scholarly journals A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Fixed Frequency ◽  
Wavelet Method ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Ill Posed

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

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 HP estimation for the Cauchy problem for nonlinear elliptic equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 193-202
Author(s):  
Thang Duc Le
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Bounded Domain ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain. As we know, the problem is severely ill-posed. We apply the Fourier truncation method to regularize the problem. Error estimates between the regularized solution and the exact solution are established in Hp space under some priori assumptions on the exact solution.

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.

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 Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines

2021 ◽  
Author(s):  
Mary Nanfuka ◽  
Fredrik Berntsson ◽  
John Mango
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Smoothing Splines ◽  
Cauchy Data ◽  
Rectangular Domain ◽  
Stability Estimates ◽  
Second Derivative ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Approximate Derivative

AbstractWe consider the Cauchy problem for the Helmholtz equation defined in a rectangular domain. The Cauchy data are prescribed on a part of the boundary and the aim is to find the solution in the entire domain. The problem occurs in applications related to acoustics and is illposed in the sense of Hadamard. In our work we consider regularizing the problem by introducing a bounded approximation of the second derivative by using Cubic smoothing splines. We derive a bound for the approximate derivative and show how to obtain stability estimates for the method. Numerical tests show that the method works well and can produce accurate results. We also demonstrate that the method can be extended to more complicated domains.

On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation

2016 ◽  
Vol 24 (5) ◽  
Author(s):  
Phan Trung Hieu ◽  
Pham Hoang Quan
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
The Cauchy Problem

AbstractIn this paper, we consider the modified inhomogeneous Helmholtz equation Δ

scholarly journals Regularization of a Cauchy problem for the heat equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 184-192
Author(s):  
Au Van Vo ◽  
Tuan Hoang Nguyen
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Heat Equation ◽  
Error Estimates ◽  
A Priori ◽  
Cauchy Data ◽  
Linear Source ◽  
Truncation Method ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we study a Cauchy problem for the heat equation with linear source in the form ut(x,t)= uxx(x,t)+f(x,t), u(L,t)=  φ(t), u(L,t)= Ψ (t), (x,t) ∈ (0,L) ×(0, 2π). This problem is ill-posed in the sense of Hadamard. To regularize the problem, the truncation method is proposed to solve the problem in the presence of noisy Cauchy data φε and Ψε satisfying ‖ φε - φ ‖+‖ Ψε - Ψ ‖ ≤ ε and that fε satisfying ‖ fε(x,. ) - f(x,.) ‖ ≤ ε .  We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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