Existence results and numerical solution for the Dirichlet problem for fully fourth order nonlinear equation

In this paper we study the existence and uniqueness of a solution and propose an iterative method for solving a beam problem which is described by the fully fourth order equation $$u^{(4)}(x)=f(x,u(x),u'(x),u'''(x),u'''(x)), \quad 0 < x < 1$$ associated with the Dirichlet boundary conditions. This problem was studied by several authors. Here we propose a novel approach by the reduction of the problem to an operator equation for the triplet of the nonlinear term $\varphi (x)=f(x,u(x),u'(x),u''(x),u'''(x))$ and the unknown values $u''(0), u''(1).$ Under some easily verified conditions on the function $f$ in a specified bounded domain, we prove the contraction of the operator. This guarantees the existence and uniqueness of a solution and the convergence of an iterative method for finding it. Many examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method. The advantages of the obtained results over those of Agarwal are shown on some examples.Comment: 26 pages, 6 figure

Existence results and iterative method for solving a fourth order nonlinear integro-differential equation

In this paper we consider a class of fourth order nonlinear integro-differential equations with Navier boundary conditions. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.Comment: 17 pages, 1 figur

A novel approach to fully third order nonlinear boundary value problems

In this work we propose a novel approach to investigate boundary value problems (BVPs) for fully third order differential equations. It is based on the reduction of BVPs to operator equations for the nonlinear terms but not for the functions to be sought. By this approach we have established the existence, uniqueness, positivity and monotony of solutions and the convergence of the iterative method for approximating the solutions under some easily verified conditions in bounded domains. These conditions are much simpler and weaker than those of other authors for studying solvability of the problems before by using different methods. Many examples illustrate the obtained theoretical results.Comment: 21 pages, 6 figure

A simple numerical method of second and third orders convergence for solving a fully third order nonlinear boundary value problem

In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both continuous and discrete levels. We prove that the discrete methods are of second order and third accuracy due to the use of appropriate formulas for numerical integration and obtain estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.Comment: 22 pages, 1 figur