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

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 Title: Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains Authors: Ei, Shin-Ichiro Browse this author →KAKEN DB Ochiai, Hiroyuki Browse this author Tanaka, Yoshitaro Browse this author Keywords: Method of fundamental solutions Neumann problems of the modified Helmholtz equation Numerical analysis Error analysis Issue Date: 1-Mar-2022 Publisher: Elsevier Journal Title: Journal of Computational and Applied Mathematics Volume: 402 Start Page: 113795 Publisher DOI: 10.1016/j.cam.2021.113795 Abstract: 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, N(2)a(N) 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. (C) 2021 The Author(s). Published by Elsevier B.V. Type: article URI: http://hdl.handle.net/2115/83255 Appears in Collections: 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)