Abdullah Kablan - Academia.edu (original) (raw)

Papers by Abdullah Kablan

Nucleation and Atmospheric Aerosols, 2018

Journal of Applied Analysis and Computation, 2023

DOAJ (DOAJ: Directory of Open Access Journals), Oct 1, 2013

We consider an inverse problem of the scattering theory for energydependent Sturm-Liouville equat... more We consider an inverse problem of the scattering theory for energydependent Sturm-Liouville equations on the half line [0, +∞) with point δinteraction and eigenparameter-dependent boundary condition. We define the scattering data of the problem first, then consider the basic equation and study an algorithm for finding the potentials with the given scattering data.

International journal of applied mathematics and statistics, Apr 14, 2018

The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville probl... more The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville problems in which the discontinuity of its derivative is not determined beforehand but occurs on its own. This paper seeks to extend that idea and construct the Green’s function for finitely many intervals case. We consider the second order scalar differential equation with its boundary conditions and convert it to its equivalent first order linear system. From this conversion, we formulate the characteristic function whose zeros are the eigenvalues of the homogeneous system. In addition, we construct the generalized matrix Green’s function from which we get the top right component as the Green’s function for finitely-many interval Sturm-Liouville problem.

International Journal of Analysis and Applications, 2018

The purpuse of this article is to show the matrix representations of Sturm-Liouville operators wi... more The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented. 1. Introduction Acording to classical spectral theory, a Sturm-Liouville problem (SLP) consisting of the equation −(py) + qy = λwy, on J = (a, b) and boundary conditions has infinite spectrum under some assumptions. Atkinson in his book [1] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues. Then in [2], Kong, Wu and Zettl obtained the following result: For every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint SLP with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals. Recently, there has been much attention paid to the SLPs with finite spectrum. For a comprehensive treatment of the subject we refer the reader to the book by Zettl [3], and the papers by Kong, Wu and Zettl [2], Ao, Sun, and Zhang [4], [5] and Ao, Bo and Sun [6], [7]. In 2009, the equivalence of SLP with

Miskolc Mathematical Notes, 2017

This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions a... more This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions and eigenvalue parameter contained in the boundary condition. Sturm-Liouville problem with discontinuities at one or two points and its different variants have already been investigated. In this study we extend these results to a finite number of point ı interactions case. The crucial part of this study is the using graph demonstration to obtain asymptotic representation of solutions.

Mediterranean Journal of Mathematics, Nov 8, 2014

In this study, we construct a certain class of matrix eigenvalue problems correspond to a class o... more In this study, we construct a certain class of matrix eigenvalue problems correspond to a class of regular fourth-order boundary value problems with transmission conditions of Atkinson type. The relation between boundary value problem and matrix eigenvalue problem is they have exactly the same eigenvalues.

Abstract and Applied Analysis, 2013

This paper deals with a Dirac system with transmission condition and eigenparameter in boundary c... more This paper deals with a Dirac system with transmission condition and eigenparameter in boundary condition. We give an operatortheoretic formulation of the problem then investigate the existence of the solution. Some spectral properties of the problem are studied.

Journal of Applied Analysis & Computation

One way to study the spectral properties of Sturm-Liouville operators is difference equations. Th... more One way to study the spectral properties of Sturm-Liouville operators is difference equations. The coefficients of the second order difference equation which is equivalent Sturm-Liouville equation can be written as a tridiagonal matrix. One investigation area for tridiagonal matrix is finding eigenvalues, eigenvectors and normalized numbers. To determine these datas, we use the solutions of the second order difference equation and this investigation is called direct spectral problem. Furthermore, reconstruction of matrix according to some arguments is called inverse spectral problem. There are many methods to solve inverse spectral problems according to selecting the datas which are generalized spectral function, spectral data of the matrix and two spectra of the matrix. In this article, we study discrete form the Sturm-Liouville equation with generalized function potential and we will focus on the inverse spectral problems of second order difference equation for spectral data and t...

Fractals

Prüfer transformation is more effective and flexible in studying the spectral analysis of boundar... more Prüfer transformation is more effective and flexible in studying the spectral analysis of boundary value problem than using the classical methods in operator theory. The goal of this paper is to study Prüfer approach to spectral analysis of periodic Sturm–Liouville problem with transmission condition. Since we are dealing with a singular problem, the characteristic function we obtained is a piecewise function. At the end of the study, the existence of eigenvalues of investigated problem by using Prüfer transformation is given.

In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic p... more In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic pencil of Strum-Liouville operarors with periodic generalized potential.

The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δinteract... more The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δinteractions. Such an equation gives us a Sturm-Liouville boundary value problem which has n transmission conditions. We show that for any positive numbers m j (j = 0, 1, ..., n) that are related to number of partition of the intervals between two successive interaction points, we can construct a Sturm-Liouville equations with δ-interactions, which have exactly d eigenvalues. Where d is the sum of m j 's.

