The intervals of oscillations in the solutions of the Legendre differential equations

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• 作者：Dimitris M Christodoulou ; James Graham-Eagle…
• 关键词：oscillations ; second ; order linear differential equations ; analytical theory ; transformations ; Legendre differential equations
• 刊名：Advances in Difference Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,431 KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847

文摘

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 \(x=\pm1\) of the Legendre differential equations. The solutions \(y_{n}(x)\) of the Legendre equation are predicted to be oscillatory in \(|x| < 1\) for \(n\geq3\) and nonoscillatory outside of that interval for all values of n. In contrast, the solutions \(y_{n}^{m}(x)\) of the associated Legendre equation are predicted to be oscillatory for \(n\geq3\) and \(m\leq n-2\) only in smaller subintervals \(|x| < x_{*} < 1\), the sizes of which are determined by n and m. Numerical integrations confirm that such subintervals are distinctly smaller than \((-1, +1)\).