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z0>#u)u!H#`uu(0auEQ7w6{_dS#_&O*Od%rEw?CuIvE0;FRAdWu+Wb}`Lt5^&4U8?4 z-4>>YQb^prlzyp_{-7VYmVTykYBAYS|K-umHWtvW*Hi2!I6M6g0;i0FRa*Igd3O4l z8X@PWEoU5f2@WiI+Od%^A0_{L(Aj{bwTSZpqTEZy=&tdcYW6#GqY2(;h1RtNb|SMj zPi 0\text{ Reproduktionsrate der Beute}\\ + &&&\beta > 0\text{ Fressrate der Räuber pro Beute}\\ + N_2' &= -\gamma N_2 + \delta N_1N_2&&\gamma > 0\text{ Sterberate der Räuber, wenn keine Beute vorhanden ist}\\ + & &&\delta > 0\text{ Reproduktionsrate der Räuber pro Beute} + \end{align*} + \item SIR - Modell aus Epidemiologie (z.B. Corona): + \item \begin{tabular}{ccc} + succeptible & infected & removed\\ + $S(t)$ & $I(t)$ & $R(t)$ + \end{tabular} + \begin{align*} + N &= I + S + R\\ + \dv{S}{t} &= \nu N - \beta \frac{SI}{N}-\mu S\\ + \dv{I}{t} &= \beta \frac{SI}{N} - \gamma I - \mu I\\ + \dv{R}{t} &= \gamma I - \mu R + \end{align*} + Dabei sei + \begin{align*} + \gamma&\text{ die Rate, mit der Infizierte genesen oder sterben,}\\ + \mu&\text{ die allgemeine Sterberate pro Person,}\\ + \nu&\text{ die Geburtsrate pro Person,}\\ + \beta& + \text{ die Anzahl neuer Infektionen, die ein erster infektiöser Fall pro Zeit verursacht und}\\ + \frac{\beta}{N}& \text{ die Transmissionsrate.} + \end{align*} + \end{enumerate} +\end{bsp} +%\begin{figure}[h] +% \caption{Veranschaulichung: Richtungsfeld +%\end{figure} +\begin{definition}[System erster Ordnung] + Sei $D = I\times \Omega \subset \R\times \R^n,\quad f\colon D\to \R^n$ stetig. Dann heißt + \begin{equation} + y'=f(t,y)\label{DGLOrd1}\tag{$\star$} + \end{equation} + ein System von $n$ Differentialgleichungen 1. Ordnung. +\end{definition} +Eine Lösung von \eqref{DGLOrd1} ist eine differenzierbare Funktion $y:I\to \R^n$ mit +\begin{enumerate}[(a)] + \item $\graph(y)\coloneqq \{(t,y(t))\in \R\times \R^n\mid t\in I\}\subset D$ und + \item $y'(t) = f(t,y(t))\quad \forall t\in I$. +\end{enumerate} +\begin{bem} + $y = \begin{pmatrix} + y_1\\\vdots\\y_n + \end{pmatrix}$ und $f=\begin{pmatrix} + f_1\\\vdots\\f_n + \end{pmatrix}$ Dann ist + \begin{align*} + \eqref{DGLOrd1} \Leftrightarrow y_1'&= f_1(t,y_1,\dots,y_n)\\ + \vdots&\\ + y_n'&= f_n(t,y_1,\dots,y_n) + \end{align*} +\end{bem} +\begin{definition}[Anfangswertaufgabe/Anfangswertproblem] + AWA zu \eqref{DGLOrd1} ist: + \begin{align*} + y' &= f(t,y),\quad t\in I \\ + y(t_0) &= y_0&&\text{Anfangsbedingung} + \end{align*} + Gesucht wird eine differenzierbare Funktion $y\colon I\to \R^n$ derart, dass + \begin{enumerate} + \item $\graph(y) \subset D$ + \item $y'(t) = f(t,y(t)),\;t\in I$ + \item $y(t_0) = y_0$ + \end{enumerate} +\end{definition} +\begin{satz}[DGL $\leftrightarrow$ Integralgleichung] + Sei $D\subset \R\times \R^n,\; f\colon D\to \R^n$ stetig, $(t_0,y_0)\in D$ und $y\colon I\to \R^n$ stetig mit $\graph(y)\subset D,\; t_0\in I$. Dann gilt + \[ + y\text{ löst AWA }y'=f(t,y),\;y(t_0)=y_0\Leftrightarrow y(t) = y_0 + \int_{t_0}^t f(s,y(s))\d s \quad \forall t\in I + \] +\end{satz} +\begin{proof} + "$\Rightarrow$". Sei $y$ eine Lösung von AWA. Dann ist $y$ diffbar mit $y'(t) = f(t,y(t))$. + \[\implies \int_{t_0}^t f(s,y(s)) \d s= \int_{t_0}^t y'(s)\d s \oldstackrel{\text{HDI}}{=} y(t) - y(t_0) = y(t)-y_0.