# On the Extension of Adams--Bashforth--Moulton Methods for Numerical Integration of Delay Differential Equations and Application to the Moon's Orbit

## TL;DR

This paper extends Adams--Bashforth--Moulton methods to enable accurate forward and backward numerical integration of delay differential equations, with applications to modeling the Moon's orbit considering tidal delays.

## Contribution

It introduces a modified Adams--Bashforth--Moulton method capable of integrating DDEs both forwards and backwards in time, addressing a key challenge in celestial mechanics.

## Key findings

- Successful backward and forward integration of Moon's DDEs demonstrated
- Enhanced numerical methods for delay differential equations developed
- Application to lunar orbit modeling with improved accuracy

## Abstract

One of the problems arising in modern celestial mechanics is the need of precise numerical integration of dynamical equations of motion of the Moon. The action of tidal forces is modeled with a time delay and the motion of the Moon is therefore described by a functional differential equation (FDE) called delay differential equation (DDE). Numerical integration of the orbit is normally being performed in both directions (forwards and backwards in time) starting from some epoch (moment in time). While the theory of normal forwards-in-time numerical integration of DDEs is developed and well-known, integrating a DDE backwards in time is equivalent to solving a different kind of FDE called advanced differential equation, where the derivative of the function depends on not yet known future states of the function. We examine a modification of Adams--Bashforth--Moulton method allowing to perform integration of the Moon's DDE forwards and backwards in time and the results of such integration.