Author Topic: HA3 problem 6  (Read 701 times)

Victor Ivrii

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HA3 problem 6
« on: February 05, 2015, 07:38:31 PM »
a.  Consider the heat equation on $J=(-\infty,\infty)$ and prove that an energy
\begin{equation}
E(t)=\int_J u^2 (x,t)\,dx
\label{eq-HA3.5}
\end{equation}
does not increase; further, show that it really decreases unless $u(x,t)=\operatorname{const}$;

b.  Consider the heat equation on $J=(0,l)$ with the Dirichlet or Neumann boundary conditions and prove that an $E(t)$ does not increase; further, show that it really decreases unless $u(x,t)=\operatorname{const}$;

c. Consider the heat equation on $J=(0,l)$ with the Robin boundary conditions
\begin{gather}
u_x(0,t)-a_0u(0,t)=0,\\[4pt]
u_x(l,t)+a_lu(l,t)=0.
\end{gather}
If $a_0>0$ and $a_l>0$, show that the endpoints contribute to the decrease of $E(t)=\int_0^l u^2 (x,t)\,dx$.

This is interpreted to mean that part of the  energy is lost at the boundary, so we call the boundary conditions  radiating or dissipative.

Hint. To prove decrease of $E(t)$ consider it derivative by $t$, replace $u_t$ by $ku_{xx}$ and integrate by parts.

Remark. In the case of heat (or diffusion) equation an *energy* given by (\ref{eq-HA3.5}) is rather mathematical artefact.

Yang Liu

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Re: HA3 problem 6
« Reply #1 on: February 06, 2015, 02:32:43 AM »
Attached :)

Victor Ivrii

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Re: HA3 problem 6
« Reply #2 on: February 06, 2015, 11:54:26 AM »
Difficult to read, not very useful. Please do not use upright "d" for partial derivatives (use $\partial$)