diff --git a/ana17.tex b/ana17.tex index 30f29a8..43bc572 100644 --- a/ana17.tex +++ b/ana17.tex @@ -58,6 +58,42 @@ \text{Graph}(y) = \{ (t, y(t)) , t \in I_{\text{max}}\} \] unbeschränkt ist, weil $t \to t_0 + T^{*} = \infty$ oder $\Vert y(t) \Vert \xrightarrow{t \to t_0 + T^{*}} \infty$. + + \begin{figure}[h] + \centering + \begin{tikzpicture}[declare function={f(\x) = tan(deg(\x-2));}] + \begin{axis}% + [default 2d plot, + grid=none, + ymax=4, + ymin=-4, + xmin=0, + xmax=4, + xtick=\empty, ytick=\empty, + ] + \addplot[domain=0.56:3.56,samples=100,smooth,red] {f(x)}; + \draw[dashed] (0.56, 5) -- (0.56, -5); + \draw[dashed] (3.45, 5) -- (3.45, -5); + \draw (2.56, {f(2.56)}) node[fill,inner sep=1pt]{}; + \draw (2.56, {f(2.56)}) node[draw,shape=rectangle,minimum width=10mm, minimum height=7mm, + anchor=center] {}; + \draw (2.85, {f(2.85)}) node[fill,inner sep=1pt]{}; + \draw (2.85, {f(2.85)}) node[draw,shape=rectangle,minimum width=9mm, minimum height=7mm, + anchor=center] {}; + \draw (3.02, {f(3.02)}) node[fill,inner sep=1pt]{}; + \draw (3.02, {f(3.02)}) node[draw,shape=rectangle,minimum width=6mm, minimum height=6mm, + anchor=center] {}; + \draw (3.12, {f(3.12)}) node[fill,inner sep=1pt]{}; + \draw (3.12, {f(3.12)}) node[draw,shape=rectangle,minimum width=5mm, minimum height=5mm, + anchor=center] {}; + \draw (3.18, {f(3.18)}) node[fill,inner sep=1pt]{}; + \draw (3.18, {f(3.18)}) node[draw,shape=rectangle,minimum width=4mm, minimum height=4mm, + anchor=center] {}; + \draw (3.65, -4) node{$\partial D$}; + \end{axis} + \end{tikzpicture} + \caption{Schrittweise Fortsetzung einer Lösung bis zum Rand von $D$} + \end{figure} \end{bem} \begin{korollar}[Globale Existenz] @@ -227,6 +263,7 @@ \end{tikzpicture} \caption{Für $y_0 = 0$ existieren beliebig viele zusammengesetzte Lösungen.} \end{subfigure} + \caption{Zur Uneindeutigkeit von AWA} \end{figure} Beobachtung: $f(t,x)$ ist stetig auf $\R \times \R$, aber $f(t,x)$ ist nicht Lipschitz stetig diff --git a/analysisII.pdf b/analysisII.pdf index 814834c..08eba11 100644 Binary files a/analysisII.pdf and b/analysisII.pdf differ