Next Article in Journal
Impact of Fractional Calculus on Correlation Coefficient between Available Potassium and Spectrum Data in Ground Hyperspectral and Landsat 8 Image
Next Article in Special Issue
A Numerical Method for Filtering the Noise in the Heat Conduction Problem
Previous Article in Journal
The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative
Previous Article in Special Issue
A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems
Open AccessArticle

A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation

by Shangqin He 1,2 and Xiufang Feng 1,*
1
School of Mathematics and Statistics, NingXia University, Yinchuan 750021, China
2
College of Mathematics and Information Science and Technology, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 487; https://doi.org/10.3390/math7060487
Received: 20 April 2019 / Revised: 20 May 2019 / Accepted: 23 May 2019 / Published: 28 May 2019
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems - Theory and Applications)
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. View Full-Text
Keywords: three-dimensional Laplace equation; ill-posed; de la Vallée Poussin kernel; mollification method; regular parameter; error estimate three-dimensional Laplace equation; ill-posed; de la Vallée Poussin kernel; mollification method; regular parameter; error estimate
Show Figures

Figure 1

MDPI and ACS Style

He, S.; Feng, X. A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation. Mathematics 2019, 7, 487.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop