This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a Banach space framework which requires particular assumptions while the study focuses on the ${\mathfrak{K}}_1<1$ condition which leads to the existence of a controllable solution. The proposed criteria are demonstrated through a numerical example which tests the theoretical results. The real-world case study uses artificial neural network (ANN) technology to predict Litecoin prices through the application of the fractional controllability model which analyzes historical financial data. The hybrid framework enables precise forecasting of nonlinear time series because it combines fractional calculus mathematical principles with ANN learning abilities. The proposed method demonstrates its predictive efficiency. The method shows robust performance through experimental results using cross-validation and performance metrics. The proposed model demonstrates competitive performance while providing additional advantages such as incorporation of memory effects and theoretical controllability. The research establishes a novel connection between fractional dynamical systems and machine learning which serves as an essential tool for studying complicated systems in theoretical research and practical applications.

TOPSIS (technique for order preference by similarity to ideal solution) is a mathematical approach employed in the context of multi-criteria decision-making. Its primary purpose is to establish a ranking among various alternatives by taking into consideration multiple attributes or criteria. This method utilizes distance metrics to gauge the nearness of each alternative to both the ideal and anti-ideal solutions. Ultimately, its goal is to facilitate the identification of the most optimal alternative. In this era of uncertainty, many parents grapple with myriad uncertainties when it comes to selecting the right educational institution for their children. This chapter provides an initial glimpse into the neutrosophic pythagorean plithogenic hypersoft set (N_pphs) along with its fundamental definition. Additionally, an algorithm has been devised for the TOPSIS methodology, accompanied by a practical numerical illustration. This is intended to assist parents in making an informed decision when choosing the most suitable school for their children with their set of criteria.

The study uses the Upadhyaya transform to find exact solutions of three physical models which include the Atwood machine the Beer--Lambert law and electron motion in electromagnetic fields. The proposed framework demonstrates how exact analytical solutions derived via the Upadhyaya transform can be utilized as physically consistent priors, constraints, and validation references within learning-based models, thereby enhancing interpretability, numerical stability, and reproducibility while reducing dependence on large empirical datasets. The research presents an application-oriented method which connects integral transform theory with contemporary machine learning techniques to simulate and study complex physical and chemical systems.

This work presents and elaborates the new notion of Neutrosophic Fermatean Soft Plithogenic Sets (NFSP-sets), which provides a new sophisticated mathematical framework of managing uncertainty, indeterminacy, and contradiction in decision-making systems. The suggested framework is a systematic study of the fundamental operations, algebraic properties, and laws of interaction of NFSP-sets. In order to illustrate its usefulness in practice, an elaborate application is implemented to assess and rank football teams using actual match statistics. The performance data of several games is combined with the AND-PRODUCT rule based on Neutrosophic Fermatean Soft Plithogenic matrices, which provides an accurate and comprehensive evaluation of the performance of a team in terms of teamwork, defense, and scoring efficiency. Numerical tests also prove that the given approach is more accurate and reliable than the traditional ranking models. Altogether, the results confirm the strength and relevance of NFSP-sets in sports analytics, and they may become a strong decision-support tool of systematic and evidence-based performance analysis.

This study investigated an impulsive neutral Volterra–Fredholm integro-differential equation (VFIDE) involving the Caputo fractional derivative and developed a mathematical framework for establishing the existence and controllability of solutions. Under appropriate assumptions on the fractional order, nonlinear functions, and integral operators, the existence of solutions was established using the Schauder fixed-point theorem, while controllability was analyzed through an admissible control formulation. The proposed model incorporates fractional memory effects, Volterra–Fredholm nonlocal interactions, and abrupt state changes through impulsive terms. A fully specified numerical example was presented using a product-integration scheme, numerical quadrature, and fixed-point iteration, with grid refinement used to examine numerical stability. The framework was further illustrated using daily Solana cryptocurrency closing-price data from January 1 to December 1, 2025. The results demonstrate the applicability of the proposed framework to fractional impulsive dynamical modeling.