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6 Well Posed PDE Problems

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

1.
a solution to the problem exists
2.
the solution is unique, and
3.
the solution depends continuously on the problem data.

(In a PDE problem the problem data consists of the coefficients in the PDE; the functions appearing in boundary and initial conditions; and the region on which the PDE is required to hold.)