Numerical Solutions for Fractional Bagley–Torvik Equation with Integral Boundary Conditions

The Bagley–Torvik equation (BTE) is an important model in mathematical physics and mechanics, but obtaining its analytical solution remains challenging. For its numerical treatment, the presence of composite functions in the generalized BTE poses additional difficulties, and efficient approaches for handling nonlinear terms are still lacking in the literature. This study proposes an improved numerical method for the fractional BTE with integral boundary conditions. By employing an integration technique, the original problem is transformed into a weakly singular Fredholm–Hammerstein (F–H) integral equation of the second kind. To address the nonlinear terms, an enhanced piecewise Taylor expansion scheme is developed to construct the discrete form, while the uniqueness of the solution is proven using the contraction mapping theorem in Banach spaces. The convergence and error analyses are rigorously carried out, and numerical experiments confirm the accuracy and efficiency of the proposed approach.