Summary of Separation of Variables — Question 10

Question 10 – Solution

Strategy. Apply the same coefficient envelope to three different modal factors, keeping the location where each error bound is valid.

Step 1: Construct the fields and their traces. The three factors give $$H=\sum a_ne^{-n^2t}\sin(nx),\quad W=\sum a_n\cos(nt)\sin(nx),\quad P=\sum a_n\frac{\sinh(ny)}{\sinh(nb)}\sin(nx).$$ The summable coefficient envelope ensures uniform displacement and boundary-series convergence. Heat derivatives converge away from $t=0$, and harmonic derivatives away from the top edge. The wave has finite energy since $\sum n^2a_n^2<\infty$; its data and PDE have the corresponding energy/weak meaning without an automatic higher-smoothness claim.

Step 2: Bound the three tails. Using $\sum_{n>N}n^{-2}\le 1/N$ gives $$\boxed{B_H(N)=\frac{e^{-(N+1)^2\tau}}N,\qquad B_W(N)=\frac 1N.}$$ For $y\le b-d$, write the hyperbolic ratio as $e^{-n(b-y)}(1-e^{-2ny})/(1-e^{-2nb})$. It is at most $e^{-nd}/(1-e^{-2b})$, so $$\boxed{B_P(N)=\frac{e^{-(N+1)d}}{N(1-e^{-2b})}.}$$ These respectively bound the heat error for $t\ge\tau$, the wave displacement error at all times, and the harmonic error in the stated lower strip.

Step 3: Choose certified integer cutoffs. Each bound decreases with $N$. With the specified parameters, $$\begin{array}{c|cc|c} &\text{preceding bound}&\text{accepted bound}&N\\ \hline H&B_H(4)\approx.02052&B_H(5)\approx.005465&5\\ W&B_W(99)\approx.01010&B_W(100)=.01000&100\\ P&B_P(9)\approx.01055&B_P(10)\approx.007394&10 \end{array}$$ These are the smallest integers certified by these bounds, not necessarily the fewest modes required by particular data. The graph uses continuous bound envelopes on logarithmic axes.

Step 4: Interpret the filtering mechanisms. Heat damps high modes by $e^{-n^2t}$; distance from the driven harmonic edge gives spatial attenuation of order $e^{-nd}$. Wave modes retain their amplitudes and only oscillate: at every $t=2m\pi$, the entire displacement returns to $f$. Consequently waiting cannot supply a uniform future decay factor for every wave tail. The heat and harmonic improvements require positive waiting time or positive distance; they cannot be silently extended to $t=0$ or to the top trace.