On the relationship between the turning and singular points in Sturm-Liouville equations

In this paper, we present some results about the Sturm-Liouville equation with turning points and singularities and transform them to each other. By applying a change of a variable, we can transform the differential equation with a turning point to the differential equation with a singularity. Also we will prove that a differential equation with a singularity will be transformed to a differential equation with a turning point in some cases.


Introduction
Differential equations with turning points and singularities have emerged as an effective and powerful tool to study a wide class of related problems arising in various branches of mathematics, mechanics, physics, geophysics, computer sciences and other branches of natural sciences. Better knowledge and understanding from these equations provide us to study a wide class of problems arising in mathematical physics, radio electronics, and other fields of sciences and technologies (see Constantin, 1998;Conway, 1995;Dehghan & Jodayree, 2013;Hryniv and Mykytyuk, 2004;Lapwood & Usami, 1981;Litvinenko & Soshnikov, 1964). Differential equations with turning points and singularities have been studied in  ;Neamaty & Khalili (2013); and Yurko (1997). But the relationship between turning points and singularities has not yet been studied basically. In , differential equations with a turning point of the second type are transformed to a singularity. To the best of our knowledge, it does not exist an exact study on the relevance between a turning and singular point. For this reason, we would like to point out the transformation to convert two points to each other. This transformation is like the Liouville transformation, but with this difference that the Liouville transformation transforms the Sturm-Liouville equations with the positive weight function to the Sturm-Liouville equations with the weight function r 2 ðxÞ ¼ 1 and continuous potential function. Indeed we extend this transformation and would like to prove that the extended transformation can be applied for Sturm-Liouville equations with the turning point. We gave here only the main idea and refer the interested reader to Miller & Michel (1982) for further details. Therefore, we show that this transformation will omit the turning point and commute a turning point to a singularity. Moreover, we will prove that a singular point does not simply vanish. In other words, we will transform a singular point of second order to a turning point by the same transformation which indeed we use the reverse of this transformation to omit the singularity.
As mentioned earlier, the main goal of this article is to study the relationship between a turning point and a singularity in the Sturm-Liouville equations. We will apply a change of a variable to transform a turning point into a singularity. Taking the same transformation, we will convert merely a singularity of the second order to a turning point and give the interesting results. We mention that the approach considered in this article can serve as a tool for various studies connected with the spectral theory of Sturm-Liouville equations and topics connected with these problems, like, for example, direct and inverse spectral problems. We note that direct and inverse spectral problems for differential equations having singularities and/or turning points were studied in Dehghan & Jodayree (2013), Neamaty & Khalili (2013, 2014, and other works.

Main results
We consider the differential equation À y 00 ðxÞ þ qðxÞyðxÞ ¼ λr 2 ðxÞyðxÞ; x 2 ½0; T; (2:1) with the weight function r 2 ðxÞ: We assume that the weight function has the zeros in an interior point x ¼ a which is called a turning point. The functions qðxÞ and r 2 ðxÞ are real valued.
Lemma 1. Consider the Equation (2.1). Let two real functions f and g be piecewise continuous. If the following change of the variable omits the turning point, then f 2 ðxÞ ¼ gðxÞ: Proof. After obtaining y 00 and substituting in the Equation (2.1), we get where f n :¼ ðf ðxÞÞ n for n 2 Z. To eliminate du dz , its coefficient must be deleted. So By integrating from two sides of the previous relation, we get The Lemma 1 is proved.
Theorem 2 is proved.

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It seems that singular points of other orders do not vanish. In this case, we only mention the following theorem.
Theorem 3. The following transformation made on the discontinuous function (2:35) does not eliminate the singular point in Sturm-Liouville equations.

Conclusions
In this article, we verify a transformation to omit the turning points in Sturm-Liouville equations. In other words, we proposed a transformation and proved that this transformation can eliminate a turning point (Theorem 1). Also we showed that differential equations with a singular point of the second order are transformed to a differential equation with a turning point by using this transformation.
Moreover, we proved that any transformation of the form (2.35) cannot omit the singularity.