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.

Aydemir, Hilal; Merdan, Mehmet; Demir, Ümit

The Elzaki-Adomian decomposition method (EADM) is intended to serve as an efficient analytical method for the resolution of these original fractional-order Riccati differential equations. This can be accomplished permanently by incorporating the Adomian decomposition method with Elzaki. The local fractional derivative is implemented in this format. Particularly in the context of nonlinear differential equations (ODE), this approach is preferred over digital gaps. Additionally, the method’s convergence. Random individuals with uniform, beta, normal, and gamma distributions are used to select the initial conditions or coefficients of the equations. The variance, confidence interval, and expected value of the solutions that are obtained will be determined. MATLAB (2013a) package software will be employed to display the individuals that were brought together, and the results will be analyzed randomly.

Köseoğlu, Ali; Şahin, Rıdvan; Demir, Ümit

The Dempster–Shafer evidence theory is a very practical concept for handling uncertain information. The foundation of this theory lies in the basic probability assignment (BPA), which exclusively accounts for the degree of support attributed to focal elements (FEs). In this study, neutrosophic evidence sets (NESs) are defined to introduce additional probabilistic measures, aimed at addressing the uncertainty, imprecision, incompleteness, and inconsistency present in real-world information. The basic element of NESs is a neutrosophic basic probability assignment (NBPA), which consists of three components. The truth degree of FEs is represented by the first BPA, the second BPA represents the indeterminacy degree of FEs, and the last BPA characterizes the falsity degree of FEs. In NESs, each support degree of FEs is shown separately without any limitation. Therefore, the general concept of NESs is broader compared to traditional evidence sets and intuitionistic fuzzy evidence sets. Unlike the neutrosophic set (NS), the NBPA method assigns truth-support, uncertainty-support, and false-support degrees, as well as these support degrees, to single and multiple subsets in a discriminative framework. This paper aimed to develop some information measures for NESs, such as neutrosophic Deng entropy (NDE), neutrosophic cosine similarity measure, and neutrosophic Jousselme distance. Then, an improved method based on NDE and neutrosophic cosine similarity measure was established to combine contradictory evidence to increase the influence of reliable evidence on the one hand and to reduce the influence of unreliable evidence on the other hand. Finally, a case involving sensor data integration for target identification was studied to highlight the importance of these innovative ideas. The numerical example demonstrates that the proposed method provides more reliable and superior fusion performance compared to classical models, particularly in scenarios involving high conflict and uncertain information. However, the effectiveness of the method is partially influenced by the structure of the similarity matrix and the entropy parameters, which necessitates careful parameter tuning to achieve optimal results. These limitations are explicitly highlighted to serve as a guide for future improvements and broader applications of the method.

Alkan, Aslı; Anaç, Halil

In this research, we propose a new numerical method that combines with the Caputo-Fabrizio Elzaki transform and the q-homotopy analysis transform method. This work aims to analyze the Caputo-Fabrizio fractional Newell-Whitehead-Segel (NWS) equation utilizing the Caputo-Fabrizio q-Elzaki homotopy analysis transform method. The Newell-Whitehead-Segel equation is a partial differential equation employed for modeling the dynamics of reaction-diffusion systems, specifically in the realm of pattern generation in biological and chemical systems. A convergence analysis of the proposed method was performed. Two-dimensional and three-dimensional graphs of the solutions have been drawn with the Maple software. It is seen that the resulting proposed method is more powerful and effective than the Aboodh transform homotopy perturbation method and conformable Laplace decomposition method in the results.

Günen, Mehmet Akif; Pérez-Delgado, María-Luisa; Beşdok, Erkan

Earth observation satellites capture panchromatic images at high spatial resolution and multispectral images at lower resolution to optimize the use of their onboard energy sources. This results in a technical necessity to synthesize high-resolution multispectral images from these data. Pansharpening techniques aim to combine the spatial detail of panchromatic images with the spectral information of multispectral images. However, due to the discrete nature of these images and their varying local statistical properties, many pansharpening methods suffer from numerical artifacts such as chromatic and spatial distortions. This paper introduces the L0-Norm-based pansharpening method (L0pan), which addressed these challenges by maximizing the number of similar pixels between the synthesized pansharpened image and the original panchromatic and multispectral images. L0pan was optimized using a population-based colony search algorithm, enabling it to effectively balance both chromatic fidelity and spatial resolution. Extensive experiments across nine different datasets and comparison with nine other pansharpening methods using ten quality metrics demonstrated that L0pan significantly outperformed its counterparts. Notably, the colony search algorithm yielded the best overall results, highlighting the algorithm’s strength in refining pansharpening accuracy. This study contributed to the advancement of pansharpening techniques, offering a method that preserved both chromatic and spatial details more effectively than existing approaches.

Alkan, Aslı; Anaç, Halil

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.

Cona, Lale

Picard's three step iteration algorithm was one of the iteration algorithms that was recently shown to be faster than some other iterative algorithms in the existing literature. The purpose of this paper was to study using this iteration algorithm for the solution of nonlinear Volterra integral equations. It was investigated that the sequences obtained from this iteration algorithm converged to the solution of nonlinear Volterra integral equations. Moreover, data dependence was obtained for nonlinear Volterra integral equations. An example was given that confirmed the applicability of the newly proven theorems.

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.