A unified approach to study the existence and numerical solution of functional differential equation

TL;DR

This paper presents a unified theoretical framework and a second-order accurate numerical method for solving third order nonlinear functional differential equations, with potential applicability to equations of any order.

Contribution

It introduces a reduction to operator equations, proves existence and uniqueness, and develops an efficient numerical method with error estimates.

Findings

The numerical method is of second order accuracy.

Theoretical results are validated through examples.

The approach can be extended to functional differential equations of any order.

Abstract

In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. 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. The approach used for the third order nonlinear functional differential equation can be applied to functional differential equations of any orders.