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.

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.

Dedeturk, Mutlu; Cakalli, H.; Kocinac, L.D.R.; Ashyralyev, A.; Harte, R.; Dik, M.; Canak, I.; Kandemir, H.S.; Tez, M.; Gurtug, O.

In this work, inverse problem for an elliptic equation is studied. We gave a numerical solution of inverse elliptic problem by differential transform method in test example.

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.

Ashyralyyev, Charyyar; Akyuz, Gulzipa; Dedeturk, Mutlu

In this work, we consider an inverse elliptic problem with Bitsadze-Samarskii type multipoint nonlocal and Neumann boundary conditions. We construct the first and second order of accuracy difference schemes (ADSs) for problem considered. We stablish stability and coercive stability estimates for solutions of these difference schemes. Also, we give numerical results for overdetermined elliptic problem with multipoint Bitsadze-Samarskii type nonlocal and Neumann boundary conditions in two and three dimensional test examples. Numerical results are carried out by MATLAB program and brief explanation on the realization of algorithm is given.

Ashyralyyev, Charyyar; Dedeturk, Mutlu; Lukashov, A.; Ashyralyev, A.

Approximation of Dirichlet type overdetermined multidimensional elliptic problem with Dirichlet-Neumann boundary conditions are discussed. A third order of accuracy difference scheme for its approximate solution is proposed. The stability, almost coercive stability and coercive stability inequalities for the solution of constructed difference scheme are established. Test example for a two-dimensional elliptic problem is presented.

Ashyralyyev, Charyyar; Dedeturk, Mutlu

In the present study, the inverse problem for a multidimensional elliptic equation with mixed boundary conditions and overdetermination is considered. The first and second orders of accuracy in t and the second order of accuracy in space variables for the approximate solution of this inverse problem are constructed. Stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are established. For the two-dimensional inverse problems with mixed boundary value conditions, numerical results are presented in test examples.

Ashyralyyev, Charyyar; Dedeturk, Mutlu; Ashyralyev, A; Malkowsky, E

The inverse problem for the multidimensional elliptic equation with Neumann-Dirichlet conditions are presented. For the approximate solution of this inverse problem the first and second order of accuracy in t and in space variables difference schemes are constructed. The stability, almost coercive stability and coercive stability estimates for the solution of these difference schemes are obtained. The algorithm for approximate solution is tested in a two-dimensional inverse problem.

Ashyralyyev, Charyyar; Dedeturk, Mutlu

Dirichlet şartıyla verilen eliptik diferansiyel denklem için ters problemin iyi tanımlığı incelendi. Ters problemin yaklaşık çözümü için bir sonlu farklar yöntemi uygulandı. Birinci ve ikinci derecede kesinlikte olan fark şemalarının çözümü için kararlılık ve koersiv kararlılık kestirimleri elde edildi. Uygulamada çok boyutlu eliptik denklem için ters problem çalışıldı. Teorik ifadeler iki boyutlu eliptik denklem ile verilen bir sayısal örnekle desteklendi.

Ashyralyyev, Charyyar; Dedeturk, Mutlu

A finite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the first and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.