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1.
孙萍  冯晓莉 《数学杂志》2011,31(4):756-762
本文研究了无界带形区域Ω={(x,y)|0相似文献   

2.
Laplace方程Cauchy问题u_(xx)(x,y)+u_(yy)(x,y)=0,0x1,y∈R,u(0,y)=g(y),y∈R,u_x(0,y)=0,y∈R是一个严重不适定问题,数据g的微小变化可以引起解的巨大误差,本文通过构造一个在频域具紧支集的小波并在尺度空问上展开数据和解,滤除高频部分,并结合Galerkin方法,建立了一种逼近准确解的正则化方法,恢复解对初值的连续依赖性,建立误差估计.  相似文献   

3.
本文主要研究Laplace方程的Cauchy问题,该问题在很多领域有广泛的应用.众所周知,Laplace方程的Cauchy问题是严重不适定问题,即其解不连续依赖于所给的Cauchy数据.本文应用一个高阶Tikhonov正则化方法求解矩形区域上的Laplace方程的Cauchy问题,在对精确解的适当的先验界假设和正则化参数选取下,得到了相应的收敛性估计,数值结果表明所提的方法是高效稳定的.  相似文献   

4.
凌岭 《数学学报》1980,23(4):624-631
<正> 对于 Laplace 双曲型方程Lz≡z_(xy)+a(x,y)z_x+b(x,y)z_y+c(x,y)z=0 (1)考虑一曲线弧 L,假设 L 被任何与 x 轴相平行或与 y 轴相平行的直线仅交于一点,而在L 上给定 Cauchy 数据,则方程(1)的解就唯一确定.若 L 由两根单调弧组成,而在 L 上给定 Cauchy 数据,Picard 指出,由于数据过多,Cauchy 问题一般是不可能的.J.Ha-  相似文献   

5.
研究了带阻尼的三维Navier-Stokes方程Cauchy问题解的正则性,得到了在Besov空间中关于速度场的对数改进的正则性准则.  相似文献   

6.
孙瑶  陈博 《计算数学》2018,40(3):254-270
 本文处理二维和三维Helmholtz方程的边界数据复原问题.通过利用位势理论近似问题的解,导出了解决Cauchy问题的一种非迭代积分方程方法.为了处理形成问题的不适定性,采用了Tikhonov正则化结合Morozov偏差原理的方法,并且给出了算法的收敛性和误差估计,最后给出了二维和三维的数值算例.通过数值算例我们检验了源点和边界之间距离的关系,算法关于噪声、源点数目的数值收敛性,稳定性.  相似文献   

7.
本文处理二维和三维Helmholtz方程的边界数据复原问题.通过利用位势理论近似问题的解,导出了解决Cauchy问题的一种非迭代积分方程方法.为了处理形成问题的不适定性,采用了Tikhonov正则化结合Morozov偏差原理的方法,并且给出了算法的收敛性和误差估计,最后给出了二维和三维的数值算例·通过数值算例我们检验了源点和边界之间距离的关系,算法关于噪声、源点数目的数值收敛性,稳定性.  相似文献   

8.
在本文中, 首先给出了超空间中次正则函数(sandwich方程 DxfDx=0的解)的一些性质, 然后证明了超空间中的Cauchy-Pompeiu公式, 最后得到了超空间中的Cauchy积分公式和Cauchy积分定理.  相似文献   

9.
主要考虑以有限Borel测度为初值的非线性双曲方程的Cauchy问题ut (um)x=up,其中0<m<1,0<p≤m是给定常数.特别地,得到了上述问题BV解的存在性.  相似文献   

10.
在П(L0)n R≠θ的条件下,本文讨论了具有中间亏指数的对称微分算式l(y)的自共轭域,其中П(L0)是由l(y)生成的最小算子L0的正则型域.使用方程l(y)=λ0y,(λ0∈П(L0)∩R)的实参数L2-解,我们对最大算子域DM进行新的分解,由此得到l(y)的自共轭域新的完全解析刻画,其中自共轭边界条件中矩阵M,N的确定与l(y)=λ0y在无穷远点的性质无关,仅与其在t=0点初始值的选择有关.由于自共轭箅子谱是实的,使用实参数λ0不仅有利于我们找到方程的显解,更重要的是可以得到谱的有关信息.  相似文献   

11.
This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0 < x≤ 1, y ∈ R. The Cauchy data at x = 0 is given and the solution is then sought for the interval 0 < x ≤1. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution. Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.  相似文献   

12.
We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.  相似文献   

13.
This Note is concerned with the severely ill-posed Cauchy–Helmholtz problem. This Cauchy problem being rephrased through an “interfacial” equation, we resort to an Aitken–Schwarz method for solving this equation. Numerical trials highlight the efficiency of the present method.  相似文献   

14.
Hadamard‐type instability has been known for over a century as a cause of ill‐posedness of the Cauchy problem for elliptic PDEs. This ill‐posedness manifests itself as evanescent modes growing exponentially when propagated in the reverse direction. Since every oscillating mode of the Laplace equation is evanescent, the ill‐posedness of its Cauchy problem is solely due to Hadamard‐type instability. The presence of the propagating modes and beams for the Helmholtz equation gives rise to an entirely different type of ill‐posedness, hitherto unknown to the practice, and untreated by the theory, of inverse scattering. We will present this fundamental phenomenon of ill‐posedness for the Helmholtz equation. © 2007 Wiley Periodicals, Inc.  相似文献   

15.
In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.  相似文献   

16.
Numerical Algorithms - We investigate the Cauchy problem associated with the Helmholtz equation in three dimensions, namely the numerical reconstruction of the primary field (Dirichlet data) and...  相似文献   

17.
In this paper, we consider a Cauchy problem for the Helmholtz equation in a rectangle. An optimal filtering method is presented for approximating the solution of this problem, and the Hölder type error estimate is obtained. Numerical illustration shows that the method works effectively.  相似文献   

18.
We investigate a Cauchy problem for the Helmholtz equation. A modified boundary method is used for solving this ill-posed problem. Some Hölder-type error estimates are obtained. The numerical experiment shows that the modified boundary method works well.  相似文献   

19.
This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method.  相似文献   

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