Decoding computational complexity: a fractional-order clique-based approach for solving Hilfer fractal-fractional differential equations

In the current investigation, we propose a novel and computationally efficient numerical framework for solving Hilfer fractal-fractional differential equations (HF-FDEs) by introducing a new class of basis functions termed fractional-order clique functions (FCFs). In contrast to conventional fractional models, which primarily rely on classical kernels and often fall short in encapsulating the intricate interplay between memory-dependent behavior and fractal geometries, the adopted Hilfer fractal-fractional derivative offers a unified formulation that inherently incorporates both non-locality and fractality-features essential for accurately modeling complex real-world processes. To the best of our knowledge, this work marks the first development and implementation of FCFs within a numerical solution framework. The distinctive analytical properties of FCFs facilitate precise, adaptable, and computationally stable representations of HF-FDE solutions. By integrating the FCFs-based approximation with a collocation technique and Newton’s iterative algorithm, the under-study problem is efficiently transformed into a system of nonlinear algebraic equations. A thorough convergence analysis is presented to ensure the theoretical soundness of the approach, and its practical performance is validated through five numerical examples. The results decisively demonstrate the enhanced accuracy and effectiveness of the proposed method in capturing the multifaceted behavior of fractional dynamic systems when compared to traditional approaches.

کلیدواژه ها:

Hilfer fractal-fractional differential equations ، Fractional-order clique functions ، Convergence analysis ، Numerical method