Matematik Bölümü

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Citation - WoS: 2
Citation - Scopus: 3
Analysis of Mixed-Order Caputo Fractional System With Nonlocal Integral Boundary Condition
(Tubitak Scientific & Technological Research Council Turkey, 2018) Khodabakhshi, Neda; Baleanu, Dumitru; Akman Yildiz, Tugba
This paper deals with a mixed-order Caputo fractional system with nonlocal integral boundary conditions. This study can be considered as an extension of previous studies, since the orders of the equations lie on different intervals. We discuss the existence and uniqueness of the solution using fixed point methods. We enrich the study with an example.
Citation - Scopus: 26
A Caputo-Fabrizio Fractional-Order Cholera Model and Its Sensitivity Analysis
(Mehmet Yavuz, 2023) Akgül, A.; Jarad, F.; Kumam, P.; Nonlaopon, K.; Ahmed, I.
In recent years, the availability of advanced computational techniques has led to a growing emphasis on fractional-order derivatives. This development has enabled researchers to explore the intricate dynamics of various biological models by employing fractional-order derivatives instead of traditional integer-order derivatives. This paper proposes a Caputo-Fabrizio fractional-order cholera epidemic model. Fixed-point theorems are utilized to investigate the existence and uniqueness of solutions. A recent and effective numerical scheme is employed to demonstrate the model’s complex behaviors and highlight the advantages of fractional-order derivatives. Additionally, a sensitivity analysis is conducted to identify the most influential parameters. © 2023 by the authors.
Citation - WoS: 16
Citation - Scopus: 20
Common Fixed Point Theorems in Cone Banach Spaces
(Hacettepe Univ, Fac Sci, 2011) Abdeljawad, Thabet; Tas, Kenan; Karapınar, Erdal
Recently, E. Karapınar (Fixed Point Theorems in Cone Banach Spaces, Fixed Point Theory Applications, Article ID 609281, 9 pages, 2009) presented some fixed point theorems for self-mappings satisfying certain contraction principles on a cone Banach space. Here we will give some generalizations of this theorem.
Citation - WoS: 6
The Complementary Nabla Bennett-Leindler Type Inequalities
(Ankara Univ, Fac Sci, 2022) Kayar, Zeynep; Kaymakcalan, Billur
We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from 0 < zeta < 1 to zeta > 1. Different from the literature, the directions of the new inequalities, where zeta > 1, are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for 0 < zeta < 1. By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents. The dual results for the delta approach and the special cases for the discrete and continuous ones are obtained as well. Some of our results are novel even in the special cases.
Citation - WoS: 27
Citation - Scopus: 26
Completion of Cone Metric Spaces
(Hacettepe Univ, Fac Sci, 2010) Abdeljawad, Thabet
In this paper a completion theorem for cone metric spaces and a com- pletion theorem for cone normed spaces are proved. The completion spaces are defined by means of an equivalence relation obtained by convergence via the scalar norm of the Banach space E.
Coset Algebras of the Maxwell-Einstein Supergravities
(Tubitak Scientific & Technological Research Council Turkey, 2006) Yılmaz, Nejat Tevfik
The general structure of the scalar cosets of the Maxwell-Einstein supergravities is given. Following an introduction of the non-linear coset formalism of the supergravity theories a comparison of the coset algebras of the Maxwell-Einstein supergravities in various dimensions is discussed.
Citation - WoS: 2
Citation - Scopus: 6
Diamond Alpha Hardy-Copson Type Dynamic Inequalities
(Hacettepe Univ, Fac Sci, 2022) Kaymakcalan, Billur; Kayar, Zeynep
In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.
Edelstein-Type Fixed Point Theorems in Compact Tvs-Cone Metric Spaces
(Hacettepe University, 2014) Abdeljawad, Thabet
In this paper we prove two fixed point theorems in compact cone metricspaces over normal cones. The first theorem generalizes Edelstein theorem [8] and is different from the generalization obtained in [11]. Thesecond theorem generalizes the main result in [10] and the first theorem.However, the two theorems fail in different categories. Moreover, different versions of the two theorems are proved in TVS-cone metric spacesby making use of the nonlinear scalarization function used very recentlyby Wei-Shih Du in [A note on cone metric fixed point theory and itsequivalence, Nonlinear Analysis,72(5),2259-2261 (2010).] to prove theequivalence of the Banach contraction principle in cone metric spacesand usual metric spaces.
