Dang, Quang A.; Nguyen, Thanh Huong Solving the Dirichlet problem for fully fourth order nonlinear differential equation. (English) ✄ ✂  ✁ Zbl 1438.34088 Afr. Mat. 30, No. 3-4, 623-641 (2019). Summary: In this paper we study the existence and uniqueness of a solution and propose an iterative method for solving a beam problem which is described by the fully fourth order equation u (4) (x)= f (x, u(x),u ′ (x),u ′′ (x),u ′′′ (x)), a < x < b, associated with the Dirichlet boundary conditions. This problem was well studied by Agarwal by the reduction of it to a nonlinear operator equation for the unknown function u(x). Here we propose a novel approach by the reduction of the problem to an operator equation for the nonlinear term φ(x)= f (x, u(x),u ′ (x),u ′′ (x),u ′′′ (x)). Under some easily verifiable conditions on the function f in a specified bounded domain, we prove the existence, uniqueness and positivity of a solution and the convergence of an iterative method for finding it. Some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method. The advantages of the proposed approach to the problem over the well-known approach of Agarwal is shown on examples. MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equa- tions 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations Cited in 3 Documents Keywords: beam equation; existence and uniqueness of solution; fully fourth order nonlinear equation; iterative method Full Text: DOI References: [1] Agarwal, RP; Chow, YM, Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., 10, 203-217, (1984) · Zbl 0541.65055 · doi:10.1016/0377-0427(84)90058-X [2] Agarwal, R.P.: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986) · Zbl 0619.34019 · doi:10.1142/0266 [3] Bai, Z., The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal., 67, 1704-1709, (2007) · Zbl 1122.34010 · doi:10.1016/j.na.2006.08.009 [4] Dang, QA; Dang, QL; Ngo, TKQ, A novel efficient method for nonlinear boundary value problems, Numer. Algorithm, 76, 427-439, (2017) · Zbl 1378.65149 · doi:10.1007/s11075-017-0264-6 [5] Dang, QA; Ngo, TKQ, Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term, Nonlinear Anal. Real World Appl., 36, 56-68, (2017) · Zbl 1362.34036 · doi:10.1016/j.nonrwa.2017.01.001 [6] Dang, QA; Ngo, TKQ, New fixed point approach for a fully nonlinear fourth order boundary value problem, Bol. Soc. Parana. Math., 36, 209-223, (2018) · Zbl 1424.34083 · doi:10.5269/bspm.v36i4.33584 [7] Doedel, EJ, Finite difference collocation methods for nonlinear two point boundary value problems, SIAM J. Numer. Anal., 16, 173-185, (1979) · Zbl 0438.65068 · doi:10.1137/0716013 [8] Du, J.; Cui, M., Constructive proof of existence for a class of fourth-order nonlinear BVPs, Comput. Math. Appl., 59, 903-911, (2010) · Zbl 1189.34038 · doi:10.1016/j.camwa.2009.10.003 [9] Ertürk, VS; Momani, S., Comparing numerical methods for solving fourth-order boundary value problems, Appl. Math. Comput., 188, 1963-1968, (2007) · Zbl 1119.65066 [10] Feng, H.; Ji, D.; Ge, W., Existence and uniqueness of solutions for a fourth-order boundary value problem, Nonlinear Anal., 70, 3561-3566, (2009) · Zbl 1169.34308 · doi:10.1016/j.na.2008.07.013 Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2022 FIZ Karlsruhe GmbH Page 1