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.

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 The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan-Hsiou Cheng
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
First Eigenvalue ◽  
Liouville Property ◽  
The First Eigenvalue ◽  
Eigenvalue Gap ◽  
Single Well ◽  
Eigenvalue Ratio ◽  
Sturm Liouville ◽  
Prüfer Substitution

AbstractIn this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

Fluxon Centering in Josephson Junctions with Exponentially Varying Width

2012 ◽  
Vol 2 (3) ◽  
pp. 204-213
Author(s):  
E. G. Semerdjieva ◽  
M. D. Todorov
Keyword(s):  
Josephson Junctions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Static Solution ◽  
Physical Parameters ◽  
Nonlinear Eigenvalue Problems ◽  
Continuous Analogue ◽  
Sturm Liouville ◽  
The Stability ◽  
The Impact

AbstractNonlinear eigenvalue problems for fluxons in long Josephson junctions with exponentially varying width are treated. Appropriate algorithms are created and realized numerically. The results obtained concern the stability of the fluxons, the centering both magnetic field and current for the magnetic flux quanta in the Josephson junction as well as the ascertaining of the impact of the geometric and physical parameters on these quantities. Each static solution of the nonlinear boundary-value problem is identified as stable or unstable in dependence on the eigenvalues of associated Sturm-Liouville problem. The above compound problem is linearized and solved by using of the reliable Continuous analogue of Newton method.

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 Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Mohammed Al-Refai ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Rayleigh Quotient ◽  
First Eigenvalue ◽  
Suggested Model ◽  
Conformable Derivative ◽  
Arbitrary Boundary ◽  
Sturm Liouville ◽  
Natural Boundary Conditions

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

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.

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