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Rahimkhani, Parisa, Samadyar, Nasrin, Razzaghi, Mohsen. (1405). Decoding computational complexity: a fractional-order clique-based approach for solving Hilfer fractal-fractional differential equations. سامانه مدیریت نشریات علمی, 14(3), 1147-1164. doi: 10.22034/cmde.2025.68006.3266
Parisa Rahimkhani; Nasrin Samadyar; Mohsen Razzaghi. "Decoding computational complexity: a fractional-order clique-based approach for solving Hilfer fractal-fractional differential equations". سامانه مدیریت نشریات علمی, 14, 3, 1405, 1147-1164. doi: 10.22034/cmde.2025.68006.3266
Rahimkhani, Parisa, Samadyar, Nasrin, Razzaghi, Mohsen. (1405). 'Decoding computational complexity: a fractional-order clique-based approach for solving Hilfer fractal-fractional differential equations', سامانه مدیریت نشریات علمی, 14(3), pp. 1147-1164. doi: 10.22034/cmde.2025.68006.3266
Rahimkhani, Parisa, Samadyar, Nasrin, Razzaghi, Mohsen. Decoding computational complexity: a fractional-order clique-based approach for solving Hilfer fractal-fractional differential equations. سامانه مدیریت نشریات علمی, 1405; 14(3): 1147-1164. doi: 10.22034/cmde.2025.68006.3266

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

Computational Methods for Differential Equations
مقاله 10، دوره 14، شماره 3، مهر 2026، صفحه 1147-1164    اصل مقاله (2.14 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.68006.3266
نویسندگان
Parisa Rahimkhani* 1؛ Nasrin Samadyar2؛ Mohsen Razzaghi3
1Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran.
2Department of Basic Science, Kermanshah University of Technology, Kermanshah, Iran.
3Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi, USA.
چکیده
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.
کلیدواژه‌ها
Hilfer fractal-fractional differential equations؛ Fractional-order clique functions؛ Convergence analysis؛ Numerical method
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