Computational analysis of fractal-fractional differential systems via Vieta-Lucas fractal-fractional operational matrices

Articles in Press

Document Type : Research Paper

Authors

1 School of Mathematics, Computers and Information Sciences, Central University of Himachal Pradesh, Shahpur Campus, Shahpur 176206, India.

2 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon.

Abstract

Fractal-fractional differential equations are important as they can help to model the real-world systems that have memory effects and thus find their existence in various real-world phenomena such as physics, engineering, biology, and biomedicine. It is always challenging to handle the fractal-fractional derivatives using traditional numerical methods, which motivates the need to develop the numerical methods that can handle these fractal-fractional models accurately and effectively. In this work, a novel numerical scheme, the Vieta-Lucas fractal fractional matrix (VLFF) method, is presented for solving the system of fractal-fractional differential equations (FFDEs). The operational matrices for derivative and fractal fractional derivatives of Vieta-Lucas polynomials over the generalized domain are constructed. Making use of the fractal-fractional operational matrix technique streamlines the computation processes and significantly reduces the challenges while dealing with fractal-fractional derivatives. This simplified matrix method is then applied with the Tau approach to find the solution of a system of FFDEs. Further, the geometric representation of the fractal-fractional derivative matrix is presented, showcasing the fractal patterns. The proposed method is validated through numerous examples, with solution curves and error analysis presented for different fractal and fractional orders. A comparison has been made with the existing numerical methods to validate the accuracy and reliability of the proposed VLFF method.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 17 April 2026

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History

  • Receive Date: 12 April 2025
  • Revise Date: 11 November 2025
  • Accept Date: 14 April 2026

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