This graduate-level text provides extensive information on the numerical solutions of equations for modelling aquifers, heat conduction and diffusion, the dynamic vibration of structures, and associated non-linear and coupled problems where the space discretization is done by the finite element method. The results are first or second order ordinary differential equations of a particular form because of their origin. Taking a completely new approach, the author discusses the mathematical implications and limitations of the most popular algorithms for their solution. The book also gives readers results of the author's extensive work with engineers and scientists along with a balanced account of other results in the literature. The mathematics is deliberately kept simple, enabling mathematicians and engineers to communicate in the solution with a wide range of problems.