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Publications (8)
We give a general proof on convergence estimates for some regularization methods to solve a Cauchy problem for the Laplace equation in a rectangular domain. The regularization methods we considered are: a non-local boundary value problem method, a boundary Tikhonov regularization method and a generalized method. Based on the conditional stability e...
In this paper, a Cauchy problem for the Helmholtz equation is considered. It is known that such a problem is severely ill-posed, i.e. the solution does not depend continuously on the given Cauchy data. We propose a quasi-reversibility regularization method to solve it. Convergence estimates are established under two different a priori assumptions f...
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.
This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0 < x <= 1, y is an element of 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 da...
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...
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...
In this paper, we propose a numerical method for solving the Cauchy problem of the modified Helmholtz equation. By using Green's formula, the Cauchy problem is transformed to a moment problem. Then we propose a Tikhonov type regularization algorithm for obtaining an approximate solution to the Neumann date on the unspecified boundary. Convergence a...
35J05 Laplace equation, reduced wave equation (Helmholtz), Poisson equation (See also 31Axx, 31Bxx)
35R30 Inverse problems (undetermined coefficients, etc.) for PDE
65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx)
65Gxx Error analysis and interval analysis