Fractional continuum mechanics has become an important framework for modeling materials and structures with memory and nonlocal interactions, which cannot be adequately described by classical integer-order elasticity models. In particular, fixed-memory fractional operators offer a physically meaningful way to limit the interaction horizon while retaining essential nonlocal effects. However, continuum models based on compositions of fixed-memory fractional derivatives still lack systematic analytical solutions and reliable numerical treatments. This study investigates a one-dimensional elastic continuum modeled by a fractional differential equation involving compositions of Caputo derivatives with fixed memory length. Such a formulation enables the consistent incorporation of nonlocal interactions and history-dependent mechanical effects, while simultaneously avoiding the excessive influence of long-range memory. The considered approach therefore provides a balanced compromise between fully nonlocal fractional models and classical local formulations. An analytical solution is derived in the form of a sine-series expansion by transforming the governing equation into an equivalent problem with homogeneous boundary conditions and exploiting the spectral properties of the associated fractional operator. In addition, a numerical discretization scheme based on the trapezoidal rule is developed and specifically tailored to the structure of finite-memory fractional derivatives. Special attention is devoted to the treatment of artificial nodes introduced by the fixed-memory framework. These auxiliary nodes are handled using reflection conditions derived from the analytical structure of the solution, ensuring stability and consistency of the numerical approximation without the need for additional boundary constraints. Numerical experiments for various fractional orders, memory lengths, discretization parameters, and load distributions confirm second-order accuracy of the scheme. The results demonstrate a clear influence of the fractional order and the memory length on the mechanical response and show smooth convergence toward the classical elastic solution. The proposed analytical–numerical framework provides a robust tool for studying nonlocal elastic behavior in materials with fading memory.