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On numerical solution of the second-order linear Fredholm–Stieltjes integral equation

In this framework, the necessary and sufficient conditions for the existence and uniqueness of the second-order linear Fredholm–Stieltjes-integral equations, u(x)=λ∫abK(x,y)u(y)dg(y)+f(x),x∈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 an exact and closed-form solution for limited cases.