Citation - Scopus: 17
Evolutionary Mathematical Science, Fractional Modeling and Artificial Intelligence of Nonlinear Dynamics in Complex Systems
(Akif AKGUL, 2022) Karaca, Yeliz; Baleanu, Dumitru
Complex problems in nonlinear dynamics foreground the critical support of artificial phenomena so that each domain of complex systems can generate applicable answers and solutions to the pressing challenges. This sort of view is capable of serving the needs of different aspects of complexity by minimizing the problems of complexity whose solutions are based on advanced mathematical foundations and analogous algorithmic models consisting of numerous applied aspects of complexity. Evolutionary processes, nonlinearity and all the other dimensions of complexity lie at the pedestal of time, reveal time and occur within time. In the ever-evolving landscape and variations, with causality breaking down, the idea of complexity can be stated to be a part of unifying and revolutionary scientific framework to expound complex systems whose behavior is perplexing to predict and control with the ultimate goal of attaining a global understanding related to many branches of possible states as well as high-dimensional manifolds, while at the same time keeping abreast with actuality along the evolutionary and historical path, which itself, has also been through different critical points on the manifold. In view of these, we put forth the features of complexity of varying phenomena, properties of evolution and adaptation, memory effects, nonlinear dynamic system qualities, the importance of chaos theory and applications of related aspects in this study. In addition, processes of fractional dynamics, differentiation and systems in complex systems as well as the dynamical processes and dynamical systems of fractional order with respect to natural and artificial phenomena are discussed in terms of their mathematical modeling. Fractional calculus and fractional-order calculus approach to provide novel models with fractional-order calculus as employed in machine learning algorithms to be able to attain optimized solutions are also set forth besides the justification of the need to develop analytical and numerical methods. Subsequently, algorithmic complexity and its goal towards ensuring a more effective handling of efficient algorithms in computational sciences is stated with regard to the classification of computational problems. We further point out the neural networks, as descriptive models, for providing the means to gather, store and use experiential knowledge as well as Artificial Neural Networks (ANNs) in relation to their employment for handling experimental data in different complex domains. Furthermore, the importance of generating applicable solutions to problems for various engineering areas, medicine, biology, mathematical science, applied disciplines and data science, among many others, is discussed in detail along with an emphasis on power of predictability, relying on mathematical sciences, with Artificial Intelligence (AI) and machine learning being at the pedestal and intersection with different fields which are characterized by complex, chaotic, nonlinear, dynamic and transient components to validate the significance of optimized approaches both in real systems and in related realms.
Citation - WoS: 7
Citation - Scopus: 7
Fourth Order Differential Operators With Distributional Potentials
(Tubitak Scientific & Technological Research Council Turkey, 2020) Bairamov, Elgiz; Ugurlu, Ekin
In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.
Citation - WoS: 13
Citation - Scopus: 15
Hardy-Copson Type Inequalities for Nabla Time Scale Calculus
(Tubitak Scientific & Technological Research Council Turkey, 2021) Kaymakcalan, Billur; Kayar, Zeynep
This paper is devoted to the nabla unification of the discrete and continuous Hardy?Copson type inequalities. Some of the obtained inequalities are nabla counterparts of their delta versions while the others are new even for the discrete, continuous, and delta cases. Moreover, these dynamic inequalities not only generalize and unify the related ones in the literature but also improve them in the special cases.
Citation - WoS: 3
Citation - Scopus: 3
Left-Definite Hamiltonian Systems and Corresponding Nested Circles
(Tubitak Scientific & Technological Research Council Turkey, 2023) Ugurlu, Ekin
This work aims to construct the Titchmarsh-Weyl M(A)-theory for an even-dimensional left-definite Hamiltonian system. For this purpose, we introduce a suitable Lagrange formula and selfadjoint boundary conditions including the spectral parameter A. Then we obtain circle equations having nesting properties. Using the intersection point belonging to all the circles we share a lower bound for the number of Dirichlet-integrable solutions of the system.
Citation - WoS: 13
Citation - Scopus: 16
Modeling the Impact of Temperature on Fractional Order Dengue Model With Vertical Transmission
(Ramazan Yaman, 2020) Defterli, Ozlem
A dengue epidemic model with fractional order derivative is formulated to an-alyze the effect of temperature on the spread of the vector-host transmitted dengue disease. The model is composed of a system of fractional order differ-ential equations formulated within Caputo fractional operator. The stability of the equilibrium points of the considered dengue model is studied. The cor-responding basic reproduction number R alpha 0 is derived and it is proved that if R alpha 0 < 1, the disease-free equilibrium (DFE) is locally asymptotically stable. L1 method is applied to solve the dengue model numerically. Finally, numerical simulations are also presented to illustrate the analytical results showing the influence of the temperature on the dynamics of the vector-host interaction in dengue epidemics.
