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

TL;DR

This paper introduces simple iterative numerical methods with second and third order convergence for solving fully third order nonlinear boundary value problems, supported by theoretical proofs and numerical examples.

Contribution

It presents novel iterative methods with proven second and third order accuracy for third order nonlinear boundary value problems, including error estimates and validation.

Findings

Methods are of second and third order accuracy.

Numerical examples confirm theoretical error estimates.

Proposed methods are efficient and reliable.

Abstract

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.