Laplace Adomian Decomposition Method to the Bagley–Torvik Fractional Differential Equation

This paper presents an analytical solution framework for the Bagley–Torvik equation, a canonical fractional differential model central to the study of viscoelastic damping. We employ the Laplace Adomian Decomposition Method (LADM) within the framework of the Caputo fractional derivative to derive accurate approximate solutions for three distinct forcing regimes: constant, linearly increasing, and piecewise-defined forces. The proposed method effectively addresses the challenges posed by the nonlocal fractional derivative, yielding systematic, computable series solutions. The accuracy and applicability of the LADM are validated through rigorous comparison with established analytical and numerical benchmarks from the literature. The findings underscore the efficacy of this approach for characterizing complex viscoelastic responses under diverse dynamic loading conditions.