Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

Journal of Function Spaces and Applications ◽

10.1155/2013/915830 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 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.

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

Journal of Applied Mathematics ◽

10.1155/2013/306917 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 2

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.

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A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition

Physics Research International ◽

10.1155/2013/159243 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Erdoğan Şen

Keyword(s):

Boundary Condition ◽

Green Function ◽

Boundary Conditions ◽

Liouville Operator ◽

Liouville Problem ◽

Fundamental Solutions ◽

Asymptotic Formulas ◽

Eigenvalues And Eigenfunctions ◽

Sturm Liouville ◽

Interior Points

We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem.

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Chapter 2. inverse sturm-liouville problem on the half-line: determining the sturm-liouville operator from spectral function

Inverse Sturm-Liouville Problems ◽

10.1515/9783110941937-004 ◽

1987 ◽

pp. 24-63

Keyword(s):

Spectral Function ◽

Liouville Operator ◽

Liouville Problem ◽

Half Line ◽

Sturm Liouville

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ASYMPTOTIC FORMULAS FOR EIGENVALUES AND EIGENFUNCTIONS OF A NONSELFADJOINT STURM-LIOUVILLE OPERATOR

Further Progress in Analysis ◽

10.1142/9789812837332_0075 ◽

2009 ◽

Author(s):

KHANLAR R. MAMEDOV ◽

HAMZA MENKEN

Keyword(s):

Liouville Operator ◽

Asymptotic Formulas ◽

Eigenvalues And Eigenfunctions ◽

Sturm Liouville

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Leaky waves in planar dielectric waveguide

Discrete and Continuous Models and Applied Computational Science ◽

10.22363/2658-4670-2019-27-4-325-342 ◽

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.

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