A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two-dimensional modified Helmholtz equation

2011 ◽  
Vol 28 (3) ◽  
pp. 899-925 ◽  
Author(s):  
Liviu Marin
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Modified Helmholtz Equation ◽  
The Cauchy Problem

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 Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 667 ◽  
Author(s):  
Hongwu Zhang ◽  
Xiaoju Zhang
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Regularization Parameter ◽  
A Priori ◽  
Conditional Stability ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Tikhonov Method ◽  
Convergence Results ◽  
The Cauchy Problem

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied.

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