Error estimates of finite-element approximations for a fourth-order differential equation

Existence and uniqueness of positive solutions for Kirchhoff type beam equations

“…The existence and multiplicity of solutions for (1.1) and its multi-dimensional case have been studied by several authors, see [5][6][7][8][9][10][11][12] and the references there in. Meanwhile, numerical methods of (1.1) have been developed in [13][14][15][16][17][18][19][20].…”

Numerical solution of the static beam problem by Bernoulli collocation method

“…For example, Shin [28] developed a finite element approximation method to approximate the problem (1.1) based on the space of piecewise linear polynomials on the uniform grid and proposed an iterative scheme to solve a system of nonlinear equations during direct implementation. Later, Ohm, Lee and Shin [21] obtained the error estimates for this finite element approximation. In 2009, Peradze [24] used a Galerkin approximation to reduce the problem (1.1) to a system of cubic algebraic equations which are solved by means of the nonlinear Jacobi iteration process and estimated the algorithm error.…”

On the Melan equation for suspension bridges

“…The Melan equation (1.1) has also attracted the interest of numerical analysts; see [8,20,24,25,30]. In these papers, several approximating procedures for the solution of (1.1) have been discussed for different forms of the term h(w).…”