A regularization method for a Cauchy problem of the Helmholtz equation

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.     

Modified regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a modified regularization method to solve it based on the solution given by the method of separation of variables. Convergence estimates are presented under two different a-priori bounded assumptions for the exact solution. Finally, numerical examples are given to show the effectiveness of the proposed numerical method.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.     

A Herglotz wavefunction method for solving the inverse Cauchy problem connected with the Helmholtz equation

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.     

A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation

In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-type equation with homogeneous Dirichlet and Neumann boundary condition. The proposed problem is ill-posed. Up to now, most investigations on this topic focus on very specific cases, and with Dirichlet boundary condition. Recently, we solve this problem in 2D for an inhomogeneous modified Helmholtz equation (2012). This work is a continuous expansion of our previous results. Herein we introduce a general filter regularization (GFR) method, and then from the GFR we deduce two concrete filters, which are a foundation to implement a numerical procedure. In addition, we develop a numerical model for solving this problem in three dimensional region. The proposed filter method has been verified by numerical experiments.     

A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation

In this paper, we study a fractional Tikhonov regularization method (FTRM) for solving a Cauchy problem of Helmholtz equation in the frequency domain. On the one hand, the FTRM retains the advantage of classical Tikhonov method. On the other hand, our method can prevent the effect of oversmoothing of classical Tikhonov method and conveniently control the amount of damping. The convergence error estimates between the exact solution and its regularization approximation are constructed. Several interesting numerical examples are provided, which validate the effectiveness of the proposed method.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.     

The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data

In the present paper, the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data in an infinite “strip” domain is considered. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A conditional stability result is given. A new a posteriori Fourier method for solving this problem is proposed. The corresponding error estimate between the exact solution and its regularization approximate solution is also proved. Numerical examples show the effectiveness of the method and the comparison of numerical effect between the a posteriori and the a priori Fourier method are also taken into account.     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

Two regularization methods for the Cauchy problems of the Helmholtz equation

In this paper, the Cauchy problems for the Helmholtz equation are investigated. We propose two regularization methods to solve them. Convergence estimates are presented under an a-priori bounded assumption for the exact solution. Finally, the numerical examples show that the proposed numerical methods work effectively.     

Two numerical methods for a Cauchy problem for modified Helmholtz equation

We consider a Cauchy problem for the modified Helmholtz equation. A conditional stability estimate is proved. Two order optimal regularization methods and error estimates are investigated. Numerical experiment shows that these methods work well.     

A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].     

A regularization method for the cauchy problem of the modified Helmholtz equation

In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The a priori and a posteriori rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.     

A regularization method for solving the Cauchy problem for the Helmholtz equation

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.     

求解椭圆方程柯西问题的修正吉洪诺夫正则化方法

研究了一类变系数椭圆方程的柯西问题,这类问题出现在很多实际问题领域.由于问题的不适定性,不可能通过经典的数值方法来求解上述问题,必须引入正则化手段.采用了一种修正吉洪诺夫正则化方法来求解上述问题.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.数值例子进一步显示方法是稳定有效的.     

Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation

In this paper, a Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.     

A regularization method for solving the radially symmetric backward heat conduction problem

This work is devoted to solving the radially symmetric backward heat conduction problem, starting from the final temperature distribution. The problem is ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified Tikhonov regularization method is proposed for solving this inverse problem. A quite sharp estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter. A numerical example is presented to verify the efficiency and accuracy of the method.