İlgi alanları

• MakaleAçık Erişim2026Scopus SJRQ3/r/gumushane/16702
  The Anageometric Calculus: Theory, Fundamental Results, and Connections With Classical Analysis

  Yalcin, Numan; Dedeturk, Mutlu

  In this study, we develop a systematic framework for anageometric calculus as an extension of non-Newtonian analysis. Fundamental concepts, including limits, continuity, differentiation, and higher order derivatives, are established, and the connection to classical calculus is clarified through a logarithmic transformation. Within this framework, we derive the properties of the anageometric derivative, inverse function relations, partial derivatives, and mean value theorems. The theory is extended to integration by establishing the first and second fundamental theorems of anageometric calculus. Furthermore, the structural implications of the anageometric derivative are investigated, demonstrating that the equation D similar to f=lambda f naturally generates power-law solutions rather than classical exponential ones. This highlights the suitability of anageometric analysis for scale-invariant phenomena and proportional growth processes, supported by a practical heat-conduction example.

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• MakaleAçık Erişim2026Scopus SJRQ3/r/gumushane/13892
  The Non-Newtonian Differential Operators Associated With Iterated Generators

  Dedeturk, Mutlu; Yalcin, Numan

  This paper investigates non-Newtonian differentiation associated with integer iterates of generator functions. For (Formula presented.), we consider the derivative operators (Formula presented.) and derive explicit first-order representation, transformation, and factorization formulas that relate operators corresponding to different iteration levels. The construction is then extended to higher-order derivatives, where the resulting operators are represented through conjugation with the classical derivative operator Dr. Faà di Bruno’s formula and exponential partial Bell polynomials are used to obtain explicit higher-order expressions. Several examples involving exponential, double-exponential, and quadratic generators illustrate how the general results specialize to familiar non-Newtonian calculi. The results provide a unified treatment of non-Newtonian differentiation based on iterated generators and clarify its relationship with classical differentiation.

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• MakaleAçık Erişim2026Scopus SJRQ2/r/gumushane/12171
  A Non-Newtonian Extension of Laplace–Sumudu–Elzaki Transforms

  Yalcin, Numan

  Classical Laplace-, Sumudu-, and Elzaki-type transforms are formulated within additive analytical frameworks and do not naturally accommodate multiplicative scaling structures arising in non-Newtonian calculus. Motivated by this limitation, this study introduces a non-Newtonian Laplace–Sumudu–Elzaki transform (NNLSET) based on logarithmic scaling mechanisms, multiplicative measures, and power-type kernels. The proposed framework is constructed by replacing the classical measure dt with the multiplicative measure (Formula presented.) and the linear scaling structure (Formula presented.) with the nonlinear scaling structure (Formula presented.). Using the logarithmic transformation (Formula presented.), a canonical kernel representation of the form (Formula presented.) is derived, establishing a correspondence between multiplicative power-type kernels and weighted exponential structures in the logarithmic domain. Within an admissible weighted function framework, several analytical properties of the transform are established, including existence, boundedness, stability, uniqueness, restricted recoverability, and a logarithmic derivative representation associated with expressions of the form (Formula presented.). A comparative analysis with the classical Laplace–Sumudu–Elzaki framework, together with illustrative differential-equation examples, a representative nonlinear MEMS oscillator, and a numerical computation, is presented. The obtained results demonstrate that the NNLSET provides a mathematically consistent framework for the analysis of multiplicative structures, logarithmic scaling phenomena, and logarithmically structured differential equations. Its applicability is further illustrated through the analysis of a representative nonlinear MEMS oscillator.

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• MakaleAçık Erişim2025Scopus SJRQ2/r/gumushane/13954
  Algebraic and topological foundations of non-Newtonian analysis via generator functions

  Civgin, Emre; Yalcin, Numan

  In this paper we explored the algebraic and topological underpinnings of non-Newtonian analysis through the framework of a generator function α. We introduced the α-number systems and the associated arithmetic operations, and we established the corresponding algebraic structures, including groups, fields, and vector spaces defined over α-real numbers. On the topological side, we developed the notions of α-metric spaces and α-sequences, thereby extending core concepts of analysis to the non-Newtonian setting. The study culminates with the formulation of star (*) analysis, which provides a systematic mechanism for transitioning between distinct arithmetic systems, together with a rigorous treatment of *-vector spaces and linear operators.

