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.
Elastic beam equation Existence and uniqueness of solution Positivity of solution Iterative method
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