Advanced solutions for nonlinear fractional equations: a Laplace-Caputo-RKDM approach

Abstract

This study proposes a novel framework that combines the Laplace Transform with the reproducing kernel discretization method (RKDM) under the principles of Caputo derivatives to address the challenges of fractional differential equations. By leveraging the Laplace Transform for problem simplification and utilizing the RKDM framework for improved numerical accuracy and stability, this integrated approach offers a robust solution process for both linear and nonlinear FDEs. Caputo derivatives provide the necessary flexibility in handling complex initial conditions, further enhancing the efficiency of the method. Numerical experiments validate the performance of the proposed framework, demonstrating its ability to deliver precise and stable solutions efficiently. The results expand the scope of practical applications for FDEs across various fields, including structural analysis, thermodynamics, and environmental modeling. This research lays a solid foundation for future advancements in fractional calculus and dynamic system modeling, encouraging the development of innovative numerical strategies.