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.
Bougoffa, L., Rach, R., & Wazwaz, A.-M. (2016). On Solutions of Boundary Value Problem for Fourth-Order Beam Equations. Mathematical Modelling and Analysis, 21(3), 304-318. https://doi.org/10.3846/13926292.2016.1155507
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