The Numerical Approximation of Caputo Fractional Derivative of Higher Orders Using A Shifted Gegenbauer Pseudospectral Method: Two-Point Boundary Value Problems of the Bagley Torvik Type Case Study

This work presents a new framework for approximating Caputo fractional derivatives (FDs) of any positive order using a shifted Gegenbauer pseudospectral (SGPS) method. By transforming the Caputo FD into a scaled integral of the $m$th-derivative of the Lagrange interpolating polynomial (with $m$ being the ceiling of the fractional order $\alpha$), we mitigate the singularity near zero, improving stability and accuracy. The method links $m$th-derivatives of shifted Gegenbauer (SG) polynomials with SG polynomials of lower degrees, allowing for precise integration using SG quadratures. We employ orthogonal collocation and SG quadratures in barycentric form to obtain an accurate and efficient approach for solving fractional differential equations. We provide error analysis showing that the SGPS method is convergent in a semi-analytic framework and conditionally convergent with exponential rate for smooth functions in finite-precision arithmetic. This exponential convergence improves accuracy compared to wavelet-based, operational matrix, and finite difference methods. The SGPS method is flexible, with adjustable SG parameters for optimal performance. A key contribution is the fractional SG integration matrix (FSGIM), which enables efficient computation of Caputo FDs via matrix-vector multiplications and accelerates the SGPS method through pre-computation and storage. The method remains within double-precision limits, making it computationally efficient. It handles any positive fractional order $\alpha$ and outperforms existing schemes in solving Caputo fractional two-point boundary value problems (TPBVPs) of the Bagley-Torvik type.