Prabakaran Raghavendran, Yamini Parthiban
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 K1<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.