Papers by Kadriye AYDEMİR
Journal of Applied Analysis & Computation, 2020
In this paper we introduce to consideration a new type boundary value problems consisting of an "... more In this paper we introduce to consideration a new type boundary value problems consisting of an "Sturm-Liouville" equation on two disjoint intervals as −p(x)y ′′ + q(x)y + By|x = µy, x ∈ [a, c) ∪ (c, b] together with two end-point conditions whose coefficients depend linearly on the eigenvalue parameter, and two supplementary so-called transmission conditions, involving linearly left-hand and right-hand values of the solution and its derivatives at point of interaction x = c, where B : L2(a, c) ⊕ L2(c, b) → L2(a, c)⊕L2(c, b) is an abstract linear operator, non-selfadjoint in general. For self-adjoint realization of the pure differential part of the main problem we define "alternative" inner products in Sobolev spaces, "incorporating" with the boundary-transmission conditions. Then by suggesting an own approaches we establish such properties as topological isomorphism and coercive solvability of the corresponding nonhomogeneous problem and prove compactness of the resolvent operator in these Sobolev spaces. Finally, we prove that the spectrum of the considered eigenvalue problem is discrete and derive asymptotic formulas for the eigenvalues. Note that the obtained results are new even in the case when the equation is not involved an abstract linear operator B.
Bulletin of the Malaysian Mathematical Sciences Society
Qualitative Theory of Dynamical Systems
journal of new results in science, Aug 31, 2021
Cornell University - arXiv, Mar 27, 2013
The purpose of this paper is to extend some spectral properties of regular Sturm-Liouville proble... more The purpose of this paper is to extend some spectral properties of regular Sturm-Liouville problems to the special type discontinuous boundaryvalue problem, which consists of a Sturm-Liouville equation together with eigenparameterdependent boundary conditions and two supplementary transmission conditions. We construct the resolvent operator and Green' s function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space L 2 [a, b].
Cornell University - arXiv, Jul 14, 2015
In this paper we propose a new version of differential transform method (we shall call this metho... more In this paper we propose a new version of differential transform method (we shall call this method as α-parameterized differential transform method), which differs from the traditional differential transform method in calculating coefficients of Taylor polynomials. Numerical examples are presented to illustrate the efficiency and reliability of own method. The result reveal that α-Parameterized differential transform method is a simple and effective numerical algorithm.
Journal of New Theory, 2021
In this paper, we study the periodic Sturm-Liouville problem, defined on three non-intersecting i... more In this paper, we study the periodic Sturm-Liouville problem, defined on three non-intersecting intervals with four supplementary conditions which are imposed at two internal points of interaction, the so-called transmission conditions. We first prove that the eigenvalues are real and the system of eigenfunctions is an orthogonal system. Secondly, some auxiliary initial-value problems are defined and transmittal-characteristic function is constructed in terms of solutions of these initial-value problems. Finally, we establish that the eigenvalues of the considered problem are the zeros of the transmittal-characteristic function.
Cornell University - arXiv, Mar 27, 2013
The purpose of this study is to investigate a new class of boundary value transmission problems (... more The purpose of this study is to investigate a new class of boundary value transmission problems (BVTP's) for Sturm-Liouville equation on two separate intervals. We introduce modified inner product in direct sum space L 2 [a, c) ⊕ L 2 (c, b] ⊕ C 2 and define symmetric linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then by suggesting an own approaches we construct Green's function for problem under consideration and find the resolvent function for corresponding inhomogeneous problem.
Cornell University - arXiv, Mar 27, 2013
One important innovation here is that for the Sturm-Liouville considered equation together with e... more One important innovation here is that for the Sturm-Liouville considered equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at one interior point. We develop Green's function method for spectral analysis of the considered problem in modified Hilbert space.
journal of new results in science, 2012
This paper deals with the asymptotic estimates of eigenvalues for SturmLiouville problems with tw... more This paper deals with the asymptotic estimates of eigenvalues for SturmLiouville problems with two supplementary transmission conditions at one interior point. By modifying some known techniques the existence and uniqueness results of solutions are obtained for the considered problem
In this study by modifying some techniques of classical Sturm-Liouville theory and suggesting own... more In this study by modifying some techniques of classical Sturm-Liouville theory and suggesting own approaches we investigate some spectral properties of one Sturm-Liouville type problem on two disjoint intervals.
