Browsing by Author "Yantir, Ahmet"

Now showing 1 - 15 of 15

Citation - Scopus: 1
Analysis on α-time scales and its applications to Cauchy-Euler equation
(Natural Sciences Publishing, 2024) Burcu Silindir; Seçil Gergün; Ahmet Yantir; Gergün, Seçil; Silindir, Burcu; Yantir, Ahmet
This article is devoted to present the α-power function calculus on α-time scale the α-logarithm and their applications on α-difference equations. We introduce the α-power function as an absolutely convergent infinite product. We state that the α-power function verifies the fundamentals of α-time scale and adheres to both the additivity and the power rule for α-derivative. Next we propose an α-analogue of Cauchy-Euler equation whose coefficient functions are α-polynomials and then construct its solution in terms of α-power function. As illustration we present examples of the second order α-Cauchy-Euler equation. Consequently we construct α-analogue of logarithm function which is determined in terms of α-integral. Finally we propose a second order BVP for α-Cauchy-Euler equation with two point unmixed boundary conditions and compute its solution by the use of Green’s function. © 2024 Elsevier B.V. All rights reserved.
Citation - Scopus: 8
Bessel equation and Bessel function on $\\mathbb{T}(q h)$
(Tubitak, 2022) Ahmet Yantir; BURCU SILINDIR YANTIR; Zehra TUNCER; Yantır, Burcu Sılındır; Tuncer, Zehra; Yantir, Ahmet
This article is devoted to present nabla $(q h)$ -analogues of Bessel equation and Bessel function. In order to construct series solution of nabla $(q h)$ -Bessel equation we present nabla $(q h)$ -analysis regarding nabla generalized quantum binomial nabla $(q h)$ -analogues of Taylor’s formula Gauss’s binomial formula Taylor series analytic functions analytic exponential function with its fundamental properties analytic trigonometric and hyperbolic functions. We emphasize that nabla $(q h)$ -Bessel equation recovers classical $h-and q-discrete$ Bessel equations. In addition we establish nabla $(q h)4 -Bessel function which is expressed in terms of an absolutely convergent series in nabla generalized quantum binomials and intimately demonstrate its reductions. Finally we develop modified nabla $(q h)$ -Bessel equation modified nabla $(q h)$ -Bessel function and its relation with nabla $(q h)$ -Bessel function.
Citation - WoS: 7
Bessel equation and Bessel function on T(q-h)
(Tubitak Scientific & Technological Research Council Turkey, 2022) Ahmet Yantir; Burcu Silindir Yantir; Zehra Tuncer; Yantir, Burcu Silindir; Tuncer, Zehra; Yantir, Ahmet
This article is devoted to present nabla (q h)-analogues of Bessel equation and Bessel function. In order to construct series solution of nabla (q h)-Bessel equation we present nabla (q h)-analysis regarding nabla generalized quantum binomial nabla (q h)-analogues of Taylor's formula Gauss's binomial formula Taylor series analytic functions analytic exponential function with its fundamental properties analytic trigonometric and hyperbolic functions. We emphasize that nabla (q h)-Bessel equation recovers classical h- and q-discrete Bessel equations. In addition we establish nabla (q h)-Bessel function which is expressed in terms of an absolutely convergent series in nabla generalized quantum binomials and intimately demonstrate its reductions. Finally we develop modified nabla (q h)-Bessel equation modified nabla (q h)-Bessel function and its relation with nabla (q h)-Bessel function.
Citation - WoS: 2
Citation - Scopus: 2
Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces
(De Gruyter Open Ltd peter.golla@degruyter.com, 2015) Ahmet Yantir; Ireneusz Kubiaczyk; Aneta Sikorska-Nowak; Yantir, Ahmet; Kubiaczyk, Ireneusz; Sikorska-Nowak, Aneta
In this paper we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose we introduce an equivalent integral operator to the SLBVP by means of Green's function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness we prove the existence of the fixed points of the equivalent integral operator. Mönch's fixed point theorem is used to prove the main result. Finally we also remark that it is straightforward to guarantee the existence of Carathéodory solutions for the SLBVP if Kuratowski measure of noncompactness is replaced by any axiomatic measure of noncompactness. © 2016 Elsevier B.V. All rights reserved.
