Volume 2, Issue 3
A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation

Numer. Math. Theor. Meth. Appl., 2 (2009), pp. 326-340.

Published online: 2009-02

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• Abstract

This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain $0 <x\le 1$, $y\in R$. The Cauchy data at $x=0$ is given and the solution is then sought for the interval $0< x\le 1$. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution. Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.

• Keywords

Cauchy problem for the modified Helmholtz equation, ill-posed problem, fourth-order modified method.

65M32

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@Article{NMTMA-2-326, author = {}, title = {A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2009}, volume = {2}, number = {3}, pages = {326--340}, abstract = {

This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain $0 <x\le 1$, $y\in R$. The Cauchy data at $x=0$ is given and the solution is then sought for the interval $0< x\le 1$. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution. Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2009.m88032}, url = {http://global-sci.org/intro/article_detail/nmtma/6027.html} }
TY - JOUR T1 - A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 326 EP - 340 PY - 2009 DA - 2009/02 SN - 2 DO - http://doi.org/10.4208/nmtma.2009.m88032 UR - https://global-sci.org/intro/article_detail/nmtma/6027.html KW - Cauchy problem for the modified Helmholtz equation, ill-posed problem, fourth-order modified method. AB -

This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain $0 <x\le 1$, $y\in R$. The Cauchy data at $x=0$ is given and the solution is then sought for the interval $0< x\le 1$. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution. Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.

R. Shi, T. Wei & H. H. Qin. (2020). A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation. Numerical Mathematics: Theory, Methods and Applications. 2 (3). 326-340. doi:10.4208/nmtma.2009.m88032
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