文章基本信息

标题：The intervals of oscillations in the solutions of the Legendre differential equations
本地全文：下载
作者：Dimitris M Christodoulou ; James Graham-Eagle ; Qutaibeh D Katatbeh 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2016
卷号：2016
期号：1
页码：46
DOI：10.1186/s13662-016-0778-6
语种：English
出版社：Hindawi Publishing Corporation
摘要：We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out around the regular singular points x = ± 1 $x=\pm1$ of the Legendre differential equations. The solutions y n ( x ) $y_{n}(x)$ of the Legendre equation are predicted to be oscillatory in x < 1 $ x < 1$ for n ≥ 3 $n\geq3$ and nonoscillatory outside of that interval for all values of n. In contrast, the solutions y n m ( x ) $y_{n}^{m}(x)$ of the associated Legendre equation are predicted to be oscillatory for n ≥ 3 $n\geq3$ and m ≤ n − 2 $m\leq n-2$ only in smaller subintervals x < x ∗ < 1 $ x < x_{*} < 1$ , the sizes of which are determined by n and m. Numerical integrations confirm that such subintervals are distinctly smaller than ( − 1 , + 1 ) $(-1, +1)$ .
关键词：oscillations ; second-order linear differential equations ; analytical theory ; transformations ; Legendre differential equations