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Artūras Stikonas - One of the best experts on this subject based on the ideXlab platform.
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investigation of characteristic curve for Sturm Liouville Problem with nonlocal boundary conditions on torus
Mathematical Modelling and Analysis, 2011Co-Authors: Artūras StikonasAbstract:In this paper, we investigate the second-order Sturm–Liouville Problem with two additional Nonlocal Boundary Conditions. Nonlocal boundary conditions depends on two parameters. We find condition for existence of zero eigenvalue in the parameters space and classified Characteristic Curves in the plane and extended plane is described as torus. The Characteristic Curve on torus may be of three types only. Some new conclusions about existence and uniqueness domain of solution are presented.
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On the generalized eigenfunctions system of the Sturm–Liouville Problem
Lietuvos matematikos rinkinys, 2008Co-Authors: Sigita Pečiulytė, Artūras StikonasAbstract:In this paperwe investigate eigenfunctions and generalized eigenfunctions system of the Sturm–Liouville Problem with classical boundary condition on the left boundary and nonlocal boundary conditionsof four types on the right boundary.
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the Sturm Liouville Problem with a nonlocal boundary condition
Lithuanian Mathematical Journal, 2007Co-Authors: Artūras StikonasAbstract:In this paper, we consider the Sturm-Liouville Problem with one classical and another nonlocal boundary condition. We investigate general properties of the characteristic function and spectrum for such a Problem in the complex case. In the second part, we investigate the case of real eigenvalues, analyze the dependence of the spectrum on parameters of the boundary condition, and describe the qualitative behavior of all eigenvalues subject to of the nonlocal boundary condition.
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Sturm Liouville Problem for stationary differential operator with nonlocal two point boundary conditions
Nonlinear Analysis-Modelling and Control, 2006Co-Authors: Sigita Pečiulytė, Artūras StikonasAbstract:The Sturm-Liouville Problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville Problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a Problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these Problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.
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Sturm Liouville Problem for stationary differential operator with nonlocal integral boundary condition
Mathematical Modelling and Analysis, 2005Co-Authors: S Peciulyte, O Stikoniene, Artūras StikonasAbstract:Abstract The Sturm‐Liouville Problem with various types of nonlocal integral boundary conditions is considered in this paper. In the first part of paper we investigate Sturm‐Liouville Problem with two cases of nonlocal integral boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such Problem in the complex case. In the second part we investigate real eigenvalues case. The spectrum depends of these Problems on boundary condition parameters is analyzed. Qualitative behaviour of all eigenvalues subject to nonlocal boundary condition parameters is described.
Zeinab S. Mansour - One of the best experts on this subject based on the ideXlab platform.
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On fractional q -Sturm–Liouville Problems
Journal of Fixed Point Theory and Applications, 2016Co-Authors: Zeinab S. MansourAbstract:In this paper, we formulate a regular q-fractional Sturm–Liouville Problem (qFSLP) which includes the left-sided Riemann–Liouville and the right-sided Caputo q-fractional derivatives of the same order \(\alpha \), \(\alpha \in (0,1)\). The properties of the eigenvalues and the eigenfunctions are investigated. A q-fractional version of the Wronskian is defined and its relation to the simplicity of the eigenfunctions is verified. We use a fixed point theorem to introduce a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions when \(\alpha >1/2\). These results are a generalization of the integer regular q-Sturm–Liouville Problem introduced by Annaby and Mansour (J Phys A Math Gen 39:8747, 2005). An example for a qFSLP whose eigenfunctions are little q-Jacobi polynomials is introduced.
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Variational methods for fractional $q$-Sturm--Liouville Problems
arXiv: Classical Analysis and ODEs, 2016Co-Authors: Zeinab S. MansourAbstract:In this paper, we formulate a regular $q$-fractional Sturm--Liouville Problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo $q$-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. We introduce the essential $q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $\alpha>1/2$ associated with the boundary condition $y(0)=y(a)=0$. A criteria for the first eigenvalue is proved. Examples are included. These results are a generalization of the integer regular $q$-Sturm--Liouville Problem introduced by Annaby and Mansour in [1].