\] + "$\Leftarrow$". Sei die Integralgleichung erfüllt. Falls $t = t_0\implies y(t_0) = y_0 \implies$ c). Aus dem HDI folgt komponentenweise $y'(t) = f(t,y(t)) \implies y$ löst AWA. +\end{proof} +\section{AWA: Existenz von Lösungen} +\begin{satz}[Existenzsatz von Peano]\ \\ + Die Funktion $f(t,x)$ sei stetig auf dem $(n+1)$-dimensionalen Zylinder + \[ + D = \{(t,x)\in \R\times \R^n\mid |t-t_0| \le \alpha,\; \norm{x-y_0}\le \beta\} + \] + Dann existiert eine Lösung $y(t)$ von AWA auf dem Intervall $I \coloneqq [t_0-T,t_0+T]$ mit \[T \coloneqq \min_{y(t)}\{\alpha,\frac{\beta}{M}\},\; M\coloneqq \max_{(t,x)\in D}\norm{f(t,x)}\] +\end{satz} +Reminder: +\begin{enumerate} + \item Gleichmäßige Stetigkeit: \[f\colon D\to \R,\; D\subset \R^n\] ist gleichmäßig stetig in $D$, falls $\forall \epsilon > 0,\;\exists \delta > 0$, sodass $\forall x,x_0\in D$ gilt \[\norm{x-x_0}< \delta \implies \norm{f(x)-f(x_0)}< \epsilon\] + \item Gleichgradige Stetigkeit: Sei $\mathcal{F} \subset C[a,b]$. Dann ist $\mathcal{F}$ gleichgradig stetig, falls $\forall \epsilon> 0\;\exists \delta > 0$, sodass $\forall f\in \mathcal{F}$ gilt \[\forall t,t'\in [a,b],\; |t-t'| <\delta \implies \norm{f(t)-f(t')}<\epsilon\] + \item Satz von Arzela-Ascoli: + Sei $(f_n)_{n\in \N}$ eine Folge in $C[a,b]$, die gleichmäßig beschränkt und gleichgradig stetig ist, d.h. + \[\sup_{n\in \N} \norm{f_n}_\infty < \infty\] und + \[\forall\epsilon > 0,\;\exists \delta > 0,\forall n\in \N\colon\; \max_{\substack{t,t'\in [a,b]\\|t-t'|\le \delta}} \norm{f_n(t)-f_n(t')} < \epsilon.\] + Dann existiert eine Teilfolge $(f_{n_k})_{k\in \N}$, welche gegen $f\in C[a,b]$ konvergiert, d.h. \[\norm{f_{n_k} - f}_\infty \to 0\] + \item Dreiecksungleichung für Integrale. Sei $y\colon [a,b] \to\R^n$ stetig, $\norm{\cdot}$ irgendeine Norm auf $\R^n$. Dann + \[\norm{\int_a^by(t)\d t} \le \int_a^b\norm{y(t)} \d t,\] hier: + \[\int_a^by(t)\d t\coloneqq \begin{pmatrix} + \int_a^by_1(t)\d t\\ + \vdots\\ + \int_a^by_n(t)\d t + \end{pmatrix}\in \R^n\] +\end{enumerate} +\begin{proof} (Satz von Peano)\\ + Idee: Konstruiere eine Folge stetiger Funktionen (Eulersches Polygonzugverfahren). Aus dem Satz von Arzela-Ascoli folgt dann, dass es eine Teilfolge gibt, die gegen eine Lösung von AWA konvergiert.\\ + O.b.d.A. betrachte Halbintervall $I = [t_0,t_0+T]$. Sei $h>0$ Schrittweitenparameter $(h\to 0)$. Wähle eine äquidistante Unterteilung des Intervalls $I$. + \[t_0 < t_1 < \dots < t_N = t_0 + T,\quad h = |t_k-t_{k-1}|\] + Eulersches Polygonzugverfahren:\begin{itemize} + \item Starte mit $y_0^h \coloneqq y_0$. + \item Für $n\ge 1$, berechne $y_n^h=y_{n-1}^h + hf(t_{n-1},y_{n-1}^h)$. + \end{itemize} + Definiere die stückweise lineare Funktion $y^h(t)$ + \[y^h(t)\coloneqq y_{n-1}^h + (t-t_{n-1})f(t_{n-1},y_{n-1}^h),\quad t\in [t_{n-1},t_n],\quad \forall n\ge 1\] + \begin{enumerate} + \item \textbf{z.Z.