Astigmatic Gaussian beam: Exact solution of the Helmholtz equation

2018 ◽  
Author(s):  
Aleksei P. Kiselev ◽  
Alexandr B. Plachenov
Keyword(s):  
Exact Solution ◽  
Helmholtz Equation ◽  
Gaussian Beam

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 The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Fan Yang ◽  
HengZhen Guo ◽  
XiaoXiao Li
Keyword(s):  
Exact Solution ◽  
Helmholtz Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Results ◽  
Tikhonov Regularization Method ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Unknown Source ◽  
Ill Posed

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

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.

Calculation of Radiation Beam Field from Phased Array Ultrasonic Transducers Using Expanded Multi-Gaussian Beam Model

2006 ◽  
Vol 110 ◽  
pp. 163-168 ◽  
Author(s):  
Joon Soo Park ◽  
Sung Jin Song ◽  
Hak Joon Kim
Keyword(s):  
Exact Solution ◽  
Gaussian Beam ◽  
Time Delays ◽  
Phased Array ◽  
Ultrasonic Transducers ◽  
Beam Model ◽  
Radiation Beam ◽  
Sommerfeld Integral ◽  
Calculation Results ◽  
Ultrasonic Phased Array

For the reliable application of ultrasonic phased array technique, it is necessary to understand the characteristics of the radiation beam field from the array transducers quantitatively. Very recently, it has been developed the expanded multi-Gaussian beam model that can calculate the radiation beam field from a single, rectangular transducer with great computational efficiency. In this paper, this model is adopted to calculate the radiation beam field from array transducers with various time delays to achieve steering and/or focusing. The calculation results are compared to those obtained by well known Rayleigh-Sommerfeld integral that provides the exact solution in order to explore the validity of the expanded multi-Gaussian beam model in this task.

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