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In this framework, the necessary and sufficient conditions for the existence and uniqueness of the second-order linear Fredholm-Stieltjes-integral equations, u(x) = lambda integral(b)(a) K(x, y) u(y) dg(y) + f(x), x is an element of[a, b], are thoroughly derived. Moreover, instead of approximating the integral equation by different numbers of partition n, the optimal number n for the given error tolerance is established. The system of equations is implemented in MAPLE for the Runge method. An efficient scheme is proposed for second-order integral equations. The solution has been compared with ...More