An Efficient and Accurate Numerical Approach for Fractional Bagley–Torvik Equations: Hermite Polynomials Combined with Least Squares

This paper proposes an efficient and accurate numerical framework for solving fractional Bagley–Torvik equations, which model viscoelastic and memory-dependent dynamic systems. The method combines the Hermite polynomial approximation with a least-squares optimization scheme to achieve high-accuracy solutions. By leveraging the analytical properties of Hermite polynomials, Caputo fractional derivatives were computed efficiently, avoiding the complexities of direct fractional differentiation. The resulting weighted least-squares formulation transforms the problem into a stable algebraic system. Numerical results confirm the method’s superior accuracy, rapid convergence, and robustness compared with existing techniques, demonstrating its potential for broader applications in fractional-order boundary value problems.