In this thesis, a model to describe the interactions between hydrogen-like atoms and intense laser pulses using the Dirac equation is developed. The modeling of these interactions can be motivated by the advances in laser technology through the last decades, which have increased the peak intensities of laser pulses by many orders of magnitude. Improved understanding of how the pulses from the new lasers affect hydrogen-like atoms might transmit to improved understanding of, and control over, various states of matter exposed to intense electromagnetic radiation. The Dirac equation for the isolated hydrogen-like atom, before interacting with the laser pulse, was solved numerically through the expansion of the solution in both the dual kinetic balance basis set and in the ideal basis set. Both basis sets were based on B-splines. The ideal basis set was chosen as the preferred one, since the solutions from the dual kinetic basis set displayed an oscillatory behavior at the boundary of the domain. A small relativistic shift in the bound states and in the positive continuum was identified. The interaction matrix between the hydrogen-like atom and the laser pulse was then calculated by expanding the time dependent solution of the system in the eigenstates of the hydrogen-like atom. This was done both in the dipole approximation, and in an approximation that goes beyond the dipole approximation. The effect of the interaction with the laser pulse was calculated for both laser pulse approximations using a short-time propagator with the Lanczos algorithm. The total and energy differential photoionization probabilities were then calculated, and compared with results from the Schrödinger equation with and without relativistic corrections. A relativistic shift in both the total photoionization probability and in the locations of the peaks in the energy differential probability plot for the dipole approximation was identified. The transmission to the negative energy continuum states during the time of propagation was shown to be the primary cause of the relativistic effects in the model
