High order numerical methods for solving high orders functional differential equations

TL;DR

This paper develops high order numerical methods for solving third and fourth order nonlinear functional differential equations using discretized iterative techniques with quadrature corrections, achieving up to sixth order accuracy.

Contribution

The paper introduces a novel approach for constructing high order numerical methods for high order nonlinear functional differential equations, applicable to any order.

Findings

Methods of $O(h^4)$ and $O(h^6)$ accuracy were developed.

Numerical experiments confirmed the theoretical accuracy.

Approach can be extended to functional differential equations of any order.

Abstract

In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the trapezoidal quadrature formulas with corrections. Depending on the number of terms in the corrections we obtain methods of $O(h^4)$ and $O(h^6)$ accuracy. Some numerical experiments demonstrate the validity of the obtained theoretical results. The approach used here for the third and fourth orders nonlinear functional differential equations can be applied to functional differential equations of any orders.