This paper investigates the dynamics of a nonlinear elastic beam governed by a space-fractional differential equation of Caputo type, where the fractional order lies in the range z ∈ (3,4). A small parameter ε is incorporated to capture singular perturbation effects, and a composite approximation is constructed through matched asymptotic expansions. The leading-order solution is derived using the Tanh method for the reduced integer-order problem. The fractional perturbation introduces algebraically decaying corrections, which are analyzed and matched with the outer-region behavior. Numerical plots illustrate how varying ε and the fractional order z affects the beam profile, especially the far-field tails. The analysis shows that higher fractional orders amplify long-range perturbative effects in the solution profile.