Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space

• Published in: Chaos, solitons and fractals Vol. 117; pp. 161 - 167
• Main Authors: Abu Arqub, Omar, Al-Smadi, Mohammed
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2018
• Subjects: Atangana–Baleanu fractional approach; Bagley–Torvik equation; Painlevé equation; Reproducing kernel Hilbert space method; Atangana–Baleanu fractional approach; Bagley–Torvik equation; Reproducing kernel Hilbert space method; Painlevé equation
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2018.10.013

•In this analysis, by developed the reproducing kernel Hilbert space method within the Atangana–Baleanu fractional approach, the Bagley–Torvik and Painlevé equations are solved with respect to initial conditions of necessity.•The solution methodology involves the constructing of two Hilbert spaces for both range and domain space.•Numerical algorithm (Pseudocode) and procedure of solution are assembled compatibility with the optimal formulation of the problem.•The optimal profiles show the performance of the numerical solutions and the effect of the Atangana–Baleanu fractional approach in the obtained results.•In this approach, computational simulations are introduced to delineate the suitability, straightforwardness, and relevance of the calculations created. In this analysis, by developed the reproducing kernel Hilbert space method within the Atangana–Baleanu fractional approach, the Bagley–Torvik and Painlevé equations are solved with respect to initial conditions of necessity. The solution methodology involves the use of two Hilbert spaces for both range and domain space. Numerical algorithm and procedure of solution are assembled compatibility with the cogent formulation of the problem. The method of solution of the utilized problems is studied under some hypotheses, which provides the theoretical structure behind the technique. The solutions profiles show the performance of the numerical solutions and the effect of the Atangana–Baleanu fractional approach in the obtained results. In this approach, computational simulations are introduced to delineate suitability, straightforwardness, and relevance of the calculations created.