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6 Well Posed PDE Problems     continued...

For second order hyperbolic PDE problems, the vibrating string is most frequently used as an example of a well posed problem. Think of as representing the vertical displacement at position x and time t of an ideal string which in static equilibrium occupies the horizontal line joining x = 0 and x = L. Then the following IBVP models the movement of the string subject to an initial displacement given by and an initial velocity given by .

A dramatic example of an ill-posed, second order hyperbolic PDE problem is given by the following BVP for the one dimensional wave equation. It can be shown that if T is irrational, then the only solution of this BVP for the wave equation is u identically zero; whereas if T is rational, the problem has infinitely many nontrivial solution. Thus the solution fails to depend continuously on the data - namely on the size of the region on which the problem is stated.

Figure 2:Ill-posed, second order hyperbolic PDE problem described in Example 13.