We consider the ionization of atomic hydrogen by a strong infrared field. By starting from the corresponding time-dependent Schr\"odinger equation in momentum space, we develop a model in which the kernel of the nonlocal Coulomb potential is replaced by a finite sum of separable potentials. Each separable potential supports one bound state of atomic hydrogen. Here, we consider only the $1s$, $2s$, and $2p$ states. In this way, the full three-dimensional Schr\"odinger equation reduces to a system of a few coupled one-dimensional linear Volterra integral equations. The objective of this first contribution is to give a detailed account of the model and to validate it in physical situations where the dynamics are well understood.