EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

Construction of Wave-free Potentials and Multipoles in a Two-layer Fluid Having Free-surface Boundary Condition with Higher-order Derivatives

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation （or Helmholtz equation） arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.

Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 ＋q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.

RADIATION BOUNDARY CONDITIONS FOR MAXWELL'S EQUATIONS:A REVIEW OF ACCURATE TIME-DOMAIN FORMULATIONS

We review time-domain formulations of radiation boundary conditions for Maxwell＇s equations, focusing on methods which can deliver arbitrary accuracy at acceptable compu-tational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on identifying and evaluating equivalent sources, and local approximations such as the perfectly matched layer and sequences of local boundary conditions. Complexity estimates are derived to assess work and storage requirements as a function of wavelength and simulation time.

Markov Chain Monte Carlo Solution of Laplace’s Equation in Axisymmetric Homogeneous Domain

With increasing complexity of today’s electromagnetic problems, the need and opportunity to reduce domain sizes, memory requirement, computational time and possibility of errors abound for symmetric domains. With several competing computational methods in recent times, methods with little or no iterations are generally preferred as they tend to consume less computer memory resources and time. This paper presents the application of simple and efficient Markov Chain Monte Carlo (MCMC) method to the Laplace’s equation in axisymmetric homogeneous domains. Two cases of axisymmetric homogeneous problems are considered. Simulation results for analytical, finite difference and MCMC solutions are reported. The results obtained from the MCMC method agree with analytical and finite difference solutions. However, the MCMC method has the advantage that its implementation is simple and fast.

Availability of the Boundary Condition for Vorticity

By applying a boundary condition for vorticity [1] in addition to that for velocity, a velocity distribution on a flat plate set in a parallel homogeneous flow has been numerically obtained through a one-way calculation from surface to infinity, without the “matching” procedure between an analysis from surface to infinity and that from infinity to surface. The numerical results obtained were in excellent agreement with those by Howarth [2]. The usage of the boundary condition for vorticity has raised the accuracy of velocity distribution near a plate’s surface and made it possible to realize the one-way calculation from surface to infinity.

The Estimates L_(1)-L_(∞) for the Reduced Radial Equation of Schrodinger

Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.

带Maxwell边界条件的定态迁移方程的临界本征值

在Ｌ１空间中证明了带Ｍａｘｗｅｌ积分型边界条件的定态球介质迁移方程临界本征值的存在性，并且给出了临界本征值的一系列分析性质。

A ROBUST AND ACCURATE SOLVER OF LAPLACE＇S EQUATION WITH GENERAL BOUNDARY CONDITIONS ON GENERAL DOMAINS IN THE PLANE

A robust and general solver for Laplace＇s equation on the interior of a simply connected domain in the plane is described and tested. The solver handles general piecewise smooth domains and Dirichlet, Neumann, and Robin boundary conditions. It is based on an integral equation formulation of the problem. Difficulties due to changes in boundary conditions and corners, cusps, or other examples of non-smoothness of the boundary are handled using a recent technique called recursive compressed inverse preconditioning. The result is a rapid and very accurate solver which is general in scope, its performance is demonstrated via some challenging numerical tests.

A Fourth-Order Upwinding Embedded Boundary Method(UEBM)for Maxwell’s Equations in Media with Material Interfaces:Part I

In this paper,we present a new fourth-order upwinding embedded boundary method(UEBM)over Cartesian grids,originally proposed in the Journal of Computational Physics[190(2003),pp.159-183.]as a second-order method for treating material interfaces for Maxwell’s equations.In addition to the idea of the UEBM to evolve solutions at interfaces,we utilize the ghost fluid method to construct finite difference approximation of spatial derivatives at Cartesian grid points near the material interfaces.As a result,Runge-Kutta type time discretization can be used for the semidiscretized system to yield an overall fourth-order method,in contrast to the original second-order UEBM based on a Lax-Wendroff type difference.The final scheme allows time step sizes independent of the interface locations.Numerical examples are given to demonstrate the fourth-order accuracy as well as the stability of the method.We tested the scheme for several wave problems with various material interface locations,including electromagnetic scattering of a plane wave incident on a planar boundary and a two-dimensional electromagnetic application with an interface parallel to the y-axis.

浅谈安培环路定理的使用

通过应用安培环路定理求解了有限长载流直导线的磁感应强度,说明了“有限长度”的数学条件和物理背景.从麦克斯韦方程的积分形式和微分形式,讨论了安培环路定理的使用场景,阐明了非对称电流密度分布情况下求解稳恒磁场的基本思想——边界条件确定磁场,并拓展至应用高斯定理求解非对称电荷密度分布静电场问题.

