Strong convergence analysis of a finite difference scheme for fractional Langevin equations with mixed fractional damping

This paper develops and analyzes a fully discrete numerical scheme for fractional Langevin equations involving two Caputo derivatives of distinct orders and additive white noise. The Wong–Zakai approximation is employed to construct a truncated spectral representation of the noise, leading to a semi-discrete formulation whose integral representation is derived via Laplace transform techniques. A frequency-domain error analysis is established to handle the singular kernels induced by mixed fractional damping and to quantify the approximation error precisely. The resulting finite difference scheme achieves strong mean-square convergence of order 1.5, which is shown to be optimal with respect to the solution regularity and independent of the fractional orders. Numerical experiments validate the theoretical results and illustrate the effects of fractional parameters on the long-time stochastic dynamics.