Problem: Some problems arising in the study of transverse vibrations of a hinged beam (or modeling bending equilibrium of elastic beam with right end attached to fixed torsional spring, see for example, the picture from Wikipedia below) are typically modeled by nonlinear fourth-order elliptic boundary value problems (BVPs) of Kirchhoff type
$\Delta^2 u – (a+b \int_{\Omega} |\nabla u|^2 dx) \Delta u = f(x, u) \ \text{in} \ \Omega$
$u=\nabla u =0 \ \text{on} \ \partial \Omega$
where $\Omega \in \mathbb{R}^d$ is a bounded domain, $\Delta^2$ is the biharmonic operator, $u \in C^2 (\Omega)$, and $a, b$ are positive constants, and $f(x, u) \in \mathbb{C}(\Omega \times \mathbb{R},\mathbb{R})$. The difficulty of these problems is that they are highly nonlinear, in particular, due to the present of the nonlinear term under the integral sign. A common approach is to use finite element methods (FEMs) for approximating the solution.
Contribution: I, along with Prof. Quang A Dang, proposed a new and efficient iterative method for solving it without using FEMs. The idea was to reduce the original equation to find a solution to a nonlinear operator equation. For one-dimensional problems, it is simply a nonlinear algebraic equation and thus one can apply the simplified Newton methods for solving it. This approach reduces the computational time significantly when compared to using FEMs; (see Dang and Luan (2010a-CAMWA), Dang and Luan (2010b-Proceedings of 12th National Conference on Selected ICT Problems, Hanoi), Dang, Luan, and Long (2010c-AMS).