Stability and superconvergence analysis of the FDTD scheme for the 2D Maxwell equations in a lossy medium

This paper is concerned with the stability and superconvergence analysis of the famous finite-difference time-domain (FDTD) scheme for the 2D Maxwell equations in a lossy medium with a perfectly electric conducting (PEC) boundary condition, employing the energy method. To this end, we first establish some new energy identities for the 2D Maxwell equations in a lossy medium with a PEC boundary condition. Then by making use of these energy identities, it is proved that the FDTD scheme and its time difference scheme are stable in the discrete L2 and H1 norms when the CFL condition is satisfied. It is shown further that the solution to both the FDTD scheme and its time difference scheme is second-order convergent in both space and time in the discrete L2 and H1 norms under a slightly stricter condition than the CFL condition. This means that the solution to the FDTD scheme is superconvergent. Numerical results are also provided to confirm the theoretical analysis.

Splitting Finite Difference Methods on Staggered Grids for the Three-Dimensional Time-Dependent Maxwell Equations

In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consists of only two stages for each time step.It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions.Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.

基于各向异性界面的深纳米尺度电磁边界条件

以各向异性介质所构成的界面为基础构建了光学参数渐变的过渡界面模型;利用积分形式麦克斯韦方程组,同时引入了6个反映过渡区域电磁场量变化情况的界面响应函数,并经推导获得了适用于深纳米尺度的电磁边界条件.依据麦克斯韦方程组及材料本构关系,阐明了界面响应函数的物理意义;提出了与过渡界面模型等效的突变各向异性界面模型,其界面分布着电偶极矩和磁偶极矩,并经推导获得了界面偶极矩的纳米电磁边界条件.结果表明,各向异性界面模型的纳米电磁边界条件在界面响应函数可忽略的情形下,退化为传统电磁边界条件.本工作可为深纳米尺度各向异性界面的光学或电磁学性质的研究提供理论基础及物理图像.

K-S方程的精确边界控制

运用Fourier基函数的展开以及Fourier变换的方法研究带有周期边界条件的Kuramoto-Sivashinsky方程在有限时间区间[0,T]上的精确控制.首先研究线性化K-S方程的精确控制,运用Reimann-Lebesgue收敛定理以及R iese基函数的性质证明了在给定的时间T>0,对于两个任意给定的函数u0(x),u1(x)属于一定的Sobolev空间,总能找到一个控制函数使得线性化K-S方程有一个存在于某一合适的空间的解u(x,t)使其满足u(x,0)=u0(x),u(x,T)=u1(x).然后结合线性化K-S方程的精确控制,再通过定义Fredholm算子并应用此算子的一些理论可以找到K-S方程的控制函数,使其达到精确控制.

MATLAB PDE工具箱在Stokes第一问题中的应用

Stokes第一问题在工业生产领域是常见的一类问题,运用传统的相似解法解决此类问题比较抽象,难以理解。运用Matlab PDE工具箱,可以形象地描绘Stokes第一问题,可输出所研究问题的具体数值并形象地显示出某一面上的运动状态。Matlab PDE工具箱为研究管道外流体介质的粘性流动提供一种直观、快速、准确、形象的数值求解方法。

基于Hopf-Cole变换的Burgers方程初边值问题的Sinc-Galerkin法

利用Sinc-Galerkin法数值求解Burgers方程的初边值问题。首先,用Hopf-Cole变换将二阶非线性的Burgers方程变换为二阶线性方程,同时把第一类边界条件变为第二类边界条件。时间上的导数采用θ加权格式离散,空间导数采用Sinc-Galerkin法离散,端点处分别引入权函数处理变换后的第二类边界条件。最后,通过数值算例验证了Sinc-Galerkin法的指数收敛性,与精确解相比,本文构造的数值格式精度高,能够有效捕捉激波等物理现象。

推导Boussinesq方程的一种新方法

潜水含水层中的饱和地下水流动可以用两种方式描述。一种是严格的自由潜水面模型,用线性的三维连续性方程作为控制方程,在潜水面处用非线性自由潜水面边界条件。第二种是基于Dupuit假设和非线性Boussinesq方程的垂向平均模型。其中,第一种模型中的潜水自由面边界条件可以用两种方式描述,第一种只含有水头的偏导数,第二种则同时含有水头和潜水面标高的偏导数。前人分别从不同出发点推导了这两种边界条件,但是并未说明他们的等价性。用严格的数学推导证明了这两种边界条件的等价性,进一步完善了前人的工作。另一方面,已有人从第一种边界条件出发,对三维连续性水流方程求积分推导出了Boussinesq方程,但这种方法比较抽象,物理意义不太明确。从第二种边界条件出发直接推导Boussinesq方程,所用方法简单、物理意义明确,易于理解。在自由潜水面边界条件下,Boussinesq方程可以直接通过对三维连续性方程在整个饱和含水层厚度上求积分而得到。此外,潜水含水层的隔水底板可以推广为一个曲面。阐明了两种边界条件、三维连续性水流方程、Dupuit假设、Boussinesq方程之间的内在关系。