Computing the spectrum of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions

This paper deals with the computation of the eigenvalues of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions using the \textit{regularized sampling method}. A few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available

Two-parameter Sturm-Liouville problems

This paper deals with the computation of the eigenvalues of two-parameter Sturm- Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (Singular, Non-Self-Adjoint, Non-Local, Impulsive,...). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.Comment: 9 page

Regularized trace formula for Sturm-Liouville problem with retarded argument and quadratically eigenparameter-dependent boundary condition

In this paper, a regularized trace formula for a discontinuous Sturm-Liouville equation with retarded argument is obtained for the case in which the spectral parameter occurs linearly in the equation and one of the boundary conditions and quadratically in the other one. The contour integration method is used to obtain that trace formula.Publisher's Versio

Computation of the Eigenpairs of Two-Parameter Sturm-Liouville Problems Using the Regularized Sampling Method

This paper deals with the computation of the eigenvalues of two-parameter Sturm-Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (singular, non-self-adjoint, nonlocal, impulsive, etc.). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method

Eigenvalues of Vectorial Sturm-Liouville Problems with Parameter Dependent Boundary Conditions

We generalize the regularized sampling method introduced in 2005 by the author to compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) to the case of vectorial SLP with parameter dependent boundary conditions. A few problems are worked out to illustrate the effectiveness of the method and show by the same token that we have indeed a general method capable of handling with ease very broad classes of SLPs, whether scalar or vectorial

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.Comment: 11 pages, 4 figure

Spectral Analysis of Complex Dynamical Systems

The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that lateral vibrations of such strings can be decoupled from their in-plane vibrations. We then study the eigenvalue problem originating from the lateral vibrations. We show that, unlike the well-known one string vibrations case, the eigenvalues in a multi-string vibrating system do not have to be simple. Moreover we prove that the multiplicities of the eigenvalues depend on the symmetry of the model and on the total number of the connected strings [50]. We also apply Nevanlinna functions theory to characterize the spectra and to solve the inverse problem for a discrete multi-string system in a more general setting than it was done in [71],[73], [22], [69]-[72]. We also represent multi-string vibrating systems using a coupling of non-densely defined symmetric operators acting in the infinite dimensional Hilbert space. This coupling is defined by a special set of boundary operators acting in finite dimensional Krein space (the space with indefinite inner product). The main results of this research are published in [50]. The Hypothalamic Pituitary Adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body’s cortisol level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor but does not take into account the system response delay, we first perform rigorous stability analysis of all multi-parametric steady states and secondly, by construction of a Lyapunov functional, we prove nonlinear asymptotic stability for some of multi-parametric steady states. We then take into account the additional effects of the time delay parameter on the stability of the HPA axis system. Finally we prove the existence of periodic solutions for the HPA axis system. The main results of this research are published in [51]