  AtıfScopus1·WoS1·OpenAlex1

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• BildiriAçık Erişim2023/r/gumushane/16674
  Solutions of Multiplicative Integral Equations via The Multiplicative Power Series Method

  Bal, Arzu; Yalcin, Numan; Dedeturk, Mutlu

  In this study, definitions of types of multiplicative integral equations are given. And solutions of different types of multiplicative integral equations are investigated using the multiplicative power series method. These are supported by numerical examples.

  AtıfOpenAlex4·TR Dizin2

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• MakaleAçık Erişim2023/r/gumushane/20355
  On Bigeometric Laplace Integral Transform

  Yalçın, Numan; Kaymak, Sinem

  The purpose of this study is to mention the Laplace integral transform in bigeometric analysis, which is one of the non-Newtonian analysis by using the fundamental definitions and theorems of the Laplace integral transform, which is one of the integral transform methods of classical analysis. First of all, the concept of exponential arithmetic, which forms the basis of non Newtonian analysis, is given. As in classical analysis, definitions of the concepts of bigeometric limit, bigeometric continuity, bigeometric derivative and bigeometric integral are given in bigeometric analysis. Here, the definition of the bigeometric Laplace integral transform in bigeometric analysis is given. Then, some basic concepts and theorems of the bigeometric Laplace integral transform are given. For this purpose, the definitions of the concepts of bigeometric derivative and bigeometric indefinite integral and bigeometric definite integral in bigeometric analysis and the properties of these concepts are used. In addition, the properties of the bigeometric Laplace integral transform are investigated. Finally, solutions of bigeometric linear differential equations are investigated with the help of the bigeometric Laplace integral transform.

  AtıfOpenAlex3

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• MakaleAçık Erişim2023Scopus SJRQ4/r/gumushane/14381
  Solutions of multiplicative linear differential equations via the multiplicative power series method

  Yalcin, Numan; Dedeturk, Mutlu

  In this study, fundamental concepts of multiplicative analysis is given. Also, definitions of multiplicative Taylor series (MTS), multiplicative Taylor polynomials (MTP), and multiplicative power series (MPS) are given. Solutions of higher-order multiplicative linear differential equations (MLDE) are investigated with the help of the MPS method. Applications of MPS method are done for first, second, and third-order multiplicative ordinary linear differential equations. Copyright 2021, Yıldız Technical University.

  AtıfScopus5·WoS5·OpenAlex8

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• MakaleAçık Erişim2022Scopus SJRQ3/r/gumushane/14832
  MULTIPLICATIVE CHEBYSHEV DIFFERENTIAL EQUATIONS AND MULTIPLICATIVE CHEBYSHEV POLYNOMIALS

  Yalcin, Numan

  In this work, the definition of the first and second types of multiplicative Cheby-shev differential equations is given and the solutions of these equations are inves-tigated with the help of multiplicative power series. Also, the properties of first and second type multiplicative Chebyshev polynomials are given and proved. Fi-nally, these studies are supported with numerical examples.

  AtıfScopus4·WoS2·OpenAlex9

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• MakaleAçık Erişim2021Scopus SJRQ2/r/gumushane/11511
  Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method

  Yalçın, Numan; Dedeturk, Mutlu

  In this study, fundamental definitions and theorems of the Multiplicative Differential Transform Method (MDTM) are given. First and second order multiplicative initial value problems are numerically solved with the help of MDTM.

  AtıfScopus15·WoS15·OpenAlex21

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• MakaleYalnız künye2021Scopus SJRQ3/r/gumushane/11305
  The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials

  Yalçın, Numan

  In this work, we give the power series solutions around an ordinary point, in the case of variable coefficients, homogeneous multiplicative differential equations of 2nd order. Further, we introduce the multiplicative Hermite differential equations, multiplicative Hermite polynomials and basic properties of these polynomials.

  AtıfScopus17·WoS18·OpenAlex23

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