拉普拉斯潮汐波方程的边值问题

M. Homer
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引用次数: 10

摘要

本文讨论了拉普拉斯潮汐方程(LTWE)的特征值问题,对于μ λ(- 1,1),由1−μ2μ2−τ2y ' (μ) ' +1μ2−τ2sτμ2+τ2μ2−τ2+s21+μ2y(μ)=λy(μ),(LTWE),其中s和τ为参数,s为整数且0 < τ < 1, λ决定特征值。这个常微分方程是从一个偏微分方程的线性系统推导出来的,该系统作为一个数学模型,用于在一个巨大的旋转重力球上的薄层流体的波动。对于Sturm-Liouville方程的解析和数值研究来说,该微分方程在±τ点的内部奇异性和(1 - μ2)/(μ2 - τ2)在区间(- 1,1)上的符号变化方面提出了重要的问题。Sturm-Liouville特征值问题(LTWE)的适定边值问题,该问题具有奇异端点±1,并且在±τ处具有内部奇异性。在适当的Hilbert函数空间中构造了自伴随微分算子来表示由LTWE导出的三个适定边值问题,并证明了这三个算子是酉等价的。讨论了共谱的定性性质,研究了相关微分算子域上函数的有限能量性质。这项工作延续了早期工作者对LTWE的研究,特别是霍夫、兰姆、朗盖-希金斯和林德森。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Boundary value problems for the Laplace tidal wave equation
This paper discusses the eigenvalue problem associated with the Laplace tidal wave equation (LTWE) given, for μϵ (—1,1), by 1−μ2μ2−τ2y′(μ)′+1μ2−τ2sτμ2+τ2μ2−τ2+s21+μ2y(μ)=λy(μ),(LTWE) where s and τ are parameters, with s an integer and 0 < τ < 1, and λ determines the eigenvalues. This ordinary differential equation is derived from a linear system of partial differential equations, which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive, rotating gravitational sphere. The problems raised by this differential equation are significant, for both the analytic and numerical studies of Sturm-Liouville equations, in respect of the interior singularities, at the points ± τ, and of the changes in sign of the leading coefficient (1 - μ2)/(μ2 - τ2) over the interval (-1, 1). Direct sum space methods, quasi-derivatives and transformation theory are used to determine three physically significant, well-posed boundary value problems from the Sturm-Liouville eigenvalue problem (LTWE), which has singular end-points ± 1 and, additionally, interior singularities at ± τ. Self-adjoint differential operators in appropriate Hilbert function spaces are constructed to represent each of the three well-posed boundary value problems derived from LTWE and it is shown that these three operators are unitarily equivalent. The qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied. This work continues the studies of LTWE made by earlier workers, in particular Hough, Lamb, Longuet-Higgins and Lindzen.
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