| The Diagram on the right shows 2 possible initial
conditions, i.e. Graphs of u, and their progress through T=0, T=T1,
T=T2. In the upper diagram, we can see u decaying from A to B to C:
there is a general 'decaying, flattening' trend. In the lower diagram, a
different initial condition is shown decaying from A' to B' to C'. Now,
although the initial condition here is very different from the 1st case,
we can see that at time T=T2, i.e. Graph C', the picture is not much
different from C: the irregularities at A' have been smoothed out, in
fact, that information is nearly lost.
The 'Forward' or Classical problem is, given phi, determine u(x,t) at some later time. The Inverse Problem (for example) is to try to determine phi from some later condition. As the diagram shows, this can be very difficult, or even impossible. BOTH C and C' may have come from A or A', so how can we determine which was the starting condition? Mathematically, the problem is said to be ill-posed, i.e. a small change in the state at C can cause a large change at A (if 'looking backward' in time), so any solution will be unstable, due to measurement errors. However, there are numerical and analytic techniques in existence, and still under development, to deal with this problem.
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