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.

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.

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.

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.

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.

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.

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.

Yalcin, Numan; Celik, Ercan; Gokdogan, Ahmet

In this work, taking definitions and properties of Laplace transform in classical analysis as a basis, we give some basic definitions and properties of the multiplicative Laplace transform. In addition, solutions of some multiplicative differential equations are obtained by the help of this transform.