Toronto Math Forum
APM3462018S => APM346Tests => Term Test 2 => Topic started by: Victor Ivrii on March 23, 2018, 06:14:32 AM

Solve by Fourier method
\begin{align}
& u_{tt}u_{xx}=0\qquad 0<x<\pi,\label{11}\\
& u_x_{x=0}= 0,\qquad (u_x+\alpha u)_{x=\pi}=0\label{12}\\
&u _{t=0}=\cos (x),\qquad u_t_{t=0}=0\label{13}
\end{align}
with $\alpha\in \mathbb{R}$.
Hint: We know that $\lambda_n$ are real but since we do not know the sign of $\alpha$ we do not know if it all $\lambda_n\ge 0$; so you must consider the case of some of $\lambda_n<0$.
Note: Only find equations for eigenvalues.

The associated eigenproblem is
\begin{equation}\left\{\begin{split}&X''=\lambda X,\\&X'_{x=0}=(X'\alpha X)_{x=0}=0.\end{split}\right.\label{14}\end{equation}
If $\alpha=0$ the we know the solution are integer $\cos$'s
\begin{equation}\label{error}\lambda_n=n^2, X_n(x)=\cos nx, n=0,1,....\end{equation}
If $\alpha\neq 0,\lambda>0$ then the general solution for the DE in \eqref{14} is
$$X(x)=A\cosh \gamma x + B\sinh \gamma x, \gamma>0.$$
Plugging in boundary condition we find $B=0$ and
$$\gamma A\sinh \gamma\pi+\alpha A\cosh \gamma\pi=0.$$
Hence the various eigenvalues are given by $\lambda_n=\gamma_n^2$ where $\gamma_n$ is a nonzero root of
$$\gamma\tanh \gamma\pi+\alpha=0.$$
If $\alpha\neq 0,\lambda<0$ then the general solution for the DE in \eqref{14} is
$$X(x)=A\cos \omega x + B\sin \omega x, \omega>0.$$
Plugging in boundary condition we find $B=0$ and
$$\omega A\sin\omega\pi+\alpha A\cos \omega\pi=0.$$
Hence the various eigenvalues are given by $\lambda_n=\omega_n^2$ where $\omega_n$ is a nonzero root of
$$\omega\tan \omega\pi\alpha=0.$$
If $\lambda=0$ then we have only trivial solution.

I attach pictures for $\lambda<0$ and $\lambda >0$. On the first, brown line for $\alpha >0$, red line for $\alpha<0$. On the second brown line for $\alpha> 1/\pi$, red line for $\alpha<1/\pi$
Also correct sign at $\lambda_n$

Again I don't quite see the issue of sign, please inform me the mistake.
Found it. To the posterity: my mistake was on \eqref{error}.
By the way, how can I draw with tikz a picture like this? or did you use latex at all?

I posted the answer to http://forum.math.toronto.edu/index.php?topic=658.msg3971 (http://forum.math.toronto.edu/index.php?topic=658.msg3971)