The intervals of oscillations in the solutions of the Legendre differential equations
- Dimitris M Christodoulou^{1},
- James Graham-Eagle^{1} and
- Qutaibeh D Katatbeh^{2}Email author
https://doi.org/10.1186/s13662-016-0778-6
© Christodoulou et al. 2016
Received: 25 October 2015
Accepted: 31 January 2016
Published: 17 February 2016
Abstract
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 \((-1, +1)\).
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1 Introduction
The equations studied by CGK in the second step of the program contained singularities at \(x=0\) or no singularities at all, in which cases \(c_{1}\) was set to zero in equations (3) and (4). In the present work, we extend our investigation to the study of oscillatory properties of the Legendre and associated Legendre equations [6, 7] that have regular singular points at \(x=\pm1\) (that is, away from \(x=0\)). In these cases, a horizontal shift \(c_{1}\neq0\) proves to be quite useful, since it can be chosen to circumvent one or the other singularity in the neighborhood of which the intervals of oscillations in the solutions are being sought. The other two constants in equation (3) are chosen in ways that allow for the investigation of specific intervals in x (in particular, \(0\leq x < 1\) and \(1 < x < \infty\)). We use the above equations in Section 2 and Section 3 below, where we analyze the Legendre equation and the associated Legendre equation, respectively. Finally, in Section 4, we summarize our results.
2 Legendre equation
The node counts clarify why the lowest-order oscillatory solutions occur for \(n=3\): the even solution has \({\mathcal{N}}(3) = 4\) and develops two minima, so it is oscillatory. On the other hand, the odd solution has \({\mathcal{N}}(3) = 3\) and two extrema of the same kind do not occur, so it is nonoscillatory. For \(n\geq4\), both types of solutions have enough nodes to guarantee oscillatory behavior, and the number of nodes and oscillations increases with increasing n.
3 Associated Legendre equation
The node counts clarify the borderline case with \(m=n-2\). In this case, \(n-m=2\) (even) and the even solutions all have 4 nodes (equation (20)), just enough for two extrema of the same kind to develop; so they are oscillatory. In contrast, the odd solutions have 3 nodes (equation (21)) and they are all nonoscillatory. Finally, for \(m > n-2\), the solutions have a very small number of nodes (\(1\leq{\mathcal {N}}(n, m)\leq3\)) and then oscillations cannot occur.
4 Summary
We have presented an analysis of the oscillatory properties of the solutions of the Legendre and associated Legendre differential equations [6, 7] that show two regular singular points in their coefficients away from \(x=0\) (at \(x=\pm1\)). The analysis makes use of a program that was described in [5] by CGK for investigating the x-intervals in which the solutions have an oscillatory character. According to this procedure, an attempt must first be made to transform a given equation to a form with constant coefficients. Unlike in the case of the Chebyshev equation that also has two poles at \(x=\pm1\) but can be recast into a form with constant coefficients (CGK), the Legendre equations cannot be simplified in this manner and their analysis must proceed to the second step of the program, where a transformation of x is applied in order to cast these equations into forms with constant damping (i.e., a constant coefficient of the first-derivative term). As was described in Sections 2 and 3 above, such a transformation is capable of eliminating from the analysis one or the other singularity, but not both of them at the same time. This leaves one pole in the coefficient of the canonical form of the final equation (equation (6) and equation (15)) and its presence complicates the search for oscillatory solutions. Because the coefficients (6) and (15) of the canonical form are even functions of x, it turns out that we can investigate only the interval \(x\geq0\) (which can be made free from the singularity at \(x=+1\)), and then use symmetry to derive the corresponding results for \(x < 0\).
- (a)
The solutions \(y_{n}(x)\) of the Legendre equation are predicted to be oscillatory in \(|x| < 1\) for \(n\geq3\) (according to the definition of oscillation given by CGK and adopted here as well) and nonoscillatory outside of that interval for all values of n. Numerical integrations of many cases with a variety of boundary conditions confirm these predictions. Two examples of oscillatory solutions in \((-1, +1)\) are shown in Figures 1 and 2.
- (b)
The solutions \(y_{n}^{m}(x)\) of the associated Legendre equation are predicted to be oscillatory for \(n\geq3\) only and \(m\leq n-2\) in smaller subintervals of \(|x| < x_{*} < 1\), where \(x_{*}\) is given by equation (19). (Note, however, that in the borderline case with \(m=n-2\), only the even solutions are oscillatory.) Numerical integrations, as well as a separate analysis of the alternative form (23) of the associated Legendre equation, confirm that such subintervals are predicted correctly and that they are distinctly smaller than the interval \((-1, +1)\) of the \(m=0\) case. Three examples of oscillatory solutions are shown in Figures 3, 4, and 6 and an example of a nonoscillatory solution is shown in Figure 5.
In the example of Figure 5 with \(m=n-2=10\), two extrema of the same kind do not occur in the predicted interval \(|x|<0.60\) because of the relatively large value of m. In general, in the borderline case with \(m=n-2\), some solutions are oscillatory and other solutions are not, depending on the adopted boundary conditions (as in Figures 6 and 5, respectively); but for \(m > n-2\), all solutions are nonoscillatory because they exhibit just a small number of nodes (1 to 3). Even these types of solutions are, however, characteristically nonmonotonic as they attempt to oscillate, but the small number of nodes does not permit a full ‘cycle’ to develop in the intervals \(|x| < x_{*}\) with \(x_{*}\) specified by equation (19).
Declarations
Acknowledgements
During this research project, DMC and JG-E were supported by the University of Massachusetts Lowell while QDK was on a sabbatical visit and was fully supported by the Jordan University of Science and Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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