The determination of solutions of linear differential equations with entire coefficients from their zeros

Asiri, Asim Mohammad Khalid (2012) The determination of solutions of linear differential equations with entire coefficients from their zeros. PhD thesis, University of Nottingham.

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Abstract

This thesis starts from the following observation; if v;w are solutions of

y00 + Py = 0

where P is entire, and v and w are both 0 at z0 2 C, then W(v;w) = vw0 􀀀 v0w _ 0 and v;w are linearly dependent. It is then natural to ask what happens if v;w solve different equations, but have (mostly) the same zeros. The case where the first equation is of the second order and has a polynomial coefficient while the second equation is of order greater than one with entire coefficients was investigated first, and some relations between the solutions and between the coefficients were proved. We next obtained approximately the same results when a transcendental coefficient was considered instead of a polynomial in the first equation, but with some amendments to the conditions. We then examined the case where the equations are non-homogeneous of the first order and determined what the solutions have to be. We also could determine the solutions in the case where the equations are a combination of homogeneous and non-homogeneous equations. Finally, the case where the solutions take the value 0 and a non-zero value at mostly the same points was studied, and again the solutions were determined. In order to prove our results, we used some background from Nevanlinna theory and some of its applications.

Item Type:Thesis (PhD)
Supervisors:Langley, J.K.
Feinstein, J.F.
Uncontrolled Keywords:Complex analysis, differential equations, Nevanlinna theory
Faculties/Schools:UK Campuses > Faculty of Science > School of Mathematical Sciences
ID Code:2656
Deposited By:Asim Asiri
Deposited On:20 Sep 2012 12:54
Last Modified:20 Sep 2012 12:54

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