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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 Urbana-Champaign
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 Sturm-Liouville differential equation has infinitely many real zeros. The limit of the distance between successive positive zeros of every nontrivial solution of certain Sturm-Liouville differential equations is also determined. Moreover, we prove that each solution of a certain type of second-order 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 second-order 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:2021-04-22
Type:Thesis
URI:http://hdl.handle.net/2142/110844
Rights Information:Copyright 2021 Yusuf Chebao
Date Available in IDEALS:2021-09-17
Date Deposited:2021-05


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