An attractive numerical algorithm for solving nonlinear Caputo\u2013Fabrizio fractional Abel differential equation in a Hilbert space

Mohammed Al-Smadi,Shrideh Al-Omari,Nadir Djeddi,Shaher Momani,Serkan Araci

doi:10.1186/s13662-021-03428-3

Abstract

Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space mathcal{H}^{2}[a,b]. We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

Highlights

The area of fractional calculus is an active interesting dynamic branch of theoretical and applied mathematical analysis, quantum mechanics, engineering. and physical sciences
During the past few decades, an interest of mathematicians has been focused on developing operators of fractional derivatives and some applications in dealing with solutions of the generalized classic differential equations, knowing the effect of the fractional conditioning on the quality of the established models [1,2,3,4]
This trend contributed a lot to the emergence of a number of fractional derivative definitions such as Riemann–Liouville, Caputo, Marchaud, Erdélyi–Kober, Feller, Atangana– Baleanu, Grünwald–Letnikov, Sonin–Letnikov, Hadamard, and so forth [5,6,7,8,9]. It has been shown through numerous studies on fractional analysis that different definitions are equivalent under certain circumstances. This variety of differential and integral fractional operators provides the opportunity to choose the most suitable one according to the problem associated with the initial and boundary conditions to obtain the optimal solution to the studied problem

Summary

Introduction

The area of fractional calculus is an active interesting dynamic branch of theoretical and applied mathematical analysis, quantum mechanics, engineering. and physical sciences. We employ the reproducing kernel algorithm (RKA) for investigating numerical solutions for a class of nonlinear fractional Abel differential equations (FADE) of both kinds in the sense of Caputo–Fabrizio derivative. Ω(t) ∈ H2[a, b] is an unknown analytic function to be solved, and CF Daα is a differential operator of order α in the Caputo–Fabrizio sense.

Results

Conclusion