In this paper we present a study on the analogous properties of discrete and continuous Sturm-Liouville problems arising in matrix analysis and differential equations, respectively. Green’s functions in both cases have analogous expressions in terms of the spectral data. Most of the results associated to inverse problems in both cases are identical. In particular, in both cases Weyl’s m-functio...

Abstract: In this paper, we present a novel method for computation of eigenvalues and eigenfunctions for a class of singular Sturm-Liouville boundary value problems using modified Adomian decomposition method. The proposed method can be applied to any type of regular as well as singular Sturm-Liouville problems. This current method is capable of finding any n-th eigenvalues and eigenfunctions o...

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).

We nd the adjoint of the Askey-Wilson divided di erence operator with respect to the inner product on L 2 ( 1; 1; (1 x 2 ) 1=2 dx) de ned as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic de...

The measure algebras of the title are those which are also hypergroups with some regularity conditions, Examples include the convolutions associated with Jacobi polynomial series and Fourier Bessel series, It is shown here that there is a one-to-one correspondence between these hypergroups and a class of Sturm-Liouville problems which have the characters of the hypergroup as eigenfunctions, The...

In this article we establish an oscillation theorem for second order Sturm-Liouville difference equations with general nonlinear dependence on the spectral parameter l. This nonlinear dependence on l is allowed both in the leading coefficient and in the potential. We extend the traditional notions of eigenvalues and eigenfunctions to this more general setting. Our main result generalizes the re...

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...

In recent years several high quality Sturm-Liouville software packages have been written (e.g., SLEDGE, SLEIGN, the NAG routine SL02FM). All excel at providing eigenfunction estimates at a nite set of points but only the NAG routine provides interpolation capability, and this by a fairly crude method (especially when compared with the sophistication of its eigenvalue solver). We describe an acc...

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-L...

In this study, we consider a boundary value problem generated by the Sturm-Liouville equation with frozen argument and non-separated conditions on time scale. Firstly, present some solutions characteristic function of an arbitrary bounded Secondly, prove properties eigenvalues obtain formulation for eigenvalues-number finite Finally, give asymptotic formula another special scale: $\mathbb{T}=[\...