This work is about the use of some classical spectral collocation methods as well with new software system Chebfun in order to compute eigenpairs high Sturm–Liouville eigenproblems. The analysis divided into two distinct directions. For problems clamped boundary conditions, we preconditioning differentiation matrices and for hinged end conditions equation transformed a second then conventional ...

In this paper, we study the nonlinear second-order impulsive qk-difference equations with Sturm-Liouville type, in which nonlinear team and impulsive teams are dependent on first-order qk-derivatives. We obtain the existence and uniqueness results of solutions for the problem by Banach’s contraction mapping principle and Schaefer’s fixed point theorems. Finally, we give two examples to demonstr...

The main aim of this paper is to construct quantum extension of the discrete Sturm–Liouville equation consisting of second-order difference equation and boundary conditions that depend on a quadratic eigenvalue parameter. We consider a boundary value problem (BVP) consisting of a second-order quantum difference equation and boundary conditions that depend on the quadratic eigenvalue parameter. ...

We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x)) + h(x), in (0, π), c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, where: p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0 for all x ∈ [0, π]; ci0 + ci1 > 0, i = 0, 1; h ∈ L2(0, π). We suppose that f : [0, π] × R → R is continuous and there exist increasing functions ζl, ζu : [0,∞)→ R, and positive constants A, B...

The one-dimensional harmonic oscillator wave functions are solutions to a SturmLiouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on (−∞,∞) in terms...

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...