Benchmarking Numerical Methods for Caputo Fractional Differential Equations: A Comprehensive Comparative Study

This study compares four key numerical methods for solving Caputo fractional differential equations: Homotopy Perturbation Sumudu Transform method (HPSTM), Adomian Decomposition method (ADM), Predictor-Evaluate Corrector-Evaluate method (PECE), and Grünwald-Letnikov Finite Difference method (GL-FDM). Using two benchmark problems, the fractional relaxation equation and forced fractional oscillation system, we assess accuracy, efficiency, convergence, and implementation complexity. Implemented in a standardized Python framework with adaptive precision and parallel computation, all methods show roughly second-order convergence. PECE offers the highest accuracy, HPSTM and ADM balance accuracy and efficiency, while GL-FDM is fastest computationally. Statistical analysis highlights clear performance differences, providing evidence-based guidelines for method selection. This work sets robust benchmarks and practical insights for fractional numerical methods in scientific computing.

کلیدواژه ها:

Fractional differential equations ، Numerical analysis ، Computational mathematics ، Convergence analysis