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For the wave equation $u_{tt} =c^2u_{xx}$ subject to the Robin boundary conditions $$a_0u(0, t)-b_0u_x (0, t) = 0$$ $$a_1u(1, t)+b_1u_x (1, t) = 0$$ for constants $a_0, b_0, a_1, b_1$ with $b_0\neq0$ and $b_1\neq 0$. Calculate the derivative $E'(t)$ of the energy given in the example.

How should $E(t)$ be modified in order to remain constant in time with these new boundary conditions?

What constraints should be imposed on $a_0, b_0, a_1, b_1$ to ensure the modified energy is a nonnegative function of $u$?


Attempt:

The energy $E$ associated with the wave equation $u_{tt} =c^2u_{xx}$ for $0<x<1,t>0$ is given by $$E(t)=\int_0^1 (u_t)^2+c^2(u_x)^2dx.$$

By differentiating with respect to $t$, we find, $$\frac{dE}{dt}=\int_0^1((u_t)^2+c^2(u_x)^2)dx.$$ Next integrating the second term on the right side by parts gives \begin{align*} \frac{dE}{dt} &=2\int_0^1 u_t(u_{tt}-c^2u_{xx})dx+c^2u_xu_t\Big|_0^1 \\ \end{align*} Given our PDE and boundary conditions, \begin{align*} a_0u(0,t)-b_0u_x(0,t)=0 \implies u_x(0,t)=\frac{a_0}{b_0}u(0,t) \\ a_1u(1,t)+b_1u_x(1,t)=0 \implies u_x(1,t)=-\frac{a_1}{b_1}u(1,t) \end{align*} we get \begin{align*} \frac{dE}{dt} &= 2\int_0^1 u_t(0)dx-c^2\Big[\frac{a_1}{b_1}+\frac{a_0}{b_0}\Big]uu_t\Big|_0^1 \\ &= -c^2\Big[\frac{a_1}{b_1}+\frac{a_0}{b_0}\Big]uu_t\Big|_0^1. \end{align*}


Assuming my $E'(t)$ is correct, the answer to the second question would be, let $c^2=\frac{a_1}{b_1}-\frac{a_0}{b_0}$. This will ensure $E'(t)$ to vanish - making energy constant in time.

My intuition is to show that at the boundaries the max and min are both $\ge0$. Invoking max principle, we can argue that E(t) will be non-negative. But I'm not sure what constraints to impose to the modified Energy.

EDIT: For Question 3, since I let $c^2=\frac{a_1}{b_1}-\frac{a_0}{b_0}$, we just have to let this term be positive (i.e. $\frac{a_1}{b_1}>\frac{a_0}{b_0})$

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Your calculation is slightly off at the end. You should get $$ \frac{d}{dt} \int_0^1 (u_t)^2 +(u_x)^2 = 2 c^2 [u_x(1,t) u_t(1,t) - u_x(0,t)u_t(0,t) ] \\ = -2 c^2 \left[ \frac{a_1}{b_1} u(1,t)u_t(1,t) + \frac{a_0}{b_0} u(0,t) u_t(0,t) \right] \\ = -\frac{d}{dt} c^2 \left[ \frac{a_1}{b_1} (u(1,t))^2 + \frac{a_0}{b_0} (u(0,t) )^2 \right]. $$ Now you can use this to define a new energy that is preserved in time, and once you've done this you should be able to see how to guarantee the new energy is positive.

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