The Existence Uniqueness Theorem

[bmed90]

In your own words, what exactly is the purpose of the Existence Uniqueness Theorem and why is it useful

 

[arildno]

In your own words, what exactly is the purpose of the Existence Uniqueness Theorem and why is it useful

1, Theorems have no purpose. They are either true or false.
2. Theorems in general have little usefulness for the purpose of peeling a banana, for example.

 

[HallsofIvy]

Don't you think it is nice to know whether or not a problem has a solution before you begin trying to solve it? As for uniqueness, think about the applications of differential equations to phsics. If you were to drop something repeatedly, isn't it nice to know that the uniqueness of solutions to the differential equation guarantee that it will always fall in the same way? Physics would be very complicated if experiments done repeatedly, in exactly the same way, could have many different outcomes.

Here's an example of an experiment that can: Balance a very thin rod on one end. With absolutely no "perturbation", that rod would stay balanced but, of course, there are always some kind of air current or other perturbation. And there is no way of telling in which direction the rod will fall precisely because the differential equations governing the situation do NOT satisfy the hypotheses of the "uniqueness theorem"

Here is an interesting application of the "uniqueness" property:

Suppose a taut wire is attatched to a point on the wall (the wire cannot move up or down at that point) and a single "hump" is formed on the wire, above the line of the wire when untouched, which then moves toward the wall satifying the "wave" equation. Of course, the hump is "reflected" when it hits the wall. Does it come back above or below the wall?

The wave equation is a nice, well behaved, that certainly satifies the conditions of the "existence and uniqueness" theorem for diferential equations. If we imagine the wire extending beyond the wall, so there is no wall and no fixed point there, and second hump, beyond the wall, symmetric to the first except that it is upside down, when the two humps hit the wall, they will cancel, not moving the point at the wall.

That is, that "double hump wave" satisfies the wave equation and the boundary condition that the point on the wall not move. As the original wave continues past the wall, the upside down wave continues on this side. Since the solution is unique, and that "double wave" satisfies both the equation and the boundary condition, it must be the same as the solution to the original problem. That is, the wave must reflect upside down.

You can use the same argument, looking at the slope of the wave, to show that if the point on the wall is allowed to move up and down (on a rail, perhaps) but the wire is force to have slope 0 there, the wave is reflected still above the horizontal.