Toronto Math Forum

The question says we should do it with a Fourier transform, but why don't I just take
$$u(x,y)=e^{-|y|-ix}$$? That does give me a zero Laplacian and satisfies the boundary condition, or am I wrong here?

I see, thanks.

Are we considering closed domains, i.e. $\Sigma \in \Omega$? I think we have to, otherwise the $max_{\Omega}u \geq max_{\Sigma}u$ does not pull, correct?

Hi,

what is meant by $\frac{\partial G}{\partial v_x}$ (eq. 7 in Lec. 26)? is $\frac{\partial G}{\partial v_x}=\nabla G \cdot (\nabla \cdot \vec{v})$?

Thanks!

Ok I did
$$
u(x,y)=\int_0^{\infty}\hat{u}(\omega,y)\sin(\omega x)d\omega
$$
and took the resulting eigenvalue problem as
$$
\hat{u}_{yy}-\omega^2\hat{u}=0
$$
which yields eigenvalues $\omega=i\pi n$
and eigenfunctions
$$
\hat{u}(\omega,y)=A(\omega) e^{i \pi n y}+B(\omega)e^{-i \pi n y}
$$
Is that wrong?

Dear Professor,

do we need to find a closed expression for the solution in problem 3, or can we express it as a sum over n?

Hey,

I get an ODE for problem 2, but it is not of the Euler type and I don't know how to solve it. Could anyone give me a hint?

Thanks!

I obtain the solution
$$
u(x,t)=\frac{1}{2}((x-ct)e^{x+ct}+(x-ct)e^{x-ct})+
\int_{x-ct}^{x+ct}e^{-x}dx+\int_0^{t}\int_{x-ct}^{x+ct}e^{-x-t}dxdt
$$
which becomes
$$
\frac{2ce^{-x}}{1-c^2}+\frac{e^{-t}(e^{-ct}+e^{ct})}{1+c}
$$
(plugging in $c=5$ doesn't really simplify it)

Did anyone get the same?

Dear all,

I don't know what to do with problem 3. We are asked to find conditions on the three parameters $\alpha$, $\beta$ and $\gamma$ s.t. the integral
$$
E(t)=\frac{1}{2}\int_0^L (|u_t|^2+c^2|u_x|^2+\gamma |u|^2)dx
$$

is time-independent, where u satisfies b.c. and $u_{tt}-c^2u_{xx}+\gamma u=0$.

The time independence of the integral means that
$$
\frac{\partial}{\partial t}u_tu^*_t+c^2\frac{\partial}{\partial t}u_xu^*_x+\gamma \frac{\partial}{\partial t}u u ^* =0
$$

but I can`t find $u$, as there is this $u$ term in the wave equation, and the boundary conditions do not help me with this equation neither. Any hints? Thanks a lot!