From 9e7524c25a0ba5a643fbb7555d01311f69aa603e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 3 Jun 2021 17:18:58 +0200 Subject: add slides --- vorlesungen/slides/2/hilbertraum/sturm.tex | 56 ++++++++++++++++++++++++++++++ 1 file changed, 56 insertions(+) create mode 100644 vorlesungen/slides/2/hilbertraum/sturm.tex (limited to 'vorlesungen/slides/2/hilbertraum/sturm.tex') diff --git a/vorlesungen/slides/2/hilbertraum/sturm.tex b/vorlesungen/slides/2/hilbertraum/sturm.tex new file mode 100644 index 0000000..1d772d6 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/sturm.tex @@ -0,0 +1,56 @@ +% +% sturm.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Sturm-Liouville-Problem} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Wellengleichung} +Saite mit variabler Massedichte führt auf die DGL +\[ +-y''(t) + q(t) y(t) = \lambda y(t), +\quad +q(t) > 0 +\] +mit Randbedingungen $y(0)=y(1)=0$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Sturm-Liouville-Operator} +\[ +A=-\frac{d^2}{dt^2} + q(t) = -D^2 + p +\] +auf differenzierbaren Funktionen $\Omega=[0,1]\to\mathbb{C}$ mit Randwerten +\[ +f(0)=f(1)=0 +\] +\end{block} +\end{column} +\end{columns} +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,Ag \rangle +&= +\langle f,-D^2 g\rangle + \langle f,qg\rangle += +- +\int_0^1 \overline{f}(t) \frac{d^2}{dt^2}g(t)\,dt ++\langle f,qg\rangle +\\ +&=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0} ++\int_0^1 \overline{f}'(t)g'(t)\,dt ++\langle f,qg\rangle +=-\int_0^1 \overline{f}''(t)g(t)\,dt ++\langle qf,g\rangle +\\ +&=\langle Af,g\rangle +\end{align*} +\end{block} +\end{frame} +\egroup -- cgit v1.2.1 From 680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 3 Jun 2021 18:51:36 +0200 Subject: phases --- vorlesungen/slides/2/hilbertraum/sturm.tex | 20 +++++++++++--------- 1 file changed, 11 insertions(+), 9 deletions(-) (limited to 'vorlesungen/slides/2/hilbertraum/sturm.tex') diff --git a/vorlesungen/slides/2/hilbertraum/sturm.tex b/vorlesungen/slides/2/hilbertraum/sturm.tex index 1d772d6..a6865ab 100644 --- a/vorlesungen/slides/2/hilbertraum/sturm.tex +++ b/vorlesungen/slides/2/hilbertraum/sturm.tex @@ -22,6 +22,7 @@ mit Randbedingungen $y(0)=y(1)=0$ \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% \begin{block}{Sturm-Liouville-Operator} \[ A=-\frac{d^2}{dt^2} + q(t) = -D^2 + p @@ -30,27 +31,28 @@ auf differenzierbaren Funktionen $\Omega=[0,1]\to\mathbb{C}$ mit Randwerten \[ f(0)=f(1)=0 \] -\end{block} +\end{block}} \end{column} \end{columns} +\uncover<3->{% \begin{block}{Selbstadjungiert} \begin{align*} \langle f,Ag \rangle -&= +&\uncover<4->{= \langle f,-D^2 g\rangle + \langle f,qg\rangle = - \int_0^1 \overline{f}(t) \frac{d^2}{dt^2}g(t)\,dt -+\langle f,qg\rangle ++\langle f,qg\rangle} \\ -&=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0} +&\uncover<5->{=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0} +\int_0^1 \overline{f}'(t)g'(t)\,dt -+\langle f,qg\rangle -=-\int_0^1 \overline{f}''(t)g(t)\,dt -+\langle qf,g\rangle ++\langle f,qg\rangle} +\uncover<6->{=-\int_0^1 \overline{f}''(t)g(t)\,dt ++\langle qf,g\rangle} \\ -&=\langle Af,g\rangle +&\uncover<7->{=\langle Af,g\rangle} \end{align*} -\end{block} +\end{block}} \end{frame} \egroup -- cgit v1.2.1