Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

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Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2019-0003 ◽

2020 ◽

Vol 28 (3) ◽

pp. 341-348 ◽

Cited By ~ 1

Author(s):

Ran Zhang ◽

Xiao-Chuan Xu ◽

Chuan-Fu Yang ◽

Natalia Pavlovna Bondarenko

Keyword(s):

Boundary Conditions ◽

Interior Point ◽

Liouville Operator ◽

Inverse Spectral Problem ◽

Liouville Problem ◽

Robin Boundary Conditions ◽

Jump Conditions ◽

Sturm Liouville ◽

Robin Boundary

AbstractIn this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on {(0,\pi)} with the Robin boundary conditions and the jump conditions at the point {\frac{\pi}{2}}. We prove that the potential {M(x)} on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential {M(x)} is given on {(0,\frac{(1+\alpha)\pi}{4})}; (ii) the potential {M(x)} is given on {(\frac{(1+\alpha)\pi}{4},\pi)}, where {0<\alpha<1}, respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.

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An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.50.2019.3347 ◽

2019 ◽

Vol 50 (3) ◽

pp. 207-221 ◽

Cited By ~ 2

Author(s):

Sergey Buterin

Keyword(s):

Boundary Condition ◽

Inverse Problem ◽

Boundary Conditions ◽

Differential Operators ◽

A Priori ◽

Sufficient Conditions ◽

Robin Boundary Conditions ◽

Sturm Liouville ◽

Necessary And Sufficient ◽

Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

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X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100008621 ◽

1971 ◽

Vol 69 (2) ◽

pp. 139-148

Author(s):

B. D. Sleeman

Keyword(s):

Differential Equation ◽

Eigenvalue Problem ◽

Differential Equations ◽

Boundary Conditions ◽

Ordinary Differential Equations ◽

Liouville Problem ◽

Sturm Liouville ◽

Image Position ◽

Two Parameter ◽

General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

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Expansions in terms of sets of functions with complex eigenvalues

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100024208 ◽

1948 ◽

Vol 44 (2) ◽

pp. 242-250 ◽

Cited By ~ 22

Author(s):

R. Peierls

Keyword(s):

Boundary Condition ◽

Eigenvalue Problem ◽

Boundary Conditions ◽

Real Function ◽

Liouville Problem ◽

Nuclear Collisions ◽

Complex Eigenvalues ◽

Sturm Liouville ◽

Image Position ◽

Unusual Type

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified formwhere 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditionsand

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Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Complexity ◽

10.1155/2017/3720471 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7 ◽

Cited By ~ 27

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.

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