Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials. / Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer. / Mestre em Matemática

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2007. / Bibliyografya var.

Orientador: Marta Cilene Gadotti / Banca: Suzete Maria Silva Afonso / Banca: Katia Andreia Gonçalves de Azevedo / Resumo: Este texto aborda os principais resultados sobre a Teoria de Sturm-Liouville assim como os pré-requisitos necessários para construí-los, entre eles o Teorema Espectral para Operadores Compactos e a Teoria de Fredholm. Também são apresentados alguns exemplos e uma aplicação envolvendo uma equação diferencial parcial que modela o problema da corda vibrante / Abastract: This research approaches the main results on the Sturm-Liouville Theory, as well the necessary prerequisites for constructing them, including the Spectral Theorem for Compact Operators and Fredholm Theory. It is also presented some examples and an application involving a partial differential equation that models the vibrating string problem / Mestre

We study several unbounded operators with view to extending von Neumann's theory of deficiency indices for single Hermitian operators with dense domain in Hilbert space. If the operators are non-commuting, the problems are difficult, but special cases may be understood with the use representation theory. We will further study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are defined on the common dense domain of C∞ functions with compact support, and they separately are essentially selfadjoint, but the unique selfadjoint extensions will be non-commuting. This problem is of a geometric flavor, and we study an index formulation for its solution. The applications include the study of vector fields, the theory of Dirichlet problems for second order partial differential operators (PDOs), Sturm-Liouville problems, H.Weyl's limit-point/limit-circle theory, Schrödinger equations, and more.

We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.

This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.

In this thesis, we consider the following one dimensional p-Laplacian eigenvalue problem: -((y¡¦/s)^(p-1))¡¦+(p-1)(q-£fw)y^(p-1)=0 a.e. on (0,1) (0.1) and satisfy £\y(0)+ £\ ¡¦ (y¡¦(0)/s(0))=0 £]y(1)+£]¡¦ (y¡¦(1)/s(1))=0 (0.2) where f^(p-1)=|f|^p-2 f=|f|^p-1 sgnf; £\, £\¡¦, £], £]¡¦ ∈R such that £\^2+£\¡¦^2>0 and£]^2+£]¡¦^2>0; and the functions s,q,w are required to satisfy (1) s,q,w∈L^1(0,1); (2) for 0≤x≤1, we have s≥0,w≥0 a.e.; (3) for any x∈ (0,1), ¡ì_0^1 s(t)dt>0, ¡ì_0^x w(t)dt>0,and¡ì_x^1 w(t)dt>0; (4) if for some x_1<x_2,we have¡ì_ x1^x2 w(t)dt=0,then¡ì_ x1^x2 |q(t)|dt=0; (5) for all n∈N, there is a partition {£a_i^(n)}_i=1 ^2n of [0,1] such that for any 0<k≤n-1, ¡ì_£a_2k^(n)^ £a_2k+1^(n) w>0 and ¡ì_£a_2k+1^(n)^ £a_2k+2^(n) s>0. We call the above conditions Atkinson conditions, first introduce in [1].There conditions include the case when s,q,w∈L^1(0,1) and s,w>0 a.e. We use a generalized Prufer substitution and Caratheodory theorem to prove the existence and uniqueness for the solution of the initial value problem of (0.1) above. Then we generalize the Sturm oscillation theorem to one dimensional p-Laplacian and establish the Sturm-Liouville properties of the p-Laplacian operators with L^1 coefficient functions. Our results filled up some gaps in Binding-Drabek [3].

In this thesis, we demonstrate the capability of power series, combined with numerical methods, to solve boundary value problems and Sturm-Liouville eigenvalue problems of ordinary differential equations. This kind of schemes is usually called the numerical-symbolic, numerical-analytic or semi-analytic method. In the first chapter, we develop an adaptive algorithm, which automatically decides the terms of power series to reach desired accuracy. The expansion point of power series can be chosen freely. It is also possible to combine several power series piecewisely. We test it on several models, including the second and higher order linear or nonlinear differential equations. For nonlinear problems, the same procedure works similarly to linear problems. The only differences are the nonlinear recurrence of the coefficients and a nonlinear equation, instead of linear, to be solved. In the second chapter, we use our semi-analytic method to solve singularly perturbed problems. These problems arise frequently in fluid mechanics and other branches of applied mathematics. Due to the existence of boundary or interior layers, its solution is very steep at certain point. So the terms of series need to be large in order to reach the desired accuracy. To improve its efficiency, we have a strategy to select only a few required basis from the whole polynomial family. Our method is shown to be a parameter diminishing method. A specific type of boundary value problem, called the Sturm-Liouville eigenvalue problem, is very important in science and engineering. They can also be solved by our semi-analytic method. This is our focus in the third chapter. Our adaptive method works very well to compute its eigenvalues and eigenfunctions with desired accuracy. The numerical results are very satisfactory.

The inverse nodal problem is the problem of understanding the potential function of the Sturm-Liouville operator from the set of the nodal data ( zeros of eigenfunction ). This problem was first defined by McLaughlin[12]. Up till now, the problem on finite intervals has been studied rather thoroughly. Uniqueness, reconstruction and stability problems are all solved. In this thesis, I investigate the inverse nodal problem on semi-infinite intervals q(x) is real and continuous on [0,1) and q(x)!1, as x!1. we have the following proposition. L is in the limit-point case. The spectral function of the differential operator in (1) is a step function which has discontinuities at { k} , k = 0, 1, 2, .... And the corresponding solutions (eigenfunction) k(x) = (x, k) has exactly k zeros on [0,1). Furthermore { k} forms an orthogonal set. Finally we also discuss that density of nodal points and a reconstruction formula on semiinfinite intervals.

In this thesis we present a numerical algorithm to solve the singular Inverse Sturm-Liouville problems with symmetric potential functions. The singularity, which comes from the unbounded domain of the problem, is treated by considering the limiting case of the associated problem on the symmetric finite interval. In contrast to regular problems which are considered on a finite interval the singular inverse problem has an ill-conditioned structure despite of the limiting treatment. We use the regularization techniques to overcome the ill-posedness difficulty. Moreover, since the problem is nonlinear the iterative solution procedures are needed. Direct computation of the eigenvalues in iterative solution is handled via psoudespectral methods. The numerical examples of the considered problem are given to illustrate the accuracy and convergence behaviour.