Citation - WoS: 12
EXISTENCE OF SOLUTIONS OF THE DYNAMIC CAUCHY PROBLEM IN BANACH SPACES
(DE GRUYTER POLAND SP Z O O, 2012) Mieczyslaw Cichon; Ireneusz Kubiaczyk; Aneta Sikorska-Nowak; Ahmet Yantir; Yantir, Ahmet; Cichon, Mieczyslaw; Kubiaczyk, Ireneusz; Sikorska-Nowak, Aneta
In this paper we obtain the existence of solutions and Caratheodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales x(triangle)(t) = f(tx(t)) x(0)=x(o) t is an element of I-a where f is continuous or f satisfies Caratheodory conditions and some conditions expressed in terms of measures of noncompactness. The Manch fixed point theorem is used to prove the main result which extends these obtained for real valued functions.
Citation - Scopus: 2
Existence of solutions to fractional differential inclusions with p-laplacian operator
(Texas State University - San Marcos editor@ejde.math.txstate.edu, 2014) Ahmet Yantir; Fatma Serap Topal; Topal, Fatma Serap; Yantir, Ahmet
In this article we are prove the existence of solutions for three-point fractional differential inclusions with p-Laplacian operator. We use fixed point theory for set valued upper semi-continuous maps for obtaining the solutions. © 2014 Elsevier B.V. All rights reserved.
Citation - WoS: 6
Citation - Scopus: 7
Gauss's Binomial Formula and Additive Property of Exponential Functions on T(q-h)
(UNIV NIS FAC SCI MATH, 2021) Burcu Silindir; Ahmet Yantir; Silindir, Burcu; Yantir, Ahmet
In this article we focus our attention on (q h)-Gauss's binomial formula from which we discover the additive property of (q h)-exponential functions. We state the (q h)-analogue of Gauss's binomial formula in terms of proper polynomials on T-(qT-h) which own essential properties similar to ordinary polynomials. We present (q h)-Taylor series and analyze the conditions for its convergence. We introduce a new (q h)-analytic exponential function which admits the additive property. As consequences we study (q h)-hyperbolic functions (q h)-trigonometric functions and their significant properties such as (q h)-Pythagorean Theorem and double-angle formulas. Finally we illustrate our results by a first order (q h)-difference equation (q h)-analogues of dynamic diffusion equation and Burger's equation. Introducing (q h)-Hopf-Cole transformation we obtain (q h)-shock soliton solutions of Burger's equation.
Citation - WoS: 3
Citation - Scopus: 4
Generalized Polynomials and Their Unification and Extension to Discrete Calculus
(MDPI, 2023) Mieczyslaw Cichon; Burcu Silindir; Ahmet Yantir; Secil Gergun; Silindir, Burcu; Yantir, Ahmet; Cichoń, Mieczysław; Gergün, Seçil
In this paper we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q,h)-time scales. Our proposed approach encompasses both difference and quantum problems making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator alpha that combines both jump operators in a convex manner. This allows us to generate a time scale alpha which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale alpha which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover we create some polynomials on alpha-time scales using the alpha-operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally we establish the alpha-version of the Taylor formula. Finally we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus.
Citation - WoS: 16
Citation - Scopus: 18
Generalized quantum exponential function and its applications
(University of Nis filomat@pmf.ni.ac.rs, 2019) Burcu Silindir; Ahmet Yantir; Silindir, Burcu; Yantir, Ahmet
This article aims to present (q h)-analogue of exponential function which unifies extends h-and q-exponential functions in a convenient and efficient form. For this purpose we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore we prove existence and uniqueness theorem for a first order linear homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally we present a generic dynamic wave equation which admits generalized trigonometric hyperbolic type of solutions and produces various kinds of partial differential/difference equations. © 2020 Elsevier B.V. All rights reserved.
Measure on time scales with Mathematica
(Springer Verlag, 2006) Unal Ufuktepe; Ahmet Yantir; Ufuktepe, Ünal; Yantir, Ahmet
In this paper we study the Lebesgue Δ-measure on time scales. We refer to [3 4] for the main notions and facts from the general measure and Lebesgue Δ integral theory. The objective of this paper is to show how the main concepts of Mathematica can be applied to fundamentals of Lebesgue Δ- and Lebesgue ∇- measure on an arbitrary time scale and also on a discrete time scale whose rule is given by the reader. As the time scale theory is investigated in two parts by means of σ and ρ operators we named the measures on time scales by the set function DMeasure and NMeasure respectively for arbitrary time scales. © Springer-Verlag Berlin Heidelberg 2006. © 2015 Elsevier B.V. All rights reserved.
