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In this paper, we consider the fourth-order linear differential equation u(4) +f(x)u = g(x) subject to the mixed boundary conditions u(0) = u(1) = u‘‘(0) = u‘‘(1) = 0. We first establish sufficient conditions on f(x) that guarantee a unique solution of this problem in the Hilbert space by using an a priori estimate. Accurate analytic solutions in series forms are obtained by a new variation of the Duan-Rach modified Adomian decomposition method (DRMA), and then extend this approach to some boundary value problems of fourth-order nonlinear beam equations. Also, a comparison of the two approximate solutions by the ADM with the Green function approach is presented.