In this study, we describe a linear multistep technique for solving second order integro-differential equations. Primarily, an efficient 4 Step Block Multistep Method (4SBM), which solves initial value problems of ordinary differential equations is adopted. The 4SBM method is implemented in conjunction with a family of Newton-Co ?tes integration quadrature, as a numerical solver of integro-differential equations. In addition, we derived a 3 Step Block Method (3SBM), whose derivation process employed an orthogonal polynomial as a basis function. The 3SBM is implemented with a specific member of open Newton-Co ?tes integration quadrature. Convergence properties of the derived methods are examined, to guarantee the suitability of the methods. Approximate solutions to some test problems are computed using our proposed methods. To investigate the performance of our approach, we compared our simulation result with a theoretical solution for each problem with various parameters. The outcome of the study indicates that the derived methods are efficient numerical solvers of integro-differential equations.

Keywords: Integro-differential, Newton-Cotes, multistep method, orthogonal polynomial, Fredholm, second order.