debye length is proportional to (δε)1/2 with two parameters ε and δ, and show that in the limit ε→0, shock profiles obtained in (a) can be approximated by the profiles of KdV-Burgers uniformly for 0<δ≤δ0 with some δ0>0. The proof is based on the suitable construction of the KdV-Burgers shock profiles together with the delicate analysis of a linearized variable coefficient system in exponentially weighted Sobolev spaces involving parameters ε and δ. (c) We also prove the large time asymptotic stability of traveling waves under suitably small smooth zero-mass perturbations. Note that the ions' temperature is allowed to be zero in parts (a) and (b), but necessarily required to be strictly positive in the proof of part (c).

Understanding the evolution of China's tourism industry performance: An internal–external framework

AbstractThis study traces the evolution of China's tourism industry performance. It uses an internal–external framework to evaluate external factors from the macroperspective and microperspective and examines internal factors through a supply and demand standard. We apply a variable coefficient system to optimize the index system and produce a measurement model consisting of 14 indicators, weighing the significance of each indicator via information entropy. Findings from the empirical analysis indicate that substitution and complementary indicators influence the evolution of the tourism industry, suggesting that the synergy effect of various factors should be investigated as the tourism industry continues to mature.

Solitary wave solutions for the variable-coefficient coupled nonlinear Schrödinger equations and Davey–Stewartson system using modified sine-Gordon equation method

In this study, the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts, such as nonlinear Schrödinger systems. These are of considerable importance in many fields of research, including ocean engineering and optics. As an example, we apply the modified method to variable-coefficient coupled nonlinear Schrö dinger equations and Davey–Stewartson system with variable coefficients, treating them as one-dimensional and two-dimensional systems, respectively. As a result of this application, novel solitary wave solutions are obtained for both cases. Moreover, some figures are provided to illustrate how the solitary wave propagation is determined by the different values of the variable group velocity dispersion terms, which can be used to model various phenomena.

Derivation and soliton dynamics of a new non-isospectral and variable-coefficient system

Under investigation in this paper is a new and more general non-isospectral and variable-coefficient non-linear integrodifferential system. Such a system is Lax integrable because of its derivation from the compatibility condition of a generalized linear non-isospectral problem and its accompanied time evolution equation which is generalized in this paper by embedding four arbitrary smooth enough functions. Soliton solutions of the derived system are obtained in the framework of the inverse scattering transform method with a time-varying spectral parameter. It is graphically shown the dynamical evolutions of the obtained soliton solutions possess time-varying amplitudes and that the inelastic collisions can happen between two-soliton solutions.

A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.

Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme.We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

Bootstrap AMG

We develop an algebraic multigrid (AMG) setup scheme based on the bootstrap framework for multiscale scientific computation. Our approach uses a weighted least squares definition of interpolation, based on a set of test vectors that are computed by a bootstrap setup cycle and then improved by a multigrid eigensolver and a local residual-based adaptive relaxation process. To emphasize the robustness, efficiency, and flexibility of the individual components of the proposed approach, we include extensive numerical results of the method applied to scalar elliptic partial differential equations discretized on structured meshes. As a first test problem, we consider the Laplace equation discretized on a uniform quadrilateral mesh, a problem for which multigrid is well understood. Then, we consider various more challenging variable coefficient systems coming from covariant finite-difference approximations of the two-dimensional gauge Laplacian system, a commonly used model problem in AMG algorithm development for linear systems arising in lattice field theory computations.

Analytical Modeling of Metal Transfer for GMAW in the Globular Mode

A lumped parameter dynamical model is developed to describe the metal transfer for gas metal arc welding in the globular mode. The oscillations of molten drop are modeled using a mass-spring-damper system with variable mass and spring coefficient. An analytical solution is developed for the variable coefficient system to better understand the effect of various model parameters on the drop oscillations. The effect of welding drop motion on the observed current and voltage signals is investigated, and the model agrees well with the experimental results. Furthermore, the effect of wire feeding rate (or welding current) on the metal transfer cycle time is studied, and the model successfully estimates the cycle times for different wire feeding rates. Possible regions of unstable metal transfer are investigated both by the model and experiments. The model can be used to identify the range of welding wire feed rates that results in stable metal transfer during the globular mode of metal transfer.

