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Term Test 1 / Re: TT1 = Problem 2
« on: October 16, 2012, 06:56:06 PM »
The initial conditions are not sufficient to uniquely define the solution - we need boundary conditions at x = 0.
Consider a counter-example to uniqueness:
1. u = 0 over x > 0 at t = 0
2. a large wave, moving in the -x direction, at t=0 (but still in the region x > 0).
Both 1. and 2. satisfy the given initial conditions, so the solution is not unique. Had we specified Dirichlet, Neumann, etc. conditions, however, one of the two cases would be impossible.
Consider a counter-example to uniqueness:
1. u = 0 over x > 0 at t = 0
2. a large wave, moving in the -x direction, at t=0 (but still in the region x > 0).
Both 1. and 2. satisfy the given initial conditions, so the solution is not unique. Had we specified Dirichlet, Neumann, etc. conditions, however, one of the two cases would be impossible.