Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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Article Citation - WoS: 8Citation - Scopus: 8On Some Even-Sequential Fractional Boundary-Value Problems(Springernature, 2024) Ugurlu, EkinIn this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2-sequential and 4-sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2-sequential and 4-sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2-sequential and 4-sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2-sequential fractional differential equations. At the end of the paper we show that 6-sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems.Article Left-Definite System of First-Order Equations Together With Eigenparameter-Dependent Boundary Conditions(Wiley, 2024) Ugurlu, EkinThis paper provides some information on the eigenvalues and eigenfunctions of some left-definite system of first-order differential equations subject to eigenparameter-dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left-definite equation. Using some Prufer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Frechet derivatives of eigenvalues and eigenfunctions with respect to some elements of data.Article Citation - Scopus: 1On the Zeros of Solutions of Ordinary and Fractional Differential Equations(Wiley, 2023) Ugurlu, EkinThis paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second-order Sturm-Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.Article Citation - WoS: 1Citation - Scopus: 1A New Hamiltonian System(Academic Press inc Elsevier Science, 2020) Ugurlu, EkinThis paper aims to share a new first-order differential equation that contains the continuous analogous of the orthogonal polynomials on the unit-circle. We introduce some basic results on the system and solutions of the system. Using nested-circle approach we introduce the possible number of square-integrable solutions of the system. At the end of the paper we share a limit-point criteria for the two-dimensional system of equations. (C) 2020 Elsevier Inc. All rights reserved.Article Citation - WoS: 1Citation - Scopus: 1The Spectral Analysis of a System of First-Order Equations With Dissipative Boundary Conditions(Wiley, 2021) Ugurlu, EkinThis paper aims to share some completeness theorems related with a boundary value problem generated by a system of equations and non-self-adjoint (dissipative) boundary conditions. Indeed, we consider a system of equations that contains a continuous analogous of the orthogonal polynomials on the unit circle. Constructing the characteristic function of the related dissipative operator, we share some completeness theorems. Moreover, we give an explicit form of the self-adjoint dilation of the dissipative operator.Article The Characteristic Matrix Function of a Dissipative Hamiltonian Operator(Wiley, 2021) Ugurlu, EkinIn this paper, we consider a singular dissipative even-order Hamiltonian operator with a finite number of transmission conditions. Using coordinate-free approach, we construct the characteristic matrix-function of the Cayley transform of the dissipative operator. Using the equivalence of completeness property of root functions of Cayley transform and dissipative operator, we prove some completeness theorems. Moreover, we construct an explicit form of the resolvent operator of dissipative operator.Article Citation - WoS: 1Citation - Scopus: 1Direct Approach for the Characteristic Function of a Dissipative Operator With Distributional Potentials(Springer Basel Ag, 2020) Ugurlu, EkinThe main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class C-0. Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Caratheodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established.Article Citation - WoS: 11Citation - Scopus: 11Some Singular Third-Order Boundary Value Problems(Wiley, 2020) Ugurlu, EkinIn this paper, we consider some singular formally symmetric (self-adjoint) boundary value problems generated by a singular third-order differential expression and separated and coupled boundary conditions. In particular, we consider that the minimal symmetric operator generated by the third-order differential expression has the deficiency indices (3,3). We investigate same spectral properties related with these problems, and we introduce a method to find the resolvent operator.Article Citation - WoS: 9Citation - Scopus: 10Extensions of a Minimal Third-Order Formally Symmetric Operator(Malaysian Mathematical Sciences Soc, 2020) Ugurlu, EkinIn this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax-Phillips scattering theory and Sz.-Nagy-Foias characteristic function theory we prove a completeness theorem.Article Citation - WoS: 6Citation - Scopus: 7Regular Fifth-Order Boundary Value Problems(Springernature, 2020) Ugurlu, EkinThe main purpose of this paper is to introduce a method to handle some boundary value problems generated by fifth-order formally symmetric differential equation and separated, real-coupled and complex-coupled boundary conditions. Moreover, the continuity properties of the eigenvalues of these problems on some data are studied and some Frechet derivatives of the eigenvalues are introduced.