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On Fractional q-Sturm--Liouville Problems
arXiv: Classical Analysis and ODEs, 2016Co-Authors: Zeinab S. MansourAbstract:In this paper, we formulate a regular $q$-fractional Sturm--Liouville Problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo q-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. The properties of the eigenvalues and the eigenfunctions are investigated. A $q$-fractional version of the Wronskian is defined and its relation to the simplicity of the eigenfunctions is verified. We use the fixed point theorem to introduce a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions when $\alpha>1/2$. These results are a generalization of the integer regular $q$-Sturm--Liouville Problem introduced by Annaby and Mansour in[1]. An example for a qFSLP whose eigenfunctions are little $q$-Jacobi polynomials is introduced.
Štikonas Artūras - One of the best experts on this subject based on the ideXlab platform.
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Investigation of spectrum curves for a Sturm–Liouville Problem with two-point nonlocal boundary conditions
2020Co-Authors: Bingelė Kristina, Bankauskienė Agnė, Štikonas ArtūrasAbstract:The article investigates the Sturm–Liouville Problem with one classical and another nonlocal two-point boundary condition. We analyze zeroes, poles and critical points of the characteristic function and how the properties of this function depend on parameters in nonlocal boundary condition. Properties of the Spectrum Curves are formulated and illustrated in figures for various values of parameter ξ
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On positive eigenfunctions of Sturm-Liouville Problem with nonlocal two-point boundary condition
'Vilnius Gediminas Technical University', 2020Co-Authors: Pečiulytė Sigita, Štikonas ArtūrasAbstract:Positive eigenvalues and corresponding eigenfunctions of the linear Sturm-Liouville Problem with one classical boundary condition and another non- local two-point boundary condition are considered in this paper. Four cases of non- local two-point boundary conditions are analysed. We get positive eigenfunctions existence domain for each case of these Problems. This domain depends on the pa- rameters of the nonlocal boundary Problem and it gives necessary and sufficient conditions for existing positive eigenvalues with positive eigenfunctionsMatematikos ir informatikos institutasVytauto Didžiojo universiteta
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Investigation of the spectrum for the Sturm–Liouville Problem with one integral boundary condition
'Vilnius University Press', 2020Co-Authors: Skučaitė Agnė, Pečiulytė Sigita, Skučaitė- Bingelė Kristina, Štikonas ArtūrasAbstract:In this paper, the Sturm–Liouville Problem with one classical first type boundary condition and other nonlocal integral boundary conditions of two cases is investigated. We analyze how complex eigenvalues of these Problems depend on the parameters of nonlocal integral boundary conditions. Some new results are given on complex spectra of these Problems. Many results are presented as graphs of complex characteristic functionsInformatikos fakultetasMatematikos ir informatikos institutasMatematikos ir statistikos katedraVilniaus universitetasVytauto Didžiojo universiteta
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Spectrum curves for a discrete Sturm--Liouville Problem with one integral boundary condition
'Vilnius University Press', 2019Co-Authors: Bingelė Kristina, Bankauskienė Agnė, Štikonas ArtūrasAbstract:This paper presents new results on the spectrum on complex plane for discrete Sturm–Liouville Problem with one integral type nonlocal boundary condition depending on three parameters: γ, ξ_1 and ξ_2. The integral condition is approximated by the trapezoidal rule. The dependence on parameter γ is investigated by using characteristic function method and analysing spectrum curves which gives qualitative view of the spectrum for fixed ξ_1 = m_1 / n and ξ_2 = m_2 / n, where n is discretisation parameter. Some properties of the spectrum curves are formulated and illustrated in figures for various ξ_1 and ξ_2
A. M. Akhtyamov - One of the best experts on this subject based on the ideXlab platform.
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Degenerate Boundary Conditions for the Sturm-Liouville Problem on a Geometric Graph
Differential Equations, 2019Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:We study the boundary conditions of the Sturm-Liouville Problem posed on a star-shaped geometric graph consisting of three edges with a common vertex. We show that the Sturm-Liouville Problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville Problem cannot be a nonzero constant and the set of Sturm-Liouville Problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two Problems as in the case of the Sturm-Liouville Problem on an interval.
