ASYMPTOTICS AND COMPUTATION FOR A CLASS OF FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

This work focuses on a Fredholm integro-differential equation (-epsilon(2) d(2)/dx(2) + epsilon p(x) d/dx) u(epsilon)(x) + q(x) (u(epsilon)(x) - integral(1)(0) W(z)u(epsilon)(z) dz) = f(x), where epsilon > 0 and x is an element of (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(epsilon). Focusing furthermore on epsilon down arrow 0, the boundary and interior asymptotics of u(epsilon) are predominantly influenced by the interplay among variable coefficients and boundary data, and all asymptotic behaviors of u" are fully classified into three cases: (I) integral(1)(0) W not equal 1; (II) (integral(1)(0) W, integral(1)(0) Wf/q) = (1, 0); (III) integral(1)(0) W = 1 and integral(1)(0) Wf/q not equal 0. Remarkably, when (III) is satisfied, u(epsilon) 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.