A novel caputo fractional model for english language learning: Analysis and simulation with bayesian regularization approach.

In this paper, a new Caputo discrete fractional model is introduced to capture the dynamics of English language learning. This model creates a strong foundation for examining language acquisition behaviors by including the learning process within the system. The proposed work not only presents an innovative discrete fractional model but also leverages machine learning techniques to estimate and analyze the learning process over time. To achieve numerical accuracy and stability, we employ Bayesian Regularization Artificial Neural Networks (BRA-NNs) as a machine learning-based computational solver. This approach ensures robust numerical simulations and enhances the predictive power of the model. Furthermore, the reliability of the proposed method is demonstrated through six fractional-order variants of the Fractional-Order English Language Mathematical Model (FOELMM), which are systematically derived and analyzed. The results are validated against the Fractional-Order Lotka-Volterra method, confirming the accuracy and robustness of the proposed machine learning-driven computational approach.•Development of a discrete Caputo fractional model for language learning.•Integration of machine learning techniques via Bayesian Regularization Artificial Neural Networks (BRA-NNs) for numerical simulations.•Validation of the model through the Fractional-Order Lotka-Volterra approach to ensure accuracy.