A spline construction scheme for numerically solving fractional Bagley–Torvik and Painlevé models correlating initial value problems concerning the Caputo–Fabrizio derivative approach

In this review, the well-known Bagley–Torvik and Painlevé models (PM), which are special kinds of differential problems of noninteger order ranks and have a significant role in fractional calculus implementations are utilized. These two models are solved numerically using the cubic B-spline polynomials approximation which are utilized as basis functions in a collocation plan. Stratifying the collocation points, and defining the desired solutions together with their Caputo–Fabrizio derivatives (CFD) in sum forms are the main steps of our approach. The next suffix is the use of matrix operations and fundamental linear algebra to adapt and transform the two proposed models into a computational scheme of linear and nonlinear algebraic equations. The accuracy and computational complexity of the scheme are analyzed based on a large number of independent runs and their comprehensive statistical analysis. A computational clear algorithm step for the utilized scheme concerning the two discussed models is scheduled regarding the Caputo–Fabrizio approach. Besides this, all the comparative studies on the utilized figures and obtained tables are made with Mathematica 11 package. At the end of this work, our analysis research was closed with a conclusion, a set of observations, and some recommendations.