Open vs closed notation for time-interval of the heat equation

[Krishnakumar Gopalakrishnan]

I am slightly confused by the use of open vs closed interval for denoting the time-interval of the heat equation.

For example in lecture 27 (https://www.math.colostate.edu/~bangerth/videos.676.27.html), both the video lecture and the slides seem to convey that the PDE holds for the closed interval [0,t]. However, the step-26 tutorial page denotes the same pde as being valid only in the open interval (0,t).

I acknowledge that this is a technical subtlety and may not change the results or code, but which one is more mathematically correct (closed-interval or open-interval)?

Regards,

Krishna

[Wolfgang Bangerth]

It needs to be an open interval. That's because to define the definition of a
function f(x) at some point x, you need that
f'(x0) = lim_{x->x0} (f(x)-f(x0))/(x-x0)
is well defined. That requires that f(x) is defined both to the left and to
the right of x0, and that can only be the case if x0 is an interior point of
an interval. So the derivative is *always* only well defined for interior
points, and it should have been the open interval (0,T) in the same way as we
always define Omega to be an open set.

Best
W.

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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/