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A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation

A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation
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Year:    2009

Numerical Mathematics: Theory, Methods and Applications, Vol. 2 (2009), Iss. 3 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.2009.m88032

Numerical Mathematics: Theory, Methods and Applications, Vol. 2 (2009), Iss. 3 : pp. 326–340

Published online:    2009-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    15

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

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