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One of the problems arising in modern celestial mechanics is the need ofprecise numerical integration of dynamical equations of motion of the Moon. Theaction of tidal forces is modeled with a time delay and the motion of the Moonis therefore described by a functional differential equation (FDE) called delaydifferential equation (DDE).  Numerical integration of the orbit is normally being performed in bothdirections (forwards and backwards in time) starting from some epoch (moment intime). While the theory of normal forwards-in-time numerical integration ofDDEs is developed and well-known, integrating a DDE backwards in time isequivalent to solving a different kind of FDE called advanced differentialequation, where the derivative of the function depends on not yet known futurestates of the function.  We examine a modification of Adams--Bashforth--Moulton method allowing toperform integration of the Moon's DDE forwards and backwards in time and theresults of such integration.

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