A novel approach to fully third order nonlinear boundary value problems

TL;DR

This paper introduces a new method for solving fully third order nonlinear boundary value problems by reducing them to operator equations, establishing solution properties, and demonstrating convergence with simpler conditions.

Contribution

It presents a novel approach that simplifies the analysis of third order BVPs and proves key properties under weaker conditions than previous methods.

Findings

Established existence, uniqueness, positivity, and monotony of solutions.

Proved convergence of the iterative solution method.

Validated results with multiple examples.

Abstract

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.