We consider the structure of the solution set of a nonlinear SturmLiouville boundary value problem defined on a general time scale. Using global bifurcation theory we show that unbounded continua of non-trivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and we show that certain nodal properties of the solutions are preserved along these continua. Thes...

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm theorem for indefinite systems with Dirichlet boundary conditions of the form p2m du dx + p2m−2(x) du dx + · · ·+ p1(x) du dx + p0(x)u = 0, where pi is a smooth path of matrices on the comple...

We will use the variational approach to (0.1) as described in [2] and we will stick to the notations of that paper. Given the complex n-dimensional Hermitian space (C, 〈·, ·〉), for any m ∈ N let H m := H(J,C) be the Sobolev space of all H-maps from J := [0, 1] into C. A derivative dependent Hermitian form is the form Ω(x)[u] = Pm i,j=0〈D u(x), ωi,j(x)D u(x)〉, where, each ωi,j is a smooth path o...

We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial-boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem. The horizontal profile is define...

In formal scattering theory, Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm-Liouville theory to compute Green functions within a rigorous mathematical theory. We shall show that both the Sturm-Liouville theory and the formal treatment yield the same Green functions. We shall also show how the analyticity of the Green functions as function...

We present an extension of constraint logic programming, where the admissible constraints are arbitrary first-order formulas over some domain. Constraint solving is realized by effective quantifier elimination. The current realization in our CLP(RL) system provides as possible domains R, C, and Qp for primes p. In R and C we admit arbitrary degrees in our constraints. In Qp we restrict to linea...

If a Sturm-Liouville problem is given in an open interval of the real line then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues a...

An inverse nodal problem consists in reconstructing this operator from the given zeros of their eigenfunctions. In this work, we are concerned with the inverse nodal problem of the Sturm-Liouville operator with eigenparameter dependent boundary conditions on a finite interval. We prove uniqueness theorems: a dense subset of nodal points uniquely determine the parameters of the boundary conditio...

In this paper, we present a comparative study of Sinc-Galerkin method and differential transform method to solve Sturm-Liouville eigenvalue problem. As an application, a comparison between the two methods for various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. The study outlines the significant features of the two methods. The results show that these me...

Selfadjoint Sturm-Liouville operators Hω on L2(a, b) with random potentials are considered and it is proven, using positivity conditions, that for almost every ω the operator Hω does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as Hω but in L2(ã, b̃) where (ã, b̃) ⊂ (a, b).