数列在解特殊方程与不等式中的灵活运用

中学课本中方程和不等式的解法多是常规方法,用其来解一些结构比较特殊的方程与不等式就难以奏效,或过于繁锁.本文就一些方程及不等式采用化归、转化的思想,充分利用数列有关性质来解来,供读者参考.……     

一个广义薄膜方程弱解的存在性

讨论一个广义薄膜方程,在一些初值的假定下,用时间离散化方法证明其弱解的存在性.     

一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

受迫耗散Boussinesq方程的可积性质和孤波解的探讨

考虑到耗散效应和地形外力,Rossby波的振幅可由受迫耗散Boussinesq方程来描述.当包含这两项时,模型比较复杂,不具有Painleve性质.通过将模型双线性化,双线性方法是一个可寻找孤波解和B(a|¨)cklund变换的方法.通过截断的Painleve展开式,得到了将方程双线性化的合适的因变量变换.然后得到了受迫耗散Boussinesq方程的单孤波解和B(a|¨)cklund变换.     

一类组合sinh-cosh-Gordon方程的对称约化、动力学性质与精确解（英文）

文章综合运用李对称分析、幂级数解法和动力系统法来求解组合sinh-cosh-Cordon方程的精确解.利用李对称分析得到了组合sinh-cosh-Cordon方程的向量场和相似变换,把难以求解的偏微分问题约化为常微分方程,利用幂级数解法求得了方程的精确解析解.然后用MATLAB画出了约化后方程的相图,最后利用动力系统法分析研究了解的动力学行为,并得到了方程的行波解.     

由Ricci流启发的二次微分方程

利用Uhlenbeck的一个技巧,Ricci流的曲率算子满足一个用正交Lie代数定义的漂亮的演化方程.其实这个方程也可以用任何一个Lie代数来定义.这份简要的综述里讨论了相应的二次微分方程的一些性质.     

旋转流体中圆球沿旋转轴运动产生的扰动

本文用线化理论分析了整体旋转的理想流体中有一个圆球沿旋转轴作匀速运动时流体的扰动,基于旋轴对称流动的假设导出了决定运动稳定性的扰动压力方程和扰动流函数方程.用简正模法分析了扰动流函数方程,得出了非平凡中性扰动的波数与波速必须满足的约束条件,并求出了扰动的精确表达式.文中得出结论,中性扰动共有三种可能的形式.     

不可压双曲型液晶的Ericksen-Leslie方程的大Ericksen数的极限的形式分析

该文考虑了参数化的液晶的不可压双曲型的Ericksen-Leslie方程.形式上,让参数消失该文证明了这个极限方程存在一个局部的经典解.更进一步,该文形式上得出一个关于这个参数化的液晶的双曲型方程和极限方程的解的误差估计,这对应的是关于它们的经典解在L2空间中的一个形式上的能量估计.     

非线性演化方程的切对称群分析

将优化系统的概念推广应用至切代数,并以一个二阶非线性演化方程为例,给出了方程所容许的切对称,建立了切对称的一维优化系统.并利用优化系统对所研究的方程进行了对称约化,得到了与不等价对称相对应的约化方程和不变解.     

地形效应下正压准地转模式的能量守恒

基于正压大气模式,研究了地形对大气能量的影响.首先,利用尺度分析方法,将正压模式方程组无量纲化.然后,选定一个合适的小参数,再用小参数展开,依次得到无量纲的正压模式方程组的零级和一级近似方程.获得了地形效应下的正压准地转位涡方程.最后通过新的位涡方程推导了能量守恒的结论.并进一步从能量的角度分析和解释了地形对西风的影响.     

The construction of exact solutions in the floating-plate problem

It is shown that it is possible to construct, by an inverse method, exact solutions of the problem of the flexural-gravitational oscillations of a floating elastic plate. The results obtained are used to check the accuracy of numerical solutions of the problem. It is shown that the numerical algorithm given in Ref. [Khabakhpasheva TI. The plane problem of an elastic floating plate. In Continuum Dynamics. Inst. Gidrodinamiki SO Ross Akad Nauk 2000;16:166–9.], predicts, with high accuracy, the values of the amplitudes of the oscillations of the plate and the distributions of the bending moments and hydrodynamic pressure for a wide frequency range.     

