Mathematical Analysis and Neural Learning-Based Prediction of Cryptocurrency Using Fractional Impulsive Models

The study aims to create a model that can predict results in cryptocurrency markets which show extreme price movements that include patterns of behavior and nonlinear market movements and unexpected price changes. The mathematical study first establishes the existence and uniqueness of mild solutions for a specific group of Caputo neutral fractional integrodifferential equations which include impulsive effects before using these solutions to model actual financial markets. The theoretical analysis establishes solution stability through Krasnoselskii's fixed point theorem and Banach contraction principle which demonstrates solution stability against small disturbances. The research demonstrates its real-world application through a fractional impulsive model which combines with an LSTM neural network to create a prediction system that effectively manages both extended time periods and sudden market changes. The numerical experiments which use Ethereum price data demonstrate a strong correlation between predicted values and actual values while showing low error metrics and stable convergence behavior. The findings demonstrate that fractional-order modeling combined with data-driven learning methods provides an effective approach to studying and predicting complex financial time series which exhibit extreme market fluctuations.