Abstract
In this paper, we propose a novel efficient method for a fourth-order nonlinear boundary value problem which models a statistically bending elastic beam. Differently from other authors, we reduce the problem to an operator equation for the right-hand side function. Under some easily verified conditions on this function in a specified bounded domain, we prove the contraction of the operator. This guarantees the existence and uniqueness of a solution of the problem and the convergence of an iterative method for finding it. The positivity of the solution and the monotony of iterations are also considered. We show that the examples of some other authors satisfy our conditions; therefore, they have a unique solution, while these authors only could prove the existence of a solution. Numerical experiments on these and other examples show the fast convergence of the iterative method.
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Aftabizadeh, A.R.: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116, 415–426 (1986)
Alves, E., Ma, T.F., Pelicer, M.L.: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlin. Anal. 71, 3834–3841 (2009)
Amster, P., Cárdenas Alzate, P.P.: A shooting method for a nonlinear beam equation. Nonlin. Anal. 68, 2072–2078 (2008)
Bai, Z., Ge, W., Wang, Y.: The method of lower and upper solutions for some fourth-order equations. J. Inequal. Pure Appl. Math. 5(1), Art. 13 (2004)
Dang, Q.A.: Iterative method for solving the Neumann boundary value problem for biharmonic type equation. J. Comput. Appl. Math. 196, 634–643 (2006)
Li, Y.: A monotone iterative technique for solving the bending elastic beam equations. Appl. Math. Comput. 217, 2200–2208 (2010)
Ma, R., Zhang, J., Fu, S.: The method of lower and upper solutions for fourth-order two-point boundary value problems. J. Math. Anal. Appl. 215, 415–422 (1997)
Ma, T.F., da Silva, J.: Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Appl. Math. Comput. 159, 11–18 (2004)
Ma, T.F.: Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. Appl. Numer. Math. 47, 189–196 (2003)
Pao, C.V.: Numerical methods for fourth order nonlinear elliptic boundary value problems. Numer Methods Partial Differ. Equ. 17, 347–368 (2001)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer (1984)
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A, D.Q., Long, D.Q. & Quy, N.T.K. A novel efficient method for nonlinear boundary value problems. Numer Algor 76, 427–439 (2017). https://doi.org/10.1007/s11075-017-0264-6
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DOI: https://doi.org/10.1007/s11075-017-0264-6