An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

级数展开法求静电势分布的统一公式

利用泰勒级数展开的二阶近似法求解任意边界区域内的静电泊松方程,给出了二维区域内静电势和静电场的空间分布.在物理问题的对称轴上使用罗必达法则求解.各类几何形状的边界处理中,使用了一套统一的求电势公式,从而简化了问题的复杂性,使程序编制精练化.     

Phase Transition of Ising Model on a Kind of Periodic Fractal Lattices

In this paper we introduced a periodic fractal structure (PFS) and studied the critical behavior of the nearest-neighbor-interacting Ising model within Migdal-Kadanoff bond-moving approximation. The critical curve equation and the formulas of the critical exponents were derived approximately in a special parameter region.     

快上升前沿电磁脉冲与目标腔体的孔腔共振效应研究

 利用时域有限差分方法（FD-TD Method)研究了快上升前沿电磁脉冲（FREMP）对目标腔体的孔腔共振效应。研究表明：由于FREMP的高频分量丰富,使得FREMP更容易通过目标腔体上的孔缝,与目标腔体发生孔腔共振。由于孔缝的对外辐射,振荡幅度将逐渐减弱。     

Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method

In this paper, we analyze the generalized Camassa and Holm （CH） equation by the improved element-free Galerkin （IEFG） method. By employing the improved moving least-square （IMLS） approximation, we derive the formulas for the generalized CH equation with the IEFG method. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed. The effectiveness of the IEFG method for the generalized CH equation is investigated by numerical examples in this paper.     

An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations

Poisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.     

Systems under the influence of white and colored poisson noise

We deal in this paper with systems driven by white or colored Poisson noise. For a free Brownian particle under the influence of white Poisson noise an exact generalized master equation in position space is obtained. In the Gaussian and Smoluchowski limits, known results are recovered. For a general process defined by a stochastic differential equation, with colored Poisson noise, we find an approximate generalized master equation, including first order terms in the correlation time and the first correction to the gaussianity. Under a more restrictive approximation, the stationary distribution function is given. This is used to study the phase transition in the steady state for a Verhulst model. Corrections to the gaussianity are discussed in this case.     

Root Asymptotics of Spectral Polynomials for the Lamé Operator

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass $\wpThe study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schr?dinger equation with finite gap potential given by the Weierstrass -function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integer-valued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.     

动态断裂力学的无网格流形方法

运用无网格流形方法求解动态断裂力学问题.该方法利用单位分解法和有限覆盖技术建立形函数，形函数的建立不受域内不连续的影响，可较好地求解裂纹问题.对于局部化问题，该方法的形函数构造较其他方法更为有效，避免了其他方法在建立试函数时没有考虑不连续尖端的缺点.由于采用有限覆盖技术建立试函数，该方法克服了不连续对试函数的影响，尤其当不连续变得复杂时，更能显示该方法在处理不连续方面的优点.在求解动态断裂力学问题时，弹性动力学积分弱形式的推导采用加权残数法，空间离散采用基于单位分解法的无网格流形方法，时间离散主要采用Newmark法.最后给出两个数值算例，将计算结果与解析解对比，说明该方法的正确性和可行性. 关键词： 有限覆盖 无网格流形方法 动态断裂力学 动态应力强度因子     

随机系统的概率密度函数形状调节

针对受高斯白噪声激励的非线性随机系统,提出了使状态响应的概率密度函数形状跟踪期望形状的调节方法.首先,确立了非线性随机系统的多项式反馈机制,同时对系统中的非线性部分进行多项式展开;然后,以Fokker-Planck-Kolmogorov方程为工具,导出了与控制增益相关的各阶矩递推方程,并根据跟踪问题的要求,构造了矩逼近优化问题,用梯度搜索法求解该优化问题,获得了调节函数;再依据特征函数与概率密度函数构成Fourier对的关系,对状态响应的概率密度函数进行重构;最后,通过两个例子仿真,验证了本文方法的有效性.     

Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential

In this work, we determine the Fisher and Shannon entropies, the expectation values and the squeeze state for a noncentral inversely quadratic plus exponential Mie-type potential analytically.The proposed potential is solved under the Schr?dinger equation using a special Greene Aldrich approximation to the centrifugal term to obtain a normalised wave function within the framework of the Nikiforov–Uvarov method. Numerical results are obtained for different screening parameters:α?=?0.1, 0.12 and 0.13 for varying real constant parameter(B). The numerical solutions are obtained only for ground state. The numerical results of Fisher entropy both for position and momentum spaces are in good agreement with existing literature. The normalisation constant, wave function, and probability density plots are carried out using a well designed Mathematica algorithm.The Fourier transform of position space entropy gives the momentum space entropy.     

非傍轴高斯光束传输方程解的探讨

由洛伦兹规范下光场满足的矢势和标势方程出发, 探讨了初始光场具有任意对称性的一般非傍轴光束在空间的传输情况。利用角频谱并结合傅里叶变换处理得到了非傍轴光束的传输方程。利用扰动方法并结合新近似取法得到了非傍轴横向光场的扰动形式解, 该解对于近场弱非傍轴情况下的效果比强非傍轴条件下要好。给出了光束非傍轴传输高阶(八阶)修正解。对于零阶情况, 非傍轴光束退化为傍轴光束, 传输方程也相应退化为傍轴传输的模型方程。高斯解为扰动参量的零阶修正解。     

Frequency/time domain modeling of a direct drive point absorber wave energy converter

Two different methods to model a point absorber wave energy converter(WEC)with direct drive linear power take-off(PTO)are proposed in the present study:the frequency domain(FD)method and the time domain(TD)method.In the FD analysis,the frequency response function(FRF)of the WEC device is obtained via the equation of motion,and the expressions of power capture width in regular and random waves are derived as well.In the TD modeling,based on a state space approximation of the convolution term in the motion equation,both regular wave and random wave simulations are carried out.The regular wave simulation results indicate that the state space approximation is sufficiently accurate and the capture width reaches the maximum in the vicinity of the natural frequency.In the random wave simulations,the effects of buoy size,the PTO damping and wave climate on the power capture width are discussed in detail,which leads to the conclusion that the capture widths are influenced by the natural frequency of the WEC device,peak frequency of the wave spectrum,the amplitude of FRF and PTO damping.Furthermore,the increase of the capture width is at the cost of a relatively large buoy size and PTO damping when control is not included.     

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg–de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

An improved element-free Galerkin method for solving generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

有尾焰动力飞行体电磁脉冲效应数值分析

 利用时域有限差分方法分析了动力段飞行体的等离子体高温尾焰对飞行体电磁脉冲效应的影响。计算结果表明：尾焰的存在极大地改变了飞行体蒙皮上感应电流密度的波形及分布;带尾焰飞行体感应皮电流密度不同于无尾焰时的情况;飞行体蒙皮感应电流密度的波形为衰减振铃信号波形,振铃周期等于电流波在整个飞行体和尾焰长度上流动两次的时间。     

多项式拟合铁磁材料的磁滞回线曲线

确定铁磁材料的磁滞回线的绘制属于最小二乘法问题,并确定用分段三次多项式来拟合.给出三次多项式拟合,以及在求解过程中解法方程的方法,并给出了具体算法.     

势问题的无单元Galerkin方法的误差估计

在高维情况下,首先研究了无单元Galerkin方法的形函数构造方法——移动最小二乘法在Sobolev空间W^k,p(Ω)中的误差估计.然后,在势问题的无单元Galerkin方法的基础上,研究了势问题的通过罚函数法施加本质边界条件的无单元Galerkin方法在Sobolev空间中的误差估计.当节点和形函数满足一定条件时,证明了该误差估计是最优阶的.从误差分析中可以看出,数值解的误差与权函数的影响半径密切相关.最后,通过算例验证了结论的正确性. 关键词： 无网格方法 无单元Galerkin方法 势问题 误差估计     

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer--van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.     

Generalized spectral decomposition for stochastic nonlinear problems

We present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multi-wavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low-dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional nonlinear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space.