Solving a nonlinear biharmonic boundary value problem

Abstract

In this paper we study a boundary value problem for a nonlinear biharmonic equation, which models a bending plate on nonlinear elastic foundation. We propose a new approach to existence and uniqueness and numerical solution of the problem. It is based on the reduction of the problem to finding fixed point of a nonlinear operator for the nonlinear term. The result is that under some easily verified conditions we have established the existence and uniqueness of a solution and the convergence of an iterative method for the solution. The positivity of the solution and the monotony of iterations are also considered. Some examples demonstrate the applicability of the obtained theoretical results and the efficiency of the iterative method.