12. The heat equation on Rⁿ

Corrections:

The statement on the last line of the theorem at 13:15 is wrong. It can be corrected by changing "sup φ" to "sup |φ|" (and similarly in the proof later). But actually a better statement would be "inf φ ≤ u(x,t) ≤ sup φ". This is proved simply by multiplying the obviously true inequality "inf φ ≤ φ(x) ≤ sup φ" by S(x−y,t) and integrating over all y.

And the plot of Tychonoff's example u(x,t) at 45:10 is completely wrong, as a result of using far too few terms in the approximation! For a correct approximation on all of that domain, nearly 1000 terms would have been needed, instead of just 16. Below is the incorrect plot from the video, together with a correct plot on a smaller domain, using 400 terms.

Wrong (16 terms):

Right (400 terms):

The graph is truncated at ±5; the actual values are way too huge to display at this scale, especially near the closest corner of the picture, where the function first stays very close to zero for a short while, then grows extremely fast, then just as quickly dips down to negative values of large magnitude, and then goes up and down again a few times. The values are computed at the gridpoints (x,t) = (k/25,m/200) (where k and m are integers), and the extreme ones among those values are u(4.96,0.065) ≈ 1.9×10⁸ and u(5,0.055) ≈ −4.3×10⁸. For more information, see this answer that I posted on MathOverflow.

Sidansvarig: Hans Lundmark
Senast uppdaterad: 2023-04-05