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Inverse Sturm-Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph
Differential Equations, 2019Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:The inverse Sturm-Liouville Problem with nonseparated boundary conditions on a star-shaped geometric graph consisting of three edges with a common vertex is studied. It is shown that the Sturm-Liouville Problem with general boundary conditions cannot be reconstructed uniquely from four spectra. A class of nonseparated boundary conditions is obtained for which two uniqueness theorems for the solution of the inverse Sturm-Liouville Problem are proved. In the first theorem, the data used to reconstruct the Sturm-Liouville Problem are the spectrum of the boundary value Problem itself and the spectra of three auxiliary Problems with separated boundary conditions. In the second theorem, instead of the spectrum of the Problem itself, one only deals with five of its eigenvalues. It is shown that the Sturm-Liouville Problem with these nonseparated boundary conditions can be reconstructed uniquely if three spectra of auxiliary Problems and five eigenvalues of the Problem itself are used as the reconstruction data. Examples of unique reconstruction of potentials and boundary conditions of the Sturm-Liouville Problem posed on the graph under study are given.
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On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph
Doklady Mathematics, 2018Co-Authors: V. A. Sadovnichy, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:For the first time, the inverse Sturm–Liouville Problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville Problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville Problem is proved. The spectrum of the boundary value Problem itself and the spectra of three auxiliary Problems are used as reconstruction data. It is also shown that the Sturm–Liouville Problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary Problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the Problem.
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solvability theorems for an inverse nonself adjoint Sturm Liouville Problem with nonseparated boundary conditions
Differential Equations, 2015Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:We prove theorems on the solvability of the inverse Sturm-Liouville Problem with nonseparated conditions by two spectra and one eigenvalue and theorems on the unique solvability by two spectra and three eigenvalues. We find exact and approximate solutions of the inverse Problems. Related examples and counterexample are given.
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General Inverse Sturm-Liouville Problem with Symmetric Potential
Azerbaijan Journal of Mathematics, 2015Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:The uniqueness theorems for an inverse nonselfadjoint Sturm-Liouville Problem withsymmetric potential and general boundary conditions are proved. The spectral data using forunique reconstruction of Sturm-Liouville Problems are a spectrum and six eigenvalues. The uniquenesstheorems for an inverse selfadjoint Sturm-Liouville Problem with symmetric potential andnonseparated boundary conditions are proved also. These theorems use a spectrum and two (orthree) eigenvalues for unique reconstruction of Sturm-Liouville Problems. The theorems generaliseG. Borg and N.Levinson classical results to the case Sturm-Liouville Problem with general boundaryconditions. Schemes for unique reconstruction of the Sturm-Liouville Problems with symmetricpotential and general boundary conditions are given.
Nihat Altınışık - One of the best experts on this subject based on the ideXlab platform.
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Sturm-Liouville Problem with moving discontinuity points
Boundary Value Problems, 2015Co-Authors: Fatma Hıra, Nihat AltınışıkAbstract:In this paper, we present a new discontinuous Sturm-Liouville Problem with symmetrically located discontinuities which are defined depending on a parameter in the neighborhood of an interior point in the interval. Also the Problem contains an eigenparameter in a boundary condition. We investigate some spectral properties of the eigenvalues, obtain asymptotic formulae for the eigenvalues and the corresponding eigenfunctions and construct Green’s function for the Problem. We give an illustrative example with tables and figures at the end of the paper.
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Sturm Liouville Problem with discontinuity conditions at several points
2012Co-Authors: Fatma Hıra, Nihat AltınışıkAbstract:In this paper we deal with the computation of the eigenvalues of Sturm Liouville Problem with several discontinuity conditions (transmission conditions) inside a finite interval and parameter dependent boundary condition. By using an operator theoretic interpretation we extend some classic results for regular Sturm Liouville Problems. A symmetric linear operator A is defined in an appropriate Hilbert space such that the eigenvalues of such a Problem coincide with those of A. Also, we obtained asymptotic formulaes for the eigenvalues and corresponding eigenfunctions.
