Conceptual Question About PDE Boundary Conditions

[Metronome]

In this 1D boundary value explanation, the idea seems to be that the slope of the line at the boundaries is not 0 at time t = 0, but becomes 0 immediately after time begins progressing, as a consequence of the Laplacian seeing only one neighboring point at each boundary point instead of two.

However, when an example is worked, it seems like the PDE solution satisfying the boundary condition is engineered precisely AT t = 0, the one value of t where the boundary condition DOESN'T need to be satisfied.

Is this "flattening of the boundaries at the first nonzero instant" feature incidental to the first linked animation, or is it a general feature of boundary conditions? Is it hiding somewhere in the worked example of the heat equation?

 

[HallsofIvy]

In order that such a problem have a solution, the solution must be continuous so that if the solution "becomes 0 immediately after time begins progressing" then it must be 0 at time t= 0. Does the problem actually use the phrase "at the first non-zero instant"? There is NO "first non-zero instant"!

 

[Metronome]

The phrasing the video at 9:03 uses is, "Taking the example of a straight line, whose slope at the boundary points is decidedly not 0 [the animation shows an under-labeled graph of temperature as a function of time, for the sake of concreteness we'll call it the line segment T = .8t from t = 0 to t = L], as soon as the clock starts, those boundary values will shift infinitesimally, such that the slope there suddenly becomes 0, and remains that way through the remainder of the evolution. In other word, finding a function satisfying the heat equation itself is not enough; it must also the property that it's flat at each of those endpoints for all times greater than 0."

My paraphrase was based on the phrase "slope...is decidedly not 0, as soon as the clock starts, those boundary values will shift infinitesimally, such that the slope there suddenly becomes 0." It sounds to me like you're saying this is not mathematically valid, or perhaps I introduced problematic language in my paraphrase?