} dass dieses Verfahren durchführbar ist, d.h. $\graph(y^k)\subset D$. Sei $(t,y^h(t))\subset D$ für $t_0 \le t\le t_{k-1}$. Dann gilt + \[ + \underbrace{(y^h(t))'}_{\coloneqq \dv{y^h(t)}{t}} \equiv f(t_{k-1},y_{k-1}^h),\quad t\in [t_{k-1},t_k] + \] + Nach Konstruktion gilt für $t\in [t_{k-1},t_k]$: + \begin{align*} + y^h(t)-y_0 &= y^h(t)-y_{k-1}^h + y_{k-1}^h - y_{k-2}^h+ \dots + y_1^h-y_0^h\\ + &= y^k(t)-y_{k-1}^h + \sum_{i = 1}^{k-1}(y_i^h-y_{i-1}^h)\\ + &= (t-t_{k-1})f(t_{k-1},y_{k-1}^h) + \sum_{i = 1}^{k-1}h\cdot f(t_{i-1},y_{i-1}^h)\\ + \implies \norm{y^h(t)-y_0}&\le (t-t_{k-1})\norm{f(t_{k-1},y_{k-1}^h)} + h \sum_{i = 1}^{k-1}\norm{f(t_{i-1},y_{i-1}^h)}\\ + &\le (t-t_{k-1})\cdot M + \underbrace{h(k-1)}_{=t_{k-1}-t_0} \cdot M\\ + &= (t-t_0)\cdot M\\ + &\le T\cdot M\\ + &= \min \{\alpha,\frac{\beta}{M}\}\cdot M\\ + &\le \beta + \end{align*} + Also ist $(t,y^h(t))\in D$ für $t_{k-1} \le t\le t_k$. Mit Annahme folgt $(t,y^h(t))\in D$ für $t_0\le t\le t_k \implies \graph(y^h)\subset D$. + \item \begin{enumerate}[(a)] + \item \textbf{z.Z.} dass die Funktionenfamilie $\{y^h\}_{h>0}$ gleichgradig stetig ist. Seien dafür $t,t'\in I, t'\le t$ beliebig mit $t\in [t_{k-1},t_k],\; t'\in [t_{j-1},t_j]$ für ein $t_j\le t_k$.\begin{itemize} + \item $t,t' \in [t_{k-1},t_k]$: + \begin{align*} + y^h(t)-y^h(t')&= y_{k-1}^h + (t-t_{k-1})f(t_{k-1},y_{k-1}^h)\\ + &\quad - (y_{k-1}^h + (t'-t_{k-1})f(t_{k-1},y_{k-1}^h))\\ + &= (t-t') f(t_{k-1},y_{k-1}^h)\\ + \implies \norm{y^h(t)-y^h(t')} &\le |t-t'| \cdot M + \end{align*} + \item $t_j 0$ $\delta = \frac{\epsilon}{M}$, so gilt $\forall h$ + \[|t-t'| < \delta \implies \norm{y^h(t)-y^h(t')} < \epsilon\] + Daher ist $\{y^h\}_{h>0}$ gleichgradig stetig (sogar gleichgradig Lipschitz-stetig). + \end{itemize} + \item \textbf{z.Z.} $y^h$ ist gleichmäßig beschränkt. Es gilt $\forall t\in [t_0,t_0+T]$ + \begin{align*} + \norm{y^h(t)} = \norm{y^h(t)-\underbrace{y_0}_{\mathclap{y^h(t_0) = y_0}} + y_0}\\ + &\le \underbrace{\norm{y^h(t)-y_0}}_{\text{siehe 1)}} + \norm{y_0}\\ + &\le M\cdot T + \norm{y_0} + \end{align*} + Also ist $y^h$ gleichmäßig beschränkt. + \end{enumerate} + Nach dem Satz von Arzela-Ascoli existiert eine Nullfolge $(h_i)_{i\in \N}$ und eine stetige Funktion $y\colon I\to \R^n$ so dass + \[\max_{t\in I} \norm{y^{h_i} - y(t)} \xrightarrow{i\to \infty} 0\] + Offenbar ist also $\graph(y)\subset D$. + \item \textbf{z.Z.} $y(t)$ erfüllt die Differentialgleichung $y'(t) = f(t,y(t))$ oder äquivalent dazu: $y(t)$ erfüllt die Integralgleichung \[y(t) = y_0 + \int_{t_0}^{t} f(s,y(s))\d s\] + \end{enumerate} +\end{proof} +\end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index e65db63051625afed4f4612ee35e08e35cb356f1..3f12ee9fa0646bdb50512100f196e20afb02e26b 100644 GIT binary patch delta 699687 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\input{ana15.tex} +\input{ana16.tex} \end{document}