A New Laguerre-Galerkin Operational Matrix: Highly Accurate Method for the Fractional Bagley–Torvik Equation of Variable Coefficients with Robin Boundary Conditions

Abstract This paper develops a novel numerical scheme to obtain numerical solutions (NUMSs) for the fractional Bagley-Torvik equation (FB-TE) with variable coefficients subject to Robin boundary conditions (RBCs). The novelty of our approach is due to constructing a new set of basis functions, derived from Laguerre polynomials, which are formulated to satisfy the homogeneous RBCs (HRBCs). The methodology is based on deriving the operational matrices (OMs) for ordinary derivatives (Ods) and fractional derivatives (Fds) for this new basis and implementing them within a spectral collocation method (SCM). This method converts the problem governed by the RBCs into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. We provide theoretical assurances for the algorithm’s efficacy by establishing its convergence and error analysis features. Numerical examples are presented to illustrate the accuracy and efficiency of the scheme, with approximate solutions (APPSs) in high agreement with exact solutions (ExaSs) and favorable comparisons to other methods.