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Title:  Zeros of solutions of certain differential equations 
Author(s):  Chebao, Yusuf 
Director of Research:  Hinkkanen, Aimo 
Doctoral Committee Chair(s):  Miles, Joseph 
Doctoral Committee Member(s):  Loeb, Peter; Nikolaev, Igor 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  ordinary differential equations
real functions functions of a complex variable special functions 
Abstract:  The theme of this work is the study of several interconnected aspects of zeros of solutions of certain linear ordinary differential equations. Firstly, we show that every solution of a certain kind of SturmLiouville differential equation has infinitely many real zeros. The limit of the distance between successive positive zeros of every nontrivial solution of certain SturmLiouville differential equations is also determined. Moreover, we prove that each solution of a certain type of secondorder linear homogeneous differential equation is asymptotically bounded by a function related to the coefficients of the equation. In addition, it is shown that for each nontrivial solution of Bessel’s differential equation, the limit of the distance between two successive positive zeros of its derivative of every order is always $\pi$. Furthermore, the final sets of Bessel functions of the first kind and the second kind on certain parts of the complex plane are determined. In particular, we show that for every $n \in \mathbb{Z}$, the final set of $J_n$ on the complex plane is the set $\{k\pi/2  k \in \mathbb{Z}\}$, and the final set of $Y_n$ on the positive real axis is empty. Moreover, for every $\nu \in \mathbb{R}\setminus\mathbb{Z}$, the final sets of $J_{\nu}$ and $Y_{\nu}$ on the positive real axis are the empty set. Consequently, a complete characterization of the final set of an arbitrary solution of Bessel’s differential equation with real parameter on the positive real axis is attained. Along the process of determining the final sets of Bessel functions, we encountered two special functions which satisfy certain secondorder linear nonhomogeneous differential equations. We show that under some conditions, each of these functions has infinitely many positive zeros and each such zero is simple. 
Issue Date:  20210422 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/110844 
Rights Information:  Copyright 2021 Yusuf Chebao 
Date Available in IDEALS:  20210917 
Date Deposited:  202105 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois