Abstarct: In this thesis, The Chebyshev and Legendre polynomials are used to numerical solution the Volterra-Fredholm integral equations. Then, a pseudospectral method is proposed baxsed on the integral matrix for solving integral equations and integro-differential equations. The solution of the integral equations and the integro-differential equations are obtained at the collocation points. Then, an efficient and stable approach to computing the pseudospectral integration matrix is presented through a recursive relationship of Legendre integration matrices. Furthermore, the convergence analysis of the method is given in both L^infty and L^2 spaces via a linear integral equations and the convergence rate is shown by numerical results.