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.

Alkan, Asli; Anaç, Halil

In this study, an enhanced semianalytical framework is developed for solving nonlinear fractional differential equations involving the Caputo-Fabrizio fractional operator. The proposed Caputo-Fabrizio q-Kushare homotopy analysis transform method (CFq-KHATM) combines the Caputo-Fabrizio derivative, Kushare transform, and q-homotopy analysis within a unified operator-consistent framework. Unlike conventional hybrid approaches, the method preserves the nonsingular exponential memory structure of the Caputo-Fabrizio derivative and provides additional convergence-control flexibility through the auxiliary parameter & hbar; and q-deformation parameter n. To demonstrate its applicability, a Caputo-Fabrizio fractional epidemic model with memory-dependent transmission dynamics is investigated. Numerical simulations for different fractional-order values show that memory effects significantly influence epidemic evolution, infection persistence, and environmental viral dynamics. Furthermore, comparisons with existing methods together with absolute and relative error analyses confirm the accuracy, stability, and convergence efficiency of the proposed framework. The obtained results indicate that CFq-KHATM is a flexible and computationally reliable tool for nonlinear fractional epidemic systems and other memory-dependent dynamical models.

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.