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## Initial-boundary-value Problems

This time the infinite plane is replace by the semi-infinite plane, with a boundary condition imposed on . (1.13)

Consider the case , so that the waves are travelling from small to large . The characteristics are again given by The characteristic through the origin, splits the plane into two regions, as shown in Figure 1.6. In region , ,the solution is determined by the initial condition at . In this region the characteristics originate from the -axis.

In region , , the solution is given by the boundary condition at . Here the characteristics originate from the -axis. Remembering that the solution is constant along the characteristics, this implies that the solution is determined by where the characteristics come from. Thus, the solution is (1.14)

Note that the characteristics are It is clear that (1.14) satisfies the initial condition, since implies that and , and the boundary condition since implies that and so . In addition, it is easily shown that also satisfies the PDE.

Example 1. .9 with The characteristics are given by Therefore the plane is split into three regions and the solution is A little bit of thought about this last example will soon illustrate a major problem. If the wave speed is negative, i.e. , then there is in general no solution to the initial-boundary-value problem. This is because the characteristics intersect both boundaries and cannot be defined by two different values (resulting from values of and ) on the same characteristics. This is illustrated in Figure 1.7.    Next: Method of Characteristics : Up: Method of Characteristics Previous: Initial-value Problems
Prof. Alan Hood
2000-10-30