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Keywords = de la Vallée Poussin kernel

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13 pages, 741 KiB  
Article
A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation
by Shangqin He and Xiufang Feng
Mathematics 2019, 7(6), 487; https://doi.org/10.3390/math7060487 - 28 May 2019
Cited by 2 | Viewed by 2870
Abstract
In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. [...] Read more.
In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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13 pages, 335 KiB  
Article
A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation
by Shangqin He and Xiufang Feng
Mathematics 2019, 7(4), 360; https://doi.org/10.3390/math7040360 - 20 Apr 2019
Cited by 10 | Viewed by 3073
Abstract
In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the [...] Read more.
In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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