The novel cubic B-spline method for fractional Painlevé and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense

In this analysis, we use the high order cubic B-spline method to create approximating polynomial solutions for fractional Painlevé and Bagley-Torvik equations in the Captuo, Caputo-Fabrizio, and conformable fractional sense concerning boundary set conditions. Using a piecewise spline of a 3rd-degree polynomial; the discretization of the utilized fractional model problems is gained. Taking advantage of the Taylor series expansion; the error order behavior spline theorem is proved. We demonstrate applications of our spline method to several certain kinds including the 1st(2nd) Painlevé and Bagley-Torvik fractional models. For more detail, using Mathematica 11 several drawings and many tables were calculated and their explanations were mentioned. The computational results indicate that the suggested spline approach is most acceptable in terms of cost efficiency and precision of calculations. Highlight, conclusion, and future notes are provided to extract the ability of the discussed approach and the tendency of the utilized fractional models to extrapolate new application areas in the meshless numerical training.