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

R. Shi, T. Wei & H. H. Qin

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.

  • AMS Subject Headings

65M32

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-2-326, author = {R. Shi, T. Wei and H. H. Qin}, 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 AU - R. Shi, T. Wei & H. H. Qin 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 and H. H. Qin. (2009). 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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