论文标题
磁盘域中修饰的Helmholtz方程的Neumann问题的基本解决方案方法
Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains
论文作者
论文摘要
基本溶液（MFS）的方法用于构建有界域中部分微分方程的近似解。通过组合转移到域外部点并确定线性总和的系数以满足边界有限点上的边界条件的基本溶液来证明这一点。在本文中，严格证明了MFS在磁盘结构域中修改的Helmholtz方程的Neumann问题的近似解决方案的存在。我们揭示了近似解决方案存在的足够条件。将格林定理应用于修改后的Helmholtz方程的Neumann问题，我们将近似解决方案和精确解之间的误差绑定到边界条件函数的差异以及通过边界积分的近似解决方案的正常导数。使用此错误的估计值，我们显示了MFS通过指数顺序（即$ n^2a^n $ order）的近似解决方案的收敛性，其中$ n^2a^n $订单是$ a $的正常数小于一个小于一个，$ n $是搭配点的数量。此外，在数值仿真中，误差数量趋向于$ 0 $，并倾向于$ 0 $ $ n $。
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 Green's theorem 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, $N^2a^N$ 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 $N$.
