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.

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.

scholarly journals The Sturm-Liouville Problem with Singular Potential and the Extrema of the First Eigenvalue

2013 ◽  
Vol 54 (1) ◽  
pp. 101-118
Author(s):  
Elena S. Karulina ◽  
Anton A. Vladimirov
Keyword(s):  
Singular Potential ◽  
Liouville Problem ◽  
First Eigenvalue ◽  
Constant Sign ◽  
The First Eigenvalue ◽  
Sturm Liouville

Abstract We get the infima and suprema of the first eigenvalue of the problem where q belongs to the set of constant-sign summable functions on [0,1] such that

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.

scholarly journals ON A MINIMIZATION PROBLEM FOR A FUNCTIONAL GENERATED BY THE STURM – LIOUVILLE PROBLEM WITH INTEGRAL CONDITION ON THE POTENTIAL

2017 ◽  
Vol 21 (6) ◽  
pp. 57-61
Author(s):  
S.S. Ezhak
Keyword(s):  
Boundary Value Problem ◽  
Minimization Problem ◽  
Nonlinear Boundary ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Functional ◽  
Sturm Liouville ◽  
Class Of Functions

In this article we consider the minimization problem of the functional generated by a Sturm — Liouville problem with Dirichlet boundary conditions and with an integral condition on the potential. Estimation of the infimum of functional in some class of functions y and Q(x) isreduced to estimation of a nonlinear functional non depending on the potential Q(x). This leads to related parameterized nonlinear boundary value problem. Upper and lower estimates are obtained for different values of parameter.

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.

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