This work focuses on a Fredholm integro-differential equation −ε + εp(x) uε(x) + q(x) (uε(x) − W(z)uε(z) dz ) = f(x), where ε > 0 and x ∈ (0, 1), subject to Robin boundary conditions. The contributions are summarized in twofold. For q > 0, we establish a necessary and sufficient condition for the uniqueness of uε. Focusing furthermore on ε ↓ 0, the boundary and interior asymptotics of uε are predominantly influenced by the interplay among variable coefficients and boundary data, and all asymptotic behaviors of uε are fully classified into three cases: (I) W 6= 1; (II) ( W, ) = (1, 0); (III) W = 1 and 6= 0. Remarkably, when (III) is satisfied, uε exhibits an asymptotic blow-up. Moreover, on the basis of our theoretical analysis, we propose a novel numerical algorithm for solving the above equation. The validity of our theoretical findings is substantiated through numerical examples with figures and tables.