eprintid: 1458551
rev_number: 40
eprint_status: archive
userid: 608
dir: disk0/01/45/85/51
datestamp: 2015-01-30 11:45:10
lastmod: 2020-02-12 18:14:14
status_changed: 2015-01-30 11:45:10
type: thesis
metadata_visibility: show
item_issues_count: 0
creators_name: Kecker, T
title: On the singularity structure of differential equations in the complex plane
ispublished: unpub
divisions: UCL
divisions: A01
divisions: B04
divisions: C06
abstract: In this dissertation the structure of singularities in the complex plane of solutions of certain classes of ordinary differential equations and systems of equations is studied. The thesis treats two different aspects of this topic. Firstly, we introduce the concept of movable singularities for first and second-order ordinary differential equations. On the one hand the local behaviour of solutions about their movable singularities is investigated. It is shown, for the classes of equations considered, that all movable singularities of all solutions are either poles or algebraic branch points. That means locally, about any movable singularity z0, the solutions are finitely branched and represented by a convergent Laurent series expansion in a fractional power of z-z0 with nite principle part. This is a generalisation of the Painleve property under which all solutions have to be single-valued about all their movable singularities. 

The second aspect treated in the thesis deals with the global structure of the solutions. In general, the solutions of the equations discussed in the first part have a complicated global behaviour as they will have infinitely many branches. In the second part conditions are discussed for certain equations under the existence of solutions that are globally nite-branched, leading to the notion of algebroid solutions. In order to do so, some concepts from Nevanlinna theory, the value-distribution theory of meromorphic functions and its extension to algebroid functions are introduced. Then, firstly, Malmquist's theorem for first-order rational equations with algebroid solutions is reviewed. Secondly, certain second-order equations are considered and it is examined to what types of equations they can be reduced under the existence of an admissible algebroid solution.
date: 2014-12-28
vfaculties: VMPS
oa_status: green
full_text_type: other
thesis_class: doctoral_open
language: eng
thesis_view: UCL_Thesis
primo: open
primo_central: open_green
verified: verified_manual
elements_source: Manually entered
elements_id: 999734
lyricists_name: Kecker, Thomas
lyricists_id: TKECK34
full_text_status: public
pagerange: 1 - 74
pages: 74
institution: UCL (University College London)
department: Mathematics
thesis_type: Doctoral
editors_name: Halburd, RG
citation:        Kecker, T;      (2014)    On the singularity structure of differential equations in the complex plane.                   Doctoral thesis , UCL (University College London).     Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/1458551/2/PhDthesisTK_final.pdf