This study introduced a new technique based on the combination of the least-squares method (LSM) with Euler polynomials for finding the approximate solutions of integro-differential equations (IDEs) subject to the mixed conditions. Three examples of first and second-orders linear Fredholm IDEs (FIDEs) and Volterra IDEs (VIDEs) are considered to illustrate the proposed method. The numerical results comprised to demonstrate the validity and applicability of this method comparisons with the exact solution shown that the competence and accuracy of the present technique. INTRODUCTION Presented a method for solving high-order Linear FIDE equations under the mixed conditions in terms of Legendre polynomials under mixed conditions. The method used is the Legendre collocation matrix method and then converting the equation and conditions into matrix equations, which correspond to systems of linear algebraic equations with Legendre coefficients (Yalçınbaş, 2009). Used Cauchy kernel with airfoil polynomials of the first kind, and the numerical solution for some of the integro-differential equations gets a system of linear algebraic equations. The convergence of the method gives some sufficient conditions (Mennouni and Guedjiba, 2010). Studied Bessel polynomials to find approximate solutions of high-order linear VIDEs under the mixed conditions, based on collocation points, practical matrix method, the accuracy and efficiency of the method are proven (Yüzbaşı et al., 2011). The Euler polynomial was used to solve the VIDEs of the
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