Solving the Dirichlet Problem for Fully Sixth Order Nonlinear Differential Equation

Abstract

The problem which we study in this paper is described by the fully sixth order equation ${y}^{(6)}\left(s\right)=F\left(s, y\left(s\right), y{\prime}(s), y{\prime}{\prime}\left(s\right), {y}^{(3)}\left(s\right), {y}^{(4)}\left(s\right), {y}^{(5)}\left(s\right)\right), 0<s<1$ associated with the Dirichlet boundary conditions. Firstly, we propose a novel method for investigating the existence and uniqueness of solution of the problem. It is based on the reduction of the original problem to an operator equation for the right-hand side function. Under some easily verified conditions on the function F in a specified bounded domain, we establish the existence, uniqueness and positivity of a solution. Next, we propose the simple iterative method for finding the approximate solution on both continuous and discrete levels. We prove that the discrete method is of second-order of accuracy due to the use of appropriate formula for numerical integration. The total error estimate for the discrete iterative method is also obtained. Finally, some numerical examples, where exact solution of the problem is known, are presented for illustrating the efficiency of the numerical method.