Optimal Ultraspherical Spectral Method for Fractional Differential Equations: Parameter Optimization and Application to the Bagley-Torvik Equation

This paper introduces a novel optimal ultraspherical spectral method for solving fractional differential equations, with a particular focus on the Bagley-Torvik equation. We develop a comprehensive theoretical framework based on Gegenbauer (ultraspherical) polynomials [Formula: see text], where the ultraspherical parameter [Formula: see text] is treated as an optimization variable. A rigorous optimization technique is formulated to determine the optimal parameter [Formula: see text] that minimizes the residual norm of the approximate solution for a fixed number of retained modes [Formula: see text]. We establish new operational matrices for Caputo fractional derivatives of arbitrary order in the ultraspherical basis and prove their convergence properties. The optimization problem is shown to be well-posed and the existence of an optimal parameter [Formula: see text] is guaranteed under mild regularity conditions. Extensive numerical experiments demonstrate that our method achieves superior accuracy compared to existing spectral methods, with exponential convergence rates that can be tuned through the optimal parameter selection. This work represents a significant advancement in spectral methods for fractional differential equations by introducing parameter adaptivity as a means to enhance numerical performance.