On a Fifth-Order Nonselfadjoint Boundary Value Problem
(Tubitak Scientific & Technological Research Council Turkey, 2021) Ugurlu, Ekin; Tas, Kenan
In this paper we aim to share a way to impose some nonselfadjoint boundary conditions for the solutions of a formally symmetric fifth-order differential equation. Constructing a dissipative operator related with the problem we obtain some informations on spectral properties of the problem. In particular, using coordinate-free approach we construct characteristic matrix-function related with the contraction which is obtained with the aid of the dissipative operator. In this paper we aim to share a way to impose some nonselfadjoint boundary conditions for the solutions of a formally symmetric fifth-order differential equation. Constructing a dissipative operator related with the problem we obtain some informations on spectral properties of the problem. In particular, using coordinate-free approach we construct characteristic matrix-function related with the contraction which is obtained with the aid of the dissipative
Citation - WoS: 5
Citation - Scopus: 5
On Some Fractional Operators Generated From Abel's Formula
(Tubitak Scientific & Technological Research Council Turkey, 2022) Ugurlu, Ekin
This work aims to share some fractional integrals and derivatives containing three real parameters. The main tool to introduce such operators is the corresponding Abel's equation. Solvability conditions for the Abel's equations are shared. Semigroup property for fractional integrals are introduced. Integration by parts rule is given. Moreover, mean value theorems and related results are shared. At the end of the paper, some directions for some fractional operators are given.
Citation - WoS: 10
Citation - Scopus: 9
On the Existence Interval for the Initial Value Problem of a Fractional Differential Equation
(Hacettepe Univ, Fac Sci, 2011) Mustafa, Octavian G.; Baleanu, Dumitru
We compute via a comparison function technique, a new bound for the existence interval of the initial value problem for a fractional differential equation given by means of Caputo derivatives. We improve in this way the estimate of the existence interval obtained very recently in the literature.
Citation - WoS: 8
Citation - Scopus: 11
On the Solutions of a Fractional Boundary Value Problem
(Tubitak Scientific & Technological Research Council Turkey, 2018) Ugurlu, Ekin; Baleanu, Dumitru; Tas, Kenan
This paper is devoted to showing the existence and uniqueness of solution of a regular second-order nonlinear fractional differential equation subject to the ordinary boundary conditions. The Banach fixed point theorem is used to prove the results.
Citation - WoS: 7
Citation - Scopus: 8
Representation for the Reproducing Kernel Hilbert Space Method for a Nonlinear System
(Hacettepe Univ, Fac Sci, 2019) Akgul, Ali; Khan, Yasir; Baleanu, Dumitru; Akgul, Esra Karatas
We apply the reproducing kernel Hilbert space method to a nonlinear system in this work. We utilize this technique to overcome the nonlinearity of the problem. We obtain accurate results. We demonstrate our results by tables and figures. We prove the efficiency of the method.
Citation - WoS: 1
Scattering and Characteristic Functions of a Dissipative Operator Generated by a System of Equations
(Tubitak Scientific & Technological Research Council Turkey, 2021) Ugurlu, Ekin; Bayram, Elgiz; Tas, Kenan
In this paper, we consider a system of first-order equations with the same eigenvalue parameter together with dissipative boundary conditions. Applying Lax-Phillips scattering theory and Sz.-Nagy-Foias model operator theory we prove a completeness theorem.
Citation - WoS: 17
Citation - Scopus: 16
A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel
(Hacettepe Univ, Fac Sci, 2018) Abdelkawy, Mohamed A.; Baleanu, Dumitru; Amin, Ahmed Z. M.; Bhrawy, Ali H.
This paper adapts a new numerical technique for solving twodimensional fractional integral equations with weakly singular. Using the spectral collocation method, the fractional operators of Legendre and Chebyshev polynomials, and Gauss-quadrature formula, we achieve a reduction of given problems into those of a system of algebraic equations. We apply the reported numerical method to solve several numerical examples in order to test the accuracy and validity. Thus, the novel algorithm is more responsible for solving two-dimensional fractional integral equations with weakly singular.

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