Asymptotics and computation for a class of Fredholm integro-differential equations

This work focuses on a Fredholm integro-differential equation$ \begin{align*} \left(-\varepsilon^2\frac{\mathrm{d}^2}{\mathrm{d}x^2}+{{p(x)}}_{\varepsilon}\frac{\mathrm{d}}{\mathrm{d}x}\right)u_{\varepsilon}(x)+q(x)\left(u_{\varepsilon}(x)-\int_0^1W(z) {u}_{\varepsilon}(z)\, \mathrm{d}z\right) = f(x), \end{align*} $where $ \varepsilon>0 $ and $ x\in (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 $ u_{\varepsilon} $. Focusing furthermore on $ \varepsilon\downarrow0 $, the boundary and interior asymptotics of $ {u}_{\varepsilon} $ are predominantly influenced by the interplay among variable coefficients and boundary data, and all asymptotic behaviors of $ {u}_{\varepsilon} $ are fully classified into three cases: (I) $ \int_0^1W\neq1 $; (II) $ (\int_0^1W, \int_0^1\frac{Wf}{q}) = (1, 0) $; (III) $ \int_0^1W = 1 $ and $ \int_0^1\frac{Wf}{q}\neq0 $. Remarkably, when (III) is satisfied, $ {u}_{\varepsilon} $ 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.