International journal of applied mathematics and statistics, 2018

The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville probl... more The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville problems in which the discontinuity of its derivative is not determined beforehand but occurs on its own. This paper seeks to extend that idea and construct the Green’s function for finitely many intervals case. We consider the second order scalar differential equation with its boundary conditions and convert it to its equivalent first order linear system. From this conversion, we formulate the characteristic function whose zeros are the eigenvalues of the homogeneous system. In addition, we construct the generalized matrix Green’s function from which we get the top right component as the Green’s function for finitely-many interval Sturm-Liouville problem.

International Journal of Analysis and Applications, 2018

The purpuse of this article is to show the matrix representations of Sturm-Liouville operators wi... more The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented. 1. Introduction Acording to classical spectral theory, a Sturm-Liouville problem (SLP) consisting of the equation −(py) + qy = λwy, on J = (a, b) and boundary conditions has infinite spectrum under some assumptions. Atkinson in his book [1] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues. Then in [2], Kong, Wu and Zettl obtained the following result: For every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint SLP with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals. Recently, there has been much attention paid to the SLPs with finite spectrum. For a comprehensive treatment of the subject we refer the reader to the book by Zettl [3], and the papers by Kong, Wu and Zettl [2], Ao, Sun, and Zhang [4], [5] and Ao, Bo and Sun [6], [7]. In 2009, the equivalence of SLP with

AIP Conference Proceedings, 2018

Miskolc Mathematical Notes, 2017

This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions a... more This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions and eigenvalue parameter contained in the boundary condition. Sturm-Liouville problem with discontinuities at one or two points and its different variants have already been investigated. In this study we extend these results to a finite number of point ı interactions case. The crucial part of this study is the using graph demonstration to obtain asymptotic representation of solutions.

In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic p... more In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic pencil of Strum-Liouville operarors with periodic generalized potential.

Advances in Difference Equations, 2016

In this work, we study the inverse problem for difference equations which are constructed by the ... more In this work, we study the inverse problem for difference equations which are constructed by the Sturm-Liouville equations with generalized function potential from the generalized spectral function (GSF). Some formulas are given in order to obtain the matrix J, which need not be symmetric, by using the GSF and the structure of the GSF is studied.

Electronic Journal of Qualitative Theory of Differential Equations, 2013

In this article we obtain the eigenfunction expansions of a quadratic pencil of SturmLiouville op... more In this article we obtain the eigenfunction expansions of a quadratic pencil of SturmLiouville operators with periodic coecients. The important point to note here is the given potential is a rst order generalized function. 2010 Mathematics Subject Classication. 34L10, 47A10, 47E05. Key words and phrases. Quadratic pencil of dierential operators; spectral analysis; periodic point δ-interactions, eigenfunction expansions.

Nucleation and Atmospheric Aerosols, 2018

Journal of Applied Analysis and Computation, 2023

DOAJ (DOAJ: Directory of Open Access Journals), Oct 1, 2013

We consider an inverse problem of the scattering theory for energydependent Sturm-Liouville equat... more We consider an inverse problem of the scattering theory for energydependent Sturm-Liouville equations on the half line [0, +∞) with point δinteraction and eigenparameter-dependent boundary condition. We define the scattering data of the problem first, then consider the basic equation and study an algorithm for finding the potentials with the given scattering data.

International journal of applied mathematics and statistics, Apr 14, 2018

The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville probl... more The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville problems in which the discontinuity of its derivative is not determined beforehand but occurs on its own. This paper seeks to extend that idea and construct the Green’s function for finitely many intervals case. We consider the second order scalar differential equation with its boundary conditions and convert it to its equivalent first order linear system. From this conversion, we formulate the characteristic function whose zeros are the eigenvalues of the homogeneous system. In addition, we construct the generalized matrix Green’s function from which we get the top right component as the Green’s function for finitely-many interval Sturm-Liouville problem.

International Journal of Analysis and Applications, 2018

The purpuse of this article is to show the matrix representations of Sturm-Liouville operators wi... more The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented. 1. Introduction Acording to classical spectral theory, a Sturm-Liouville problem (SLP) consisting of the equation −(py) + qy = λwy, on J = (a, b) and boundary conditions has infinite spectrum under some assumptions. Atkinson in his book [1] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues. Then in [2], Kong, Wu and Zettl obtained the following result: For every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint SLP with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals. Recently, there has been much attention paid to the SLPs with finite spectrum. For a comprehensive treatment of the subject we refer the reader to the book by Zettl [3], and the papers by Kong, Wu and Zettl [2], Ao, Sun, and Zhang [4], [5] and Ao, Bo and Sun [6], [7]. In 2009, the equivalence of SLP with

Miskolc Mathematical Notes, 2017

This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions a... more This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions and eigenvalue parameter contained in the boundary condition. Sturm-Liouville problem with discontinuities at one or two points and its different variants have already been investigated. In this study we extend these results to a finite number of point ı interactions case. The crucial part of this study is the using graph demonstration to obtain asymptotic representation of solutions.

Mediterranean Journal of Mathematics, Nov 8, 2014

In this study, we construct a certain class of matrix eigenvalue problems correspond to a class o... more In this study, we construct a certain class of matrix eigenvalue problems correspond to a class of regular fourth-order boundary value problems with transmission conditions of Atkinson type. The relation between boundary value problem and matrix eigenvalue problem is they have exactly the same eigenvalues.

