On forced oscillations of a simple model for a novel wave energy converter

The dynamics of a simple model for an ocean wave energy converter is discussed. The model for the converter is a hybrid system consisting of a pair of harmonically excited mass–spring–dashpot systems and a set of four state-dependent switching rules. Of particular interest is the response of the model to a wide spectrum of harmonic excitations. Partially because of the piecewise-smooth dynamics of the system, the response is far more interesting than the linear components of the model would suggest. As expected with hybrid systems of this type, it is difficult to establish analytical results, and hence, with the assistance of an extensive series of numerical integrations, an atlas of qualitative results on the limit cycles and other forms of bounded oscillations exhibited by the system is presented. In addition, the presence of unstable limit cycles, the stabilization of the unforced system using low-frequency excitation, the peculiar nature of the response of the system to high-frequency excitation, and the implications of these results on the energy harvesting capabilities of the wave energy converter are discussed.     

Solution of nonlinear wave equation of elastic rod

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia are studied. A nonlinear wave equation is derived. The equation is solved by the method of full approximation.     

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=f _o(X)+f ₁(X,t), where f _o(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.     

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.     

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional fractional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seepage flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.     

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.     

多尺度有限差分方法求解波动方程

小波分析是多尺度分析方法，本文利用具有紧支集的正交小波变换对有限差分方程进行空间多尺度近似，提出适合于层状介质波传问题数值计算的多尺度有限差分方法，将波动方程的求解转换到小波域中进行。利用小波基的自适应性与消失矩特性，有效减少了计算量、提高了稳定性，扩大了可求解的速度范围。地球物理勘探中的数值实例显示了算法具有良好效率。     

The research of blow-up in 2D weakly damped forced KdV equation on thin domain

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｃｏｍｐｌｅｘｉｔｙｏｆｔｈｅｄｙｎａｍｉｃｓｏｆｔｈｅｈｉｇｈｅｒ＿ｄｉｍｅｎｓｉｏｎａｌｄｉｓｓｉｐａｔｉｖｅｓｙｓｔｅｍｓｏｌｉｔａｒｙｗａｖｅ ,ｔｈｅｒｅｉｓｍｕｃｈｄｉｆｆｉｃｕｌｔｙｉｎｓｔｕｄｙｉｎｇｔｈｅｓｅｐｒｏｂｌｅｍｓ.Ｔｈｅｐａｐｅｒｓ [1 ]～ [3]ｍｅｎｔｉｏｎｅｄｔｈｅｔｗｏ＿ｄｉｍｅｎｓｉｏｎａｌＫｄＶｅｑｕａｔｉｏｎｓｄｒｉｖｅｎｂｙＬａｘａｎｄ [3]ｏｂｔａｉｎｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆｓｏｌｕｔｉｏｎｔｏｔｈｅｅｑｕａｔｉｏｎｓ…     

Longitudinal and torsional oscillations of a rod in a viscoelastic fluid

The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed. Received: 14 March 1997 Accepted: 14 May 1998     

Non-linear oscillations of a third-order differential equation

An investigation is conducted into the behavior of the solutions of a third-order non-linear differential equation which is characterized by a non-linearity depending solely upon the Euclidean norm of the associated phase space. The non-linearity represents a central restoring force, which has important applications in modern control theory. For small non-linearities, the existence of a limit cycle is established by a fixed point technique, the approach to the limit cycle is approximated by averaging methods, and the periodic solution is harmonically represented by perturbation. Computer solutions of the differential equation are provided in order to reinforce the analysis. Some related differential equations are discussed including one in which the periodic solution is explicitly prescribed.     

二维波动方程的动态有限元离散化的频散研究

本文利用二维波动方程，推导出波在单元内行进的动态形函数，在此基础上，得到了波在离散体内传播的频散关系，利用这一关系式，绘出了频谱曲线．分析表明：波在离散体内传播的频散程度不仅与波的传播方向有关而且与单元的质量分布、网格划分有密切的关系．     

On positivity preservation in nonlinear finite volume method for multi-term fractional subdiffusion equation on polygonal meshes

Nonlinear Dynamics - The positivity preserving is one of the key requirements to discrete schemes for subdiffusion equation. The main goal of our research is to explore a spatial second-order...