In this work we face hyperbolic and elliptic systems of partial differential equations with applications from health sciences to astrophysics. Some will be framed in the context of classical mechanics and other in the Theory of Relativity. Concerning the classical sector we will solve Navier-Stokes Equations to model the blood flow in aorta trying to get some relations between geometrical features and physiological magnitudes of interest. We will also discuss the Euler Equation from both Newtonian and general relativistic approach. We will derive some theoretical results with applications in the development of numerical methods for this balance law. We will propose an improved version of a Fully Constrained Formulation of the Einstein Equations. It will preserve the local uniqueness from previous versions and posses accuracy improvements with the introduction of new variables. Some preliminary test will be carried out. On the other hand, we will introduce a new numerical method to perform the time integration of stiff balance laws. The new approach present stability properties of implicit methods dealing with stiffness but with a computation cost similar to that of an explicit method. First tests in the context of Resistive Relativistic Magnetohydrodynamics and Radiation Hydrodynamics were performed. We will finish with a new algorithm to polynomial regression in several variables with applications in simulations of neutron stars.