Solution of an eigenvalue problem for the Laplace operator on a spherical surface. M.S. Thesis - Maryland Univ.
Solution of an eigenvalue problem for the Laplace operator on a spherical surface. M.S. Thesis - Maryland Univ.
复制标题
球面上拉普拉斯算子特征值问题的求解。
DOI：
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发表时间：
1974
期刊：
影响因子：
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通讯作者：
Acknowle Dgments
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文献类型：
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作者：
H. Walden;Acknowle Dgments
给出了求单位球面上Laplace-Beltrami算子基本特征值(也称为振动方程的膜特征值问题)的近似解的方法。考虑了两种特殊类型的球面区域：(1)球面三角形的内部，即由三个大圆的圆弧包围的区域；(2)在球面上延伸小于pi弧度的大圆弧的外部(带狭缝的球面)。在这两种情况下，都施加了零边界条件。为了求解两个独立变量的二阶椭圆型偏微分方程组，给出了一种有限差分近似解。发展的对称(一般为五点)有限差分方程组被写成矩阵形式，然后用点逐次超松弛迭代法求解。该迭代法收敛后，利用有限瑞利商的幂方法迭代逼近基本特征值。
Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient.