Iterative Method for Solving a Beam Equation with Nonlinear Boundary Conditions

In this paper, we propose an iterative method for solving a beam problem which is described by a nonlinear fourth-order equation with nonlinear boundary conditions. The method reduces this nonlinear fourth-order problem to a sequence of linear second-order problems with linear boundary conditions. The convergence of the method is proved, and some numerical examples demonstrate the efficiency of the method. 1. Introduction In the paper, we consider the following boundary value problem (BVP): which models bending equilibrium of elastic beams on nonlinear supports [1]. Here, represents the deflection of an elastic beam of length , subjected to a force exerted by a foundation, and and express the influence of fixed torsional springs at the ends of the beam. In [2], the problem (1) by means of a Green function was reduced to the problem of finding fixed point of a nonlinear integral equation by successive approximations. Therefore, at each iteration, it is needed to compute a single and a double integrals and a derivative. Differently from this approach in this paper following the methodology of [3, 4], we lead the solution of problem (1) to a sequence of simple linear boundary value problems (BVPs) for second-order equation, which are easily solved numerically. The convergence of the iterative method is established by the contraction principle. Some performed numerical examples demonstrate the efficiency of the method. It should be noticed that the idea of reduction of linear BVPs for fourth partial differential equations, namely, for the biharmonic and biharmonic-type equations to operator equations for investigating iterative methods for their solution was successfully used by ourselves in many works, for example, in [5, 6]. 2. Iterative Method First, we reduce problem (1) to an operator equation, and then apply the successive approximation method to the latter one. For this purpose, we set Then, due to the boundary conditions for the function we have and the function has the property By the setting (2), problem (1) is decomposed to the problems Obviously, the solution from these problems depends on the function . Consequently, its derivative also depends on . Therefore, we can represent this dependence by an operator Combining with the first relation in (2), we get the operator equation for : That is, is a fixed point of . Now, we consider properties of the operator . For this purpose, we introduce the space Next, we make the following assumptions on the given functions in Problem (1): there exist constants , , and such that for any ,？？and？？ .