Browsing by Author "Silindir, Burcu"

Now showing 1 - 6 of 6

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 - 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.
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.