In the previous sections we saw some examples of partial differential equations. We now consider some important issues regarding the formulation and solvability of PDE problems. A solution to a PDE can be described as simply a function that reduces that PDE to an identity on some region of the independent variables. In general, a PDE alone, without any auxiliary boundary or initial conditions, will either have an infinity of solutions, or have no solution. Thus, in formulating a PDE problem there are three components: (i) the PDE; (ii) the region of space-time on which the PDE is required to be satisfied; (iii) the auxiliary boundary and initial conditions to be met.
For a PDE based mathematical model of a physical system to give useful results, it is generally necessary to formulate that model as what mathematicians call a well posed PDE problem. A PDE problem is said to be well posed if