Analysis and simulations of a nonlinear elastic dynamic beam

A model for the dynamics of a Gao elastic nonlinear beam, which is subject to a horizontal traction at one end, is studied. In particular, the buckling behavior of the beam is investigated. Existence and uniqueness of the local weak solution is established using truncation, approximations, a priori estimates, and results for evolution problems. An explicit finite differences numerical algorithm for the problem is presented. Results of representative simulations are depicted in the cases when the oscillations are about a buckled state, and when the horizontal traction oscillates between compression and tension. The numerical results exhibit a buckling behavior with a complicated dependence on the amplitude and frequency of oscillating horizontal tractions.     

The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term

The inverse problem of determining an unknown source term depending on space variable in a parabolic equation is considered. A numerical algorithm is presented for recovering the unknown function and obtaining a solution of the problem. As this inverse problem is ill‐posed, Tikhonov regularization is used for finding a stable solution. For solving the direct problem, a Galerkin method with the Sinc basis functions in both the space and time domains is presented. This approximate solution displays an exponential convergence rate and is valid on the infinite time interval. Finally, some examples are presented to illustrate the ability and efficiency of this numerical method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

An Inverse Problem in Elastostatics

This paper assembles a variety of methods which have been devisedfor acoustic and elastic wave propagation inverse problems andadapts them to the problem of determining the shear modulusprofile of an elastic half-space from a knowledge of the torsionaldeflection and shear stress distributions on the surface. Methodsinclude the reduction to a Gel'fand-Levitan integral equation,for which a fast numerical algorithm is presented; a methodbased on modelling the half-space as a layered medium; identificationof the medium as a member of a family for which a closed formsolution is possible; a Green's function approach for a mediumwith small variations from uniformity.     

A numerical technique for solving IHCPs using Tikhonov regularization method

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.     

逆热传导方程的一个正则化方法

考虑了标准的一维逆热传导方程.问题是不适定的,即解不连续地依赖于数据.通过Fourier逼近的方法进行正则化处理,提出了一个新的算法,理论分析和数值实验均表明该算法是稳定的;该算法不仅保留了测量数据的部分高频成份,同时还具有相同的精度和计算复杂性.     

Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem – an inverse problem in fractal construction – is proposed. Dedicated to Walter Gautschi.     

A Stochastic Algorithm for Fault Inverse Problems in Elastic Half Space with Proof of Convergence

A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by $m$ and the regularization parameter for the linear part of the inverse problem denoted by $C$,both modeled as random variables, we derive a formula for the posterior marginal of $m.$ Modeling $C$ as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value [11]. We prove that this posterior marginal of $m$ is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Simply put, our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finite-dimensional spaces for discretizing slips is dense. Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded. We also explain how our proof can be extended to a whole class of inverse problems, as long as some basic requirements are met. Finally, we show numerical simulations that illustrate the numerical convergence of our algorithm.     

On inverse crack problems in elastostatics

In solid mechanics, nondestructive testing has been an important technique in gathering information about unknown cracks, or defects in material. From a mathematical point of view, this is described as an inverse problem of partial differential equations, that is, the problem is to extract information about the location and shape of an unknown crack from the surface displacement field and traction on the boundary of the elastic material. By using the enclosure method introduced by Prof. Ikehata we can derive the extraction formula of an unknown linear crack from a single set of measured boundary data. Then, we need to have precise properties of a solution of the corresponding boundary value problem; for instance, an expansion formula around the crack tip. In this paper we consider the inverse problem concentrating on this point. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An inverse problem for the acoustics equations: A multilevel adaptive algorithm

A numerical solution to an inverse problem for the acoustic equations using an optimization method for a stratified medium is presented. With the distribution of an acoustic wave field on the medium’s surface, the 1D distributions of medium’s density, as well as the velocity and absorption coefficient of the acoustic wave, are determined. Absorption in a Voigt body model is considered. The conjugate gradients and the Newton method are used for minimization. To increase the efficiency of the numerical method, a multilevel adaptive algorithm is proposed. The algorithm is based on a division of the whole procedure of solving the inverse problem into a series of consecutive levels. Each level is characterized by the number of parameters to be determined at the level. In moving from one level to another, the number of parameters changes adaptively according to the functional minimized and the convergence rate. The minimization parameters are chosen as illustrated by results of solving the inverse problem in a spectral domain, where the desired quantities are presented as Chebyshev polynomial series and minimization is carried out with respect to the coefficients of these series. The method is compared in efficiency with a nonadaptive method. The optimal parameters of the multilevel method are chosen. It is shown that the multilevel algorithm offers several advantages over the one without partitioning into levels. The algorithm produces primarily a more accurate solution to the inverse problem.