Two Numerical Approaches to Solving Fractional Differential Equations with a Generalized Mittag–Leffler Kernel Using Bernstein Polynomials

Abstract

This paper presents a solution to fractional differential equations containing three parameters, utilizing Bernstein polynomials through two efficient computational approaches. In the first approach, the solution is expressed as a linear combination of Bernstein polynomials. In contrast, in the second, the fractional derivative (FD) itself is represented in terms of Bernstein polynomials. The key properties of both algorithms are derived and analyzed. The generalized Atangana Baleanu Caputo (ABC) definition of FD that uses the Mittag-Leffler function as the kernel of the integration form of the FD, characterized by three tunable parameters, is adopted throughout this study. Those parameters can adjust the existence and the behavior of the solution for the FD equations. A set of initial value problems, including both linear and nonlinear fractional differential equations (FDEs), is solved using the suggested approaches. The solution profiles illustrate the performance of the numerical solutions and the impact of the ABC definition on the obtained findings, demonstrating that Bernstein polynomials provide improved accuracy and efficiency in extracting solutions for the considered fractional models. The computational simulation of this comparative analysis reveals that the second approach yields higher accuracy with smaller absolute errors and additionally provides insight into the existence of solutions, as illustrated through the studied fractional models.