Asymptotics of Eigenvalues and Eigenfunctions of the Generalized Sturm-Liouville Problem for Higher Vibration Modes

2010 ◽  
Keyword(s):  
Liouville Problem ◽  
Vibration Modes ◽  
Asymptotics Of Eigenvalues ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

scholarly journals Leaky waves in planar dielectric waveguide

2019 ◽  
Vol 27 (4) ◽  
pp. 325-342
Author(s):  
Dmitriy V. Divakov ◽  
Alexandre A. Egorov ◽  
Konstantin P. Lovetskiy ◽  
Leonid A. Sevastianov ◽  
Andrey S. Drevitskiy
Keyword(s):  
Separation Of Variables ◽  
Liouville Problem ◽  
Initial Boundary ◽  
Eigenvalues And Eigenfunctions ◽  
Leaky Modes ◽  
Initial Boundary Problem ◽  
Dielectric Media ◽  
Complex Roots ◽  
Sturm Liouville ◽  
Waveguide Problem

A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.

scholarly journals Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Erdoğan Şen ◽  
Jong Jin Seo ◽  
Serkan Araci
Keyword(s):  
Boundary Value Problem ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Coefficients ◽  
Discontinuous Boundary ◽  
Sturm Liouville ◽  
Special Case

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients and retarded argument in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

scholarly journals Reconstruction of potential function in inverse Sturm-Liouville problem via partial data

2021 ◽  
Vol 11 (2) ◽  
pp. 186-198
Author(s):  
Mehmet Açil ◽  
Ali Konuralp
Keyword(s):  
Boundary Conditions ◽  
Potential Function ◽  
Liouville Problem ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Asymptotics Of Eigenvalues ◽  
Time Saving ◽  
Partial Data ◽  
Sturm Liouville ◽  
Saving Algorithm

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively.

scholarly journals Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Erdal Bas
Keyword(s):  
Spectral Theory ◽  
Liouville Operator ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

scholarly journals A New Approach for Linear Eigenvalue Problems and Nonlinear Euler Buckling Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Meltem Evrenosoglu Adiyaman ◽  
Sennur Somali
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Numerical Results ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Eigenvalues And Eigenfunctions ◽  
New Approach ◽  
Euler Buckling ◽  
Large Step ◽  
Sturm Liouville

We propose a numerical Taylor's Decomposition method to compute approximate eigenvalues and eigenfunctions for regular Sturm-Liouville eigenvalue problem and nonlinear Euler buckling problem very accurately for relatively large step sizes. For regular Sturm-Liouville problem, the technique is illustrated with three examples and the numerical results show that the approximate eigenvalues are obtained with high-order accuracy without using any correction, and they are compared with the results of other methods. The numerical results of Euler Buckling problem are compared with theoretical aspects, and it is seen that they agree with each other.

Variational approximation for fractional Sturm–Liouville problem

2020 ◽  
Vol 23 (3) ◽  
pp. 861-874
Author(s):  
Prashant K. Pandey ◽  
Rajesh K. Pandey ◽  
Om P. Agrawal
Keyword(s):  
Fractional Order ◽  
Variational Approach ◽  
Liouville Problem ◽  
Variational Approximation ◽  
Eigenvalues And Eigenfunctions ◽  
Caputo Derivatives ◽  
Sturm Liouville ◽  
Countably Infinite

AbstractIn this paper, we consider a regular Fractional Sturm–Liouville Problem (FSLP) of order μ (0 < μ < 1). We approximate the eigenvalues and eigenfunctions of the problem using a fractional variational approach. Recently, Klimek et al. [16] presented the variational approach for FSLPs defined in terms of Caputo derivatives and obtained eigenvalues, eigenfunctions for a special range of fractional order 1/2 < μ < 1. Here, we extend the variational approach for the FSLPs and approximate the eigenvalues and eigenfunctions of the FSLP for fractional-order μ (0 < μ < 1). We also prove that the FSLP has countably infinite eigenvalues and corresponding eigenfunctions.

Sign in / Sign up

Export Citation Format

Share Document