scholarly journals An Integral Equations Method for the Cauchy Problem Connected with the Helmholtz Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yao Sun ◽  
Deyue Zhang
Keyword(s):  
Cauchy Problem ◽  
Approximate Solution ◽  
Helmholtz Equation ◽  
Numerical Method ◽  
Integral Equations ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Suitable Choice ◽  
Convergence And Stability ◽  
The Cauchy Problem

We are concerned with the Cauchy problem connected with the Helmholtz equation. We propose a numerical method, which is based on the Helmholtz representation, for obtaining an approximate solution to the problem, and then we analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

scholarly journals An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hao Cheng ◽  
Ping Zhu ◽  
Jie Gao
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Regularization Method ◽  
A Priori ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Regularization Parameters ◽  
The Cauchy Problem

A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. Thea priorianda posteriorirules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method.

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 Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

2020 ◽  
Author(s):  
Pauline Achieng ◽  
Fredrik Berntsson ◽  
Jennifer Chepkorir ◽  
Vladimir Kozlov
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Iterative Algorithm ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
General Elliptic ◽  
The Cauchy Problem

Abstract The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

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