A numerical comparative analysis of methods for solving fractional differential equations

Abstract

In this paper, we present two collocation methods utilizing subdivision schemes and Bernstein polynomials as their basis functions to solve Caputo-type fractional differential equations. These methods transform fractional differential equations into systems of linear equations, which can be solved using various suitable techniques. Notably, these methods are versatile, making them applicable to both boundary value problems and initial value problems. To demonstrate their effectiveness, we applied these methods to three test problems of fractional differential equations, including the Bagley-Torvik equation and fractional oscillation equation. Our results reveal that both methods provide a more accurate approximation to the exact solution when compared to various algorithms found in the existing literature, showcasing their efficiency, accuracy, and consistency.

Keywords:

• Boundary conditions
• collocation method
• fractional differential equation
• initial conditions
• subdivision schemes

Authors’ contributions

Conceptualization, Syeda Tehmina Ejaz; Formal analysis, Namra Zulqarnain, Jihad Younis and Saima Bibi; Methodology, Syeda Tehmina Ejaz and Namra Zulqarnain; Supervision, Syeda Tehmina Ejaz; Writing original draft, Namra Zulqarnain and Syeda Tehmina Ejaz; Writing, review and editing, Syeda Tehmina Ejaz Jihad Younis and Saima bibi.

Disclosure statement

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Data availability statement

The data used to support the findings of the study are available within this paper.