A mollification regularization method with the Dirichlet kernel for two Cauchy problems of three-dimensional Helmholtz equation

2019 ◽  
Vol 97 (11) ◽  
pp. 2320-2336 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Three Dimensional ◽  
Dirichlet Kernel ◽  
Cauchy Problems

A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation

2019 ◽  
Vol 17 (05) ◽  
pp. 1950029 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Numerical Experiment ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Dirichlet Kernel ◽  
Two Dimensional ◽  
Strip Domain ◽  
Ill Posed ◽  
Mollification Method

In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.

scholarly journals Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 705 ◽  
Author(s):  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Three Dimensional ◽  
Wave Number ◽  
Modified Helmholtz Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

scholarly journals Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

scholarly journals A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zhi-Liang Deng ◽  
Xiao-Mei Yang ◽  
Xiao-Li Feng
Keyword(s):  
Regularization Method ◽  
Heat Conduction Problem ◽  
A Priori ◽  
Dirichlet Kernel ◽  
Inverse Heat Conduction Problem ◽  
Inverse Heat Conduction ◽  
Linear Diffusion ◽  
Ill Posed ◽  
Transient Data ◽  
Parameter Choice Rules

The ill-posed problem of attempting to recover the temperature functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple regularization method based on Dirichlet kernel mollification techniques is introduced. We also proposea priorianda posterioriparameter choice rules and get the corresponding error estimate between the exact solution and its regularized approximation. Moreover, a numerical example is provided to verify our theoretical results.

Sign in / Sign up

Export Citation Format

Share Document