Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications

• Main Authors: M.I. Syam, Mohammed Al-Refai
• Format: Article
• Language: English
• Published: Elsevier 2019-06-01
• Series: Chaos, Solitons & Fractals: X
• Online Access: http://www.sciencedirect.com/science/article/pii/S2590054419300119

We study linear and nonlinear fractional differential equations of order 0 < α < 1, involving the Atangana–Baleanu fractional derivative. We establish existence and uniqueness results to the linear and nonlinear problems using Banach fixed point theorem. We then develop a numerical technique based on the Chebyshev collocation method to solve the problem. As an important application we consider the fractional Riccati equation. Two examples are presented to test the efficiency of the proposed technique, where a notable agreement between the approximate and the exact solutions is obtained. Also, the approximate solutions approach to the exact solutions of the corresponding ordinary differential equations as the fractional derivative approaches 1. Keywords: Fractional differential equations, Atangana–Baleanu derivative, Banach fixed point theorem, Collocation method, Chebyshev fractional functions, Riccati equation, MSC: 34A08, 26A33, 58C30, 65L60