scholarly journals Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function

2014 ◽  
Vol 15 (1) ◽  
pp. 197
Author(s):  
Erdoğan Şen
Keyword(s):  
Weight Function ◽  
Asymptotic Properties ◽  
Liouville Problem ◽  
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.

Asymptotic properties of a new Sturm-Liouville problem with retarded argument

2013 ◽  
Vol 37 (17) ◽  
pp. 2619-2625 ◽  
Author(s):  
Erdoğan Şen ◽  
Serkan Araci ◽  
Mehmet Acikgoz
Keyword(s):  
Asymptotic Properties ◽  
Liouville Problem ◽  
Retarded Argument ◽  
Sturm Liouville

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.

scholarly journals ON THE UPPER ESTIMATES FOR THE FIRST EIGENVALUE OF A STURM - LIOUVILLE PROBLEM WITH A WEIGHTED INTEGRAL CONDITION

2017 ◽  
Vol 21 (6) ◽  
pp. 124-129
Author(s):  
M.Yu. Telnova
Keyword(s):  
Weight Function ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Integral Condition ◽  
First Eigenvalue ◽  
Second Order Differential Equations ◽  
The First Eigenvalue ◽  
Lagrange Problem ◽  
Sturm Liouville ◽  
The Given

In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved. In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved.

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