This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties.

Cryptography, the discipline of disguising information, relies on mathematical principles for encoding and decoding data. This chapter presents an inventive approach to encrypting and decrypting messages, utilizing the R-transformation, an essential integral transform, and the congruence modulo operator. The research illustrates the pragmatic application of these mathematical tools in securing communication through the encoding of sensitive information. By investigating the integration of the R-transformation and modulo operations, this study seeks to highlight their effectiveness in reinforcing the privacy and integrity of data. Through thorough experimentation and analysis, this research introduces a groundbreaking methodology for cryptographic encoding and decoding, shedding light on its potential to fortify the security of confidential information transmission.

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept.

In this article, the Upadhyaya transform is employed in diverse chemical reaction models expressed through ordinary differential equations. The investigation reveals that this transform provides precise and efficient solutions, circumventing the necessity for complex computations. Furthermore, the integration of graphical representations enhanced the interpretability of results, offering visual insights into the temporal evolution of reactant concentrations. These findings collectively underscore the efficacy of the Upadhyaya transform in addressing ordinary differential equations within chemical reaction models.

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.