Solutions for Linear Fractional Differential Equations with Multiple Constraints Using Fractional B-Poly Bases

Md. Habibur Rahman, Muhammad I. Bhatti () and Nicholas Dimakis
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Md. Habibur Rahman: Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Muhammad I. Bhatti: Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Nicholas Dimakis: Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

Mathematics, 2025, vol. 13, issue 19, 1-17

Abstract: This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10 −18 to 10 −4 . The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques.

Keywords: Bernstein polynomials; operational matrices; initial and boundary value problems; fractional differential equations; numerical approximation; Caputo derivative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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