Citation - WoS: 4
Citation - Scopus: 4
Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces
(BELGIAN MATHEMATICAL SOC TRIOMPHE, 2013) Ahmet Yantir; Ireneusz Kubiaczyk; Aneta Sikorska-Nowak; Yantir, Ahmet; Kubiaczyk, Ireneusz; Sikorska-Nowak, Aneta
This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Winch's fixed point theorem is used to prove the main result. By the unification property of time scales our result is valid for Sturm-Liouville differential equations and difference equations but more interestingly by the extension property it is also valid for Sturm-Liouville q-difference equation.
Citation - WoS: 1
Citation - Scopus: 10
On continuous dependence of solutions of dynamic equations
(Elsevier Inc. usjcs@elsevier.com, 2015) Mieczysław Cichoń; Ahmet Yantir; Cichoń, Mieczysław; Yantir, Ahmet
The main goal of the paper is to present a new approach to the problem of continuous dependence of solutions of differential or dynamic problems on their domains. This is of particular interests when we use dynamic (difference in particular) equations as discretization of a given one. We cover a standard construction based of difference approximations for the continuous one but we are not restricted only to this case. For a given differential equation we take a sequence of time scales and we study the convergence of time scales to the domain of the considered problem. We choose a kind of convergence of such approximated solutions to the exact solution. This is a step for creating numerical analysis on time scales and we propose to replace in such a situation the difference equations by dynamic ones. In the proposed approach we are not restricted to the case of classical numerical algorithms. Moreover this allows us to find an exact solution for considered problems as a limit of a sequence of solutions for appropriate time scales instead of solving it analytically or calculating approximated solutions for the original problems. © 2015 Elsevier B.V. All rights reserved.
Citation - WoS: 3
Citation - Scopus: 3
Power function and binomial series on T(q-h)
(TAYLOR & FRANCIS LTD, 2023) Secil Gergun; Burcu Silindir; Ahmet Yantir; Gergün, Seçil; Silindir, Burcu; Yantir, Ahmet
This article is devoted to present (q h) -analogue of power function which satisfies additivity and derivative properties similar to the ordinary power function. In the light of nabla (q h) -power function we present (q h)-analogue of binomial series and conclude that such power function is (q h)-analytic. We prove the analyticity by showing that both the power function and its absolutely convergent Taylor series solve the same IVP. Finally we present the reductions of (q h)-binomial series to classical binomial series Gauss' binomial and Newton's binomial formulas.
Some special functions and cylindrical diffusion equation on α-time scale
(Walter de Gruyter GmbH, 2025) Burcu Silindir; Zehra Tuncer; Seçil Gergün; Ahmet Yantir; Silindir, Burcu; Yantir, Ahmet; Gergun, Secil; Tuncer, Zehra
This article is dedicated to present various concepts on α -time scale including power series Taylor series binomial series exponential function gamma function and Bessel functions of the first kind. We introduce the α -exponential function as a series examine its absolute and uniform convergence and establish its additive identity by employing the α -Gauss binomial formula. Furthermore we define the α -gamma function and prove α -analogue of the Bohr-Mollerup theorem. Specifically we demonstrate that the α -gamma function is the unique logarithmically convex solution of f (s + 1) = φ (s) f (s) f (1) = 1 where φ (s) refers to the α -number. In addition we present Euler's infinite product form and asymptotic behavior of α -gamma function. As an application we propose α -analogue of the cylindrical diffusion equation from which α -Bessel and modified α -Bessel equations are derived. We explore the solutions of the α -cylindrical diffusion equation using the separation of variables technique revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally we illustrate the graphs of the α -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts. © 2025 Elsevier B.V. All rights reserved.
Weak solutions of a hyperbolic-type partial dynamic equation in Banach spaces
(HACETTEPE UNIV FAC SCI, 2015) Ahmet Yantir; Duygu Soyoglu; Yantir, Ahmet; Soyoglu, Duygu
In this article we prove an existence theorem regarding the weak solutions to the hyperbolic-type partial dynamic equation z(Gamma Delta)(x y) = f(x y z(x y)) x(x 0) = 0 z(0 y) = 0 x is an element of T-1 y is an element of T-2 in Banach spaces. For this purpose by generalizing the definitions and results of Cichon et. al. we develop weak partial derivatives double integrability and the mean value results for double integrals on time scales. DeBlasi measure of weak noncompactness and Kubiaczyk's fixed point theorem for the weakly sequentially continuous mappings are the essential tools to prove the main result.