Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense

• Published in: Chaos, solitons and fractals Vol. 125; pp. 163 - 170
• Main Authors: Abu Arqub, Omar, Maayah, Banan
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2019
• Subjects: Atangana-Baleanu fractional sense; Reproducing kernel Hilbert space method; Riccati and Bernoulli equations; Riccati and Bernoulli equations; Atangana-Baleanu fractional sense; Reproducing kernel Hilbert space method
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2019.05.025

•Developing the reproducing kernel Hilbert space method within the Atangana-Baleanu fractional approach for solving the Riccati and Bernoulli equations with respect to initial conditions of necessity.•Constructing two Hilbert spaces for both range and domain space which are basis for the solution methodology of the presented method.•Framing numerical algorithm and procedure of solution with the optimal formulation of the problem.•Introducing computational simulations to delineate suitability and relevance of the calculations created.•Utilizing some useful properties of the Atangana-Baleanu fractional derivative and integral. This article is concerned with design and comprehensive study of a numerical approach for solving Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. The proposed technique is using the reproducing kernel Hilbert space to approximate the solution to those class of fractional differential equations in the form of uniformly convergent series with respect to space variables. An accurate computational algorithm is presented to confirm the gained analysis. The relevant theorems and characterizations related to Riccati and Bernoulli equations are included among the reproducing kernel theory. Supplementary problems at the end of the article serve as a complete review of the utilized method and theories. The numerical consequences of the proposed approach are practically effective and impressive, whilst the utilized theories lay the foundation stone for addressing such issues. In light of the summary, conclusions and future recommendations are also provided.