TY - JOUR

T1 - Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains

AU - Ei, Shin Ichiro

AU - Ochiai, Hiroyuki

AU - Tanaka, Yoshitaro

N1 - Funding Information:
The authors would like to thank Professor Mitsuhiro Nakao of Kyusyu University for the fruitful suggestions for the error estimation in Section 4 . This work was supported in part by JST CREST, Japan Grant Number JPMJCR14D3 to S.-I. E., JSPS, Japan KAKENHI Grant Number 15H03613 to H. O., and JSPS, Japan KAKENHI Grant Numbers 17K14228 and 20K14364 to Y. T.
Publisher Copyright:
© 2021 The Author(s)

PY - 2022/3/1

Y1 - 2022/3/1

N2 - The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, N2aN order, where a is a positive constant less than one and N is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points N.

AB - The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, N2aN order, where a is a positive constant less than one and N is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points N.

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U2 - 10.1016/j.cam.2021.113795

DO - 10.1016/j.cam.2021.113795

M3 - Article

AN - SCOPUS:85115269790

VL - 402

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 113795

ER -