The solution of Ordinary Differential Equations (ODEs) [1] and their analogues, Partial Differential Equations (PDEs) [2, 3], have a long history. The basic types of equations maybe identified in their simplest forms as the Laplace equation, the wave equation, and the diffusion equation (or heat equation) [4] and the latter two are examined in this chapter. The history of solution methods has been strongly affected by the development of computers and numerical solution methods, but only some special cases and (numerical) solution methods are discussed. The number and type of numerical algorithms are varied. However, all attempt to project the equation of interest into a discrete form on a physical spatial (or temporal) grid that is then solvable as a finite rank matrix problem. This discretization implies that continuous functions and operators on them are smooth and well defined on the interval of interest.