High performance adaptive step size fractional numerical scheme for solving fractional differential equations.

Fractional differential equations have recently gained popularity due to their ability to simulate a wide range of complex processes in various fields, including engineering, physics, biology, and finance. These equations provide a powerful framework for describing phenomena with memory effects and hereditary features that standard integer-order models cannot account for. In this study, we present fractional versions of numerical algorithms specifically designed for solving fractional-order differential equations. We thoroughly investigate the proposed approaches for stability under various fractional parameter values and compare their stability performance with existing methods. The schemes' consistency and local truncation error are calculated to ensure their accuracy. In terms of stability surface, our methods have a larger stability zone than existing fractional schemes. Two engineering applications are addressed utilizing both fixed and adaptive step-length algorithms to assess efficiency. In both cases, our methods outperform existing approaches, as evidenced by less local and global errors, reduced CPU time, and fewer function and derivative evaluations. Our newly developed fractional order technique outperforms modern high-performance algorithms in solving fractional differential equations, demonstrating superior computational efficiency and stability. These findings demonstrate the robust and efficient capabilities of the proposed methods to solve fractional-order problems.