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Solving a nonlinear biharmonic boundary value problem

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. | Journal of Computer Science and Cybernetics, V.33, N.4 (2017), 308–324 DOI 10.15625/1813-9663/33/4/11066 SOLVING A NONLINEAR BIHARMONIC BOUNDARY VALUE PROBLEM∗ DANG QUANG A1 , TRUONG HA HAI2 , NGUYEN THANH HUONG3 , NGO THI KIM QUY4 1 Centre 2 Thai for Informatics and Computing, VAST Nguyen University of Information and Communication Technology 3 College 4 Thai of Sciences, Thai Nguyen University Nguyen University of Economic and Business Administration 3 nguyenthanhhuong2806@gmail.com 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. Keywords. Nonlinear biharmonic boundary value problem, Existence and uniqueness of solution, Iterative method, Numerical solution. 1. INTRODUCTION In this paper we study the following nonlinear biharmonic boundary value problem (BVP) ∆2 u = f (x, u, ∆u), u = 0, ∆u = 0, x ∈ Ω, x ∈ Γ, (1) where Ω is a connected bounded domain in R2 , with a smooth boundary Γ, ∆ is the Laplace operator. We assume that f (x, u, v) is a function continuous in a bounded domain, which will be indicated later. The problem (1) describes the static deflection of an elastic bending plate with hinged edges rested on nonlinear foundation. For the one dimension case, the problem is of the form u(4) (x) = f (x, u(x), u00 (x)), u(0) = u(1), 00 0 0; f (x, .