Journal of Applied Analysis and Computation, 2018
By modifing the Green’s function method we study certain spectral aspects of discontinuous Sturm-... more By modifing the Green’s function method we study certain spectral aspects of discontinuous Sturm-Liouville problems with interior singularities. Firstly, we define four eigen-solutions and construct the Green’s function in terms of them. Based on the Green’s function we establish the uniform convergeness of generalized Fourier series as eigenfunction expansion in the direct sum of Lebesgue spaces L2 where the usual inner product replaced by new inner product. Finally, we extend and generalize such important spectral properties as Parseval equation, Rayleigh quotient and Rayleigh-Ritz formula (minimization principle) for the considered problem.
Rendiconti del Circolo Matematico di Palermo Series 2, 2021
Mathematical Methods in the Applied Sciences, 2021
The aim of this study is to generalize some important spectral properties of classical periodic S... more The aim of this study is to generalize some important spectral properties of classical periodic Sturm–Liouville problems to two‐linked periodic problems with additional transmission conditions at an interior point of interaction, which have not been studied previously. Although the classical periodic Sturm–Liouville problem has infinitely many real eigenvalues, there are two‐linked periodic Sturm–Liouville problems that have only a finite number of real eigenvalues. To show this, we construct a simple example of two‐linked periodic Sturm–Liouville problem with transmission conditions having only one real eigenvalue. Using our own approach, we define the characteristic determinant of a new type, find simple conditions on the coefficients of the transmission conditions that guarantee the existence of an infinite number of real eigenvalues, and derive an asymptotic formula for the eigenvalues.
Mathematical Modelling and Analysis, 2021
This work is aimed at studying some comparison and oscillation properties of boundary value probl... more This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.
Contemporary Analysis and Applied Mathematics, 2016
In this paper we present an operator-theoretic framework for the study of Sturm-Liouville system ... more In this paper we present an operator-theoretic framework for the study of Sturm-Liouville system on two separated interval together with boundary and interaction conditions. By applying the Green's function method we establish such spectral properties, as completeness of the eigenfunctions and Parseval's equality Based on these results and by using variational methods we find some extremal properties of the eigenvalues and eigenfunctions. Finally we establish the monotonicity theorem for the eigenvalues.
Contemporary Analysis and Applied Mathematics, 2012
We develop a spectral method to solve inhomogeneous boundary-value problems with singular interio... more We develop a spectral method to solve inhomogeneous boundary-value problems with singular interior point. Particularly, we derive an eigenfunction expansion formula for the Green's function for which the modified Parseval and Carleman equalities are found.
AIP Conference Proceedings, 2015
In this study we give a comprehensive treatment for a class BVP’s together with transmission (imp... more In this study we give a comprehensive treatment for a class BVP’s together with transmission (impulsive, jump or interface) conditions at the one interior singular point. A self-adjoint linear operator ℛ is defined in a suitable Hilbert space ℋ such that the eigenvalues of such a problem coincide with those of ℋ and showed that it has a compact resolvent. We construct fundamental solutions and discuss some properties of spectrum. Moreover, we obtain the representation of Green’s function for the considered problem.
This paper is devoted to the derivation of expansion a associated with a discontinuous Sturm-Liou... more This paper is devoted to the derivation of expansion a associated with a discontinuous Sturm-Liouville problems defined on $[-\pi, 0)\cup(0,\pi]$. We derive an eigenfunction expansion theorem for the Green's function of the problem as well as a theorem of uniform convergence of a certain class of functions.
Journal of Applied Analysis, 2022
The main goal of this study is the investigation of discontinuous boundary-value problems for sec... more The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions. We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.
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Papers by Kadriye AYDEMİR