Poisson方程热源识别反问题

探讨了半带状区域上二维Poisson方程只含有一个空间变量的热源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Carasso-Tikhonov正则化方法,得到了问题的一个正则近似解,并且给出了正则解和精确解之间具有Holder型误差估计.数值实验表明Carasso-Tikhonov正则化方法对于这种热源识别是非常有效的.     

求解热源识别问题的修正吉洪诺夫正则化方法

该文研究了一个热源识别问题,通过引入修正吉洪诺夫方法来处理问题的不适定性,在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步验证了方法的有效性和稳定性.     

热传导(对流-扩散)方程源项识别的粒子群优化算法

提出了利用粒子群优化(PSO)算法反演热传导方程与对流-扩散方程源项的一种新方法,在已有文献方法的基础上,求解出这两类方程正问题的解析解,再把源项识别问题转化为最优化问题,结合粒子群优化算法寻优求解.通过数值模拟与统计检验,结果表明,此方法可快速有效地实现热传导方程与对流-扩散方程源项的识别,并可推广应用到其它数学物理方程的源项或参数的反演识别.     

Moving mesh method for a reaction-diffusion equation with traveling heat source on unbounded domain

对无界区域上带移动热源的反应扩散方程提出了局部吸收边界条件.移动网格方法对导出的有界区域问题进行了求解.数值例子显示了当热源移动的速度比较慢的时候,方程会在有限时间内发生爆破现象.而当热源移动的速度足够快时,爆破现象不会发生.数值例子验证了新方法的有效性和精确性.     

修正的Helmholtz方程未知源识别的Fourier截断正则化方法

探讨半无界区域上二维修正的Helmholtz方程只含有一个空间变量的未知源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Fourier截断正则化方法,得到问题的一个正则近似解,并且给出正则解和精确解之间收敛的误差估计.数值例子表明Fourier截断正则化方法对于这种未知源识别非常有效.     

磁-微极广义热弹性介质中轴对称变形的弹性动力学

在横向磁场中,表面受机械源或热源作用时,研究电磁.微极热弹性半空间中的轴对称问题.问题的求解用到了Laplace和Hankel变换技术.作为该方法的一个应用,采用了集中源/沿圆周分布作用源(机械源和热源).对积分变换的逆变换使用数值技术,得到物理域中的应力分量和温度分布,以及感应电场和感应电磁场.对于两种不同的广义热弹性理论(Lord-shulman(L-S)理论和Green-Lindsay(G-L)理论),给出了这些物理量的表达式,并用插图显示磁场的影响.还导出了一个感兴趣的特例.     

基于分段常值水平集的抛物型方程的参数识别算法

研究了线性抛物型方程不连续参数的识别算法.根据原有算法对于加噪观测数据计算不收敛的问题,本文基于分段常值水平集方法,根据水平集函数和优化过程的特点,修正原有Uzawa型算法中的带有总变差(TV)正则化的极小化模型和对常值向量的极小化模型,并且利用分裂Bregman迭代算法处理TV范数的优越性,构造一种新的参数识别算法格式.数值实验结果显示,新算法具有计算时间短、精度高、抗噪性强的优点.     

高维Dirchlet问题数值解的概率方法

本文用概率方法求得高维Dirichlet内问题和外问题在一般区域上的数值解.高维漂移布朗族对停时具有强马氏性,它在球面上的击中时和位置分布已知,再利用Dirichlet问题解的随机表达式,我们可以获得高维Dirichlet问题的数值解.     

周期Stokes方程源的识别问题(英文)

本文研究了周期Stokes方程中源的识别问题.利用能观性不等式,得到了识别问题解的存在性,同时给出了该问题的逼近过程.     

基于积极集技术求解无约束极大极小问题的摄动SQP方法

讨论无约束极大极小(minimax)问题,基于积极集识别技术,结合摄动的序列二次规划(SQP)方法,建立问题的一个数值方法.在相当弱的条件下,算法具有弱全局收敛性,并对算法进行了初步的数值试验.     

ITERATIVE METHODS FOR THE FORWARD-BACKWARD HEAT EQUATION

In this paper we propose the finite difference method for the forward-backward heatequation.We use a coarse-mesh second-order central difference scheme at the middleline mesh points and derive the error estimate.Then we discuss the iterative methodbased on the domain decomposition for our scheme and derive the bounds for the rates ofconvergence.Finally we present some numerical experiments to support our analysis.     

A NEW PSEUDOSPECTRAL APPROXIMATION FOR THE FOWARD-BACKWARD HEAT EQUATION

ＡＮＥＷＰＳＥＵＤＯＳＰＥＣＴＲＡＬＡＰＰＲＯＸＩＭＡＴＩＯＮＦＯＲＴＨＥＦＯＷＡＲＤ－ＢＡＣＫＷＡＲＤＨＥＡＴＥＱＵＡＴＩＯＮ￥ＹｅＸｉｎｇｄｅ（叶兴德）ＪｉａｎｇＪｉｎｓｈｅｎｇ（江金生）（Ｄｅｐｔ．ｏｆＭａｔｈ．＆Ｉｎｆｏｒ．Ｓｃｉ．，Ｈａｎｇ...     

APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL HEAT EQUATION

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.     

Moving collocation method for a reaction‐diffusion equation with a traveling heat source

We consider a reaction‐diffusion equation with a traveling heat source on an unbounded domain. The numerical simulation of the problem is difficult because of the moving singularity, the blow‐up phenomenon, and the delta function in the equation. Because we are only interested in the solution behavior near the heat source, we choose a bounded moving domain which contains the heat source and has the same speed as the source. Local absorbing boundary conditions are constructed on the boundaries of the moving domain. Then, we transform the moving domain to a fixed one. At last, a special moving collocation method is adopted. The new method is much simpler than the existing moving finite difference methods. Moreover, numerical experiments illustrate the accuracy and efficiency of our moving collocation method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The Laplace transform and polynomial Trefftz method for a class of time dependent PDEs

In this paper, we present a new method for solving 1D time dependent partial differential equations based on the Laplace transform (LT). As a result, the problem is converted into a stationary boundary value problem (BVP) which depends on the parameter of LT. The resulting BVP is solved by the polynomial Trefftz method (PTM), which can be regarded as a meshless method. In PTM, the source term is approximated by a truncated series of Chebyshev polynomials and the particular solution is obtained from a recursive procedure. Talbot’s method is employed for the numerical inversion of LT. The method is tested with the help of some numerical examples.     

A numerical investigation of blow-up in reaction-diffusion problems with traveling heat sources

This paper studies the numerical solution of a reaction-diffusion differential equation with traveling heat sources. According to the fact that the locations of heat sources are known, we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem. Several examples are provided to demonstrate the efficiency of the new moving mesh method, especially in the case of two or three traveling heat sources. Moreover, numerical results illustrate that the speed of the movement of the heat source is critical for blow-up when there is only one traveling heat source. For the case of two traveling heat sources, blow-up depends not only on the speed but also on the distance between the two traveling heat sources.     

Distribution algorithm in finite difference method and its application to a 2D simulation of temperature inversion

In this article we present a numerical algorithm in a finite difference method coupled with the Schwartz distribution on a surface and its application to 2D simulation of a temperature inversion in meteorology. This phenomenon has been observed in Kyoto in Japan frequently and one of the reasons why the inversion occurs is the radiative cooling on the ground. In order to describe it in a mathematical model, we introduce transmission conditions on the ground and show its distribution formula. The characteristics of our numerical method are based on the direct method and the finite difference approximation of the Schwartz distribution defined on the ground. Through our discussion, a numerical experiment for the inversion of the temperature gradient is presented.     

修正局部Crank-Nicolson法对于二维热传导方程的应用

1．关于一线热传导方程解法的描述考虑一维空间的热传导方程的数值解法，这里Q＝（0，1）．对于（1．1）的微分项用中心差商代替，就得到半差分方程式。（；，O的近似解，并且A是三对角矩阵古典显格式、隐格式和Crank－Nicolson法可以表示为从这些近似式可以看出，为能得到未来的近似解法，应该考虑exP（兴A）的近似式．只要推—“————一’—’—”“—”—－’””’”“‘”—“”’～’””“——‘””’””一’———”“”－－””h‘”一’—‘—”—”””—”一出exP（兴川的新近似式，就能得到新的近似解法．因此以下给…     

Boundary Identification for a Blast Furnace

In this paper,the authors discuss an inverse boundary problem for the axi- symmetric steady-state heat equation,which arises in monitoring the boundary corrosion for the blast-furnace.Measure temperature at some locations are used to identify the shape of the corrosion boundary. The numerical inversion is complicated and consuming since the wear-line varies during the process and the boundary in the heat problem is not fixed.The authors suggest a method that the unknown boundary can be represented by a given curve plus a small perturbation,then the equation can be solved with fixed boundary,and a lot of computing time will be saved. A method is given to solve the inverse problem by minimizing the sum of the squared residual at the measuring locations,in which the direct problems are solved by axi- symmetric fundamental solution method. The numerical results are in good agreement with test model data as well as industrial data,even in severe corrosion case.     

THE NUMERICAL SOLUTION FOR A PARTIAL INTEGRO-DIFFERENTIAL EQUATION WITH A WEAKLY SINGULAR KERNEL

In this paper, a first order semi-discrete method of a partial integro-differential equation with a weakly singular kernel is considered. We apply Galerkin spectral method in one direction, and the inversion technique for the Laplace transform in another direction, the result of the numerical experiment proves the accuracy of this method.