A reduced model for internal waves interacting with topography at intermediate depth

A reduced strongly nonlinear model is derived for the evolution of one-dimensional internal waves over an arbitrary bottom topography. Two layers containing inviscid, immiscible, irrotational fluids of different densities are defined. The upper layer is shallow compared with the characteristic wavelength at the interface, while the bottom region’s depth is comparable to the wavelength. The nonlinear evolution equations obtained are in terms of the internal wave elevation and the mean upper-velocity for the configuration described. The system is a generalization of the one proposed by Choi and Camassa for the flat bottom case in the same physical settings. Due to the presence of a topography a variable coefficient system of partial differential equations arises. These Boussinesq-type equations contain the Intermediate Long Wave (ILW) equation and the Benjamin- Ono (BO) equation when restricted to the unidirectional propagation regime. We intend to use this model to study the interaction of waves with the bottom profile. The dynamics include wave scattering, dispersion and attenuation, among other phenomena.

Solving more general index-2 differential algebraic equations

The paper deals with a class of quasilinear index-2 differential algebraic equations, which covers both linear variable coefficient systems as well as Hessenberg form equations. Supposing low smoothness only, the solvability of initial value problems is stated via classical analytical techniques. For that class of differential algebraic equations, backward differentiation formulas and Runge-Kutta methods as well as projected versions are discussed with respect to feasibility, (in)stability, convergence, and asymptotical behaviour.

Analytical and numerical solutions to higher index linear variable coefficient DAE systems

Analytical and numerical solutions to the singular linear system of differential equations A( t) x′( t) + B( t) x( t) = f( t) are investigated. Closed-form analytical solutions of systems transformable to standard canonical form, SCF (see Campbell (1983)) are derived. Equivalent ways to compute the analytic solution are presented which are of interest in a symbolic/numeric computing environment. Many numerical methods for solving DAE systems are based on Gear's backward difference formula (BDF) method. It is proved that the BDF method and the modified BDF method, MBDF (see Clark (1986)) are stable and O( h k ) accurate after a maximum of ( s − 1) k + 1 steps for linear variable coefficient systems in SCF, where s is the dimension of the algebraic part. These new results are valid for arbitrarily higher index systems and generalize well-known results for linear constant coefficient singular systems to the variable coefficient case. Numerical experiments confirm the order and stability results.

Stability with large step sizes for multistep discretizations of stiff ordinary differential equations

In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter ε \varepsilon characterizes the stiffness of these systems. We approximate a system of ODE’s in this set by a general class of multistep discretizations which includes both one-leg and linear multistep methods. We determine sufficient conditions under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of ODE’s, when the step size resolves the slow time scale but not the fast one. We call this property stability with large step sizes. The theory presented in this paper lets us compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff ODE’s. In particular, we show that one-leg methods have better stability properties with large step sizes than their linear multistep counterparts. This observation is consistent with results obtained by Dahlquist and Lindberg [11], Nevanlinna and Liniger [33] and van Veldhuizen [42]. Our theory also allows us to relate the concept of D-stability (van Veldhuizen [42]) to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes.

An application of analog computers to the statistical analysis of time-variable networks

IN RECENT years an extensive body of mathematical techniques has been developed for the analysis of the response of linear constant coefficient control systems to stationary random processes as inputs. In many cases it is possible to achieve direct synthesis of the optimum system for an assigned task. For nonstationary inputs or variable coefficient systems, no corresponding theory exists, even though problems of this nature arise quite often in practice. In the present paper an analog method is presented for the rms error analysis of a class of nonstationary problems. However, no attempt is made at the synthesis of an optimum system. Following a brief discussion of analog methods applicable to the general nonstationary case, our attention is concentrated on the special problem of a variable coefficient linear system with a stationary random input. Exploitation of the properties of the adjoint system in this case is shown to reduce considerably the labor in computing rms errors in comparison with the general method for nonstationary inputs. The simulation of the adjoint system is shown to be readily obtainable from the simulation of the original system.