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 - <p style="text-align: justify;">This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain $0 &lt;x\le 1$, $y\in R$. The Cauchy data at $x=0$ is given and the solution is then sought for the interval $0&lt; x\le 1$. 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.</p>