Abstract and Applied Analysis, 2013

This paper deals with a Dirac system with transmission condition and eigenparameter in boundary c... more This paper deals with a Dirac system with transmission condition and eigenparameter in boundary condition. We give an operatortheoretic formulation of the problem then investigate the existence of the solution. Some spectral properties of the problem are studied.

Journal of Applied Analysis & Computation

One way to study the spectral properties of Sturm-Liouville operators is difference equations. Th... more One way to study the spectral properties of Sturm-Liouville operators is difference equations. The coefficients of the second order difference equation which is equivalent Sturm-Liouville equation can be written as a tridiagonal matrix. One investigation area for tridiagonal matrix is finding eigenvalues, eigenvectors and normalized numbers. To determine these datas, we use the solutions of the second order difference equation and this investigation is called direct spectral problem. Furthermore, reconstruction of matrix according to some arguments is called inverse spectral problem. There are many methods to solve inverse spectral problems according to selecting the datas which are generalized spectral function, spectral data of the matrix and two spectra of the matrix. In this article, we study discrete form the Sturm-Liouville equation with generalized function potential and we will focus on the inverse spectral problems of second order difference equation for spectral data and t...

Fractals

Prüfer transformation is more effective and flexible in studying the spectral analysis of boundar... more Prüfer transformation is more effective and flexible in studying the spectral analysis of boundary value problem than using the classical methods in operator theory. The goal of this paper is to study Prüfer approach to spectral analysis of periodic Sturm–Liouville problem with transmission condition. Since we are dealing with a singular problem, the characteristic function we obtained is a piecewise function. At the end of the study, the existence of eigenvalues of investigated problem by using Prüfer transformation is given.

In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic p... more In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic pencil of Strum-Liouville operarors with periodic generalized potential.

The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δinteract... more The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δinteractions. Such an equation gives us a Sturm-Liouville boundary value problem which has n transmission conditions. We show that for any positive numbers m j (j = 0, 1, ..., n) that are related to number of partition of the intervals between two successive interaction points, we can construct a Sturm-Liouville equations with δ-interactions, which have exactly d eigenvalues. Where d is the sum of m j 's.

International journal of applied mathematics and statistics, 2018

The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville probl... more The Green’s function has been constructed for one-interval and two-interval Sturm-Liouville problems in which the discontinuity of its derivative is not determined beforehand but occurs on its own. This paper seeks to extend that idea and construct the Green’s function for finitely many intervals case. We consider the second order scalar differential equation with its boundary conditions and convert it to its equivalent first order linear system. From this conversion, we formulate the characteristic function whose zeros are the eigenvalues of the homogeneous system. In addition, we construct the generalized matrix Green’s function from which we get the top right component as the Green’s function for finitely-many interval Sturm-Liouville problem.

International Journal of Analysis and Applications, 2018

The purpuse of this article is to show the matrix representations of Sturm-Liouville operators wi... more The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented. 1. Introduction Acording to classical spectral theory, a Sturm-Liouville problem (SLP) consisting of the equation −(py) + qy = λwy, on J = (a, b) and boundary conditions has infinite spectrum under some assumptions. Atkinson in his book [1] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues. Then in [2], Kong, Wu and Zettl obtained the following result: For every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint SLP with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals. Recently, there has been much attention paid to the SLPs with finite spectrum. For a comprehensive treatment of the subject we refer the reader to the book by Zettl [3], and the papers by Kong, Wu and Zettl [2], Ao, Sun, and Zhang [4], [5] and Ao, Bo and Sun [6], [7]. In 2009, the equivalence of SLP with

AIP Conference Proceedings, 2018

Miskolc Mathematical Notes, 2017

This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions a... more This paper deals with the Sturm-Liouville equation with a finite number of point ı interactions and eigenvalue parameter contained in the boundary condition. Sturm-Liouville problem with discontinuities at one or two points and its different variants have already been investigated. In this study we extend these results to a finite number of point ı interactions case. The crucial part of this study is the using graph demonstration to obtain asymptotic representation of solutions.

In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic p... more In this paper, we obtain conditions for the absence of spectral gaps in spectrum of a quadratic pencil of Strum-Liouville operarors with periodic generalized potential.

Advances in Difference Equations, 2016

In this work, we study the inverse problem for difference equations which are constructed by the ... more In this work, we study the inverse problem for difference equations which are constructed by the Sturm-Liouville equations with generalized function potential from the generalized spectral function (GSF). Some formulas are given in order to obtain the matrix J, which need not be symmetric, by using the GSF and the structure of the GSF is studied.

Electronic Journal of Qualitative Theory of Differential Equations, 2013

In this article we obtain the eigenfunction expansions of a quadratic pencil of SturmLiouville op... more In this article we obtain the eigenfunction expansions of a quadratic pencil of SturmLiouville operators with periodic coecients. The important point to note here is the given potential is a rst order generalized function. 2010 Mathematics Subject Classication. 34L10, 47A10, 47E05. Key words and phrases. Quadratic pencil of dierential operators; spectral analysis; periodic point δ-interactions, eigenfunction expansions.