This paper introduces a novel computational approach for solving higher-order multi-indexed fractional differential equations (FDEs) in the Caputo sense. By leveraging a power series polynomial collocation method, the proposed technique reformulates the FDEs into an equivalent integral form, enabling accurate and stable numerical solutions. The method addresses key challenges in fractional calculus, including the handling of multi-indexed derivatives and ensuring computational efficiency. Numerical experiments demonstrate the approach’s superior accuracy, with error analyses revealing significant improvements over existing methods. The results underscore the method’s applicability to real-world problems in physics, engineering, and biology, where fractional derivatives model memory-dependent phenomena. This work advances the toolkit for computationally solving complex FDEs while maintaining robustness and convergence.

Keywords: Fractional Differential Equations, Caputo Derivative, Multi-Indexed Systems, Power Series Method, Collocation Method, Numerical Approximation, Fractional Calculus, Computational Mathematics, Stability Analysis, Convergence Analysis, Error Estimation.

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