The univariate multinode Shepard method for the Caputo fractional derivatives: from approximation to the solution of the Bagley–Torvik equation

In this paper, we propose a numerical approach based on the univariate multinode Shepard operator for the approximation of Caputo fractional derivatives. The fractional integral is evaluated through a Gauss–Jacobi quadrature formula, and the resulting approximation is then employed within a collocation scheme for Bagley–Torvik boundary and initial value problems. Approximation and convergence properties are discussed, together with the influence of the conditioning of the associated collocation systems. Numerical experiments involving different node distributions show high accuracy for both polynomial and non-polynomial solutions. The method is also applied to variable-coefficient and nonsmooth Bagley–Torvik problems, and comparisons with previously published numerical methods confirm its competitive accuracy.