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4be2cb9..3607fad 100644 --- a/ana16.tex +++ b/ana16.tex @@ -1,7 +1,7 @@ \documentclass{lecture} \begin{document} -\newcommand{\dv}[2]{\frac{\d #1}{\d #2}} +\newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} \newcommand{\graph}{\operatorname{Graph}} \chapter{Systeme gewöhnlicher Differentialgleichungen} @@ -13,14 +13,15 @@ Differentialgleichungen (DGLn) sind Gleichungen der Form \[ y^{(n)} = f(t,y,y',\dots,y^{(n-1)})\quad\text{explizite Form} \] -für eine gesuchte Funktion $y = y(t),\; t\in I,\; I\subset \R$ "Zeitinvervall"% $(y^{(k)} = \frac{\d[k]}{\d t^k} y)$. -Differentialgleichungen $n$-ter Ordnung sind äquivalent zu speziellen Systemen DGL 1. Ordnung. Betrachte $y^{(n)} = f(t,y,y',\dots,y^{(n-1)})$ (sei $y\colon I\to \R,I\subset \R)$. Definiere Hilfsvariablen: +für eine gesuchte Funktion $y = y(t),\; t\in I,\; I\subset \R$ ,,Zeitinvervall``. % $(y^{(k)} = \frac{\d[k]}{\d t^k} y)$. +Differentialgleichungen $n$-ter Ordnung sind äquivalent zu speziellen Systemen von Differentialgleichungen 1. Ordnung. Betrachte $y^{(n)} = f(t,y,y',\dots,y^{(n-1)})$ (sei $y\colon I\to \R,I\subset \R)$. Definiere Hilfsvariablen: \begin{align*} x_1 &\coloneqq y\\ x_2 &\coloneqq y'\\ &\vdots\\ - x_n\coloneqq y^{(n-1)} -\end{align*}, also $x\in \R^n$. Ein äquivalentes System von Differentialgleichungen 1. Ordnung ist dann + x_n &\coloneqq y^{(n-1)}, +\end{align*} +also $x\in \R^n$. Ein äquivalentes System von Differentialgleichungen 1. Ordnung ist dann \[ x' = \tilde{f}(t,x),\quad \tilde{f} = \begin{pmatrix} x_2\\x_3\\\vdots\\f(t,x) @@ -58,10 +59,12 @@ Notationen: $x' = f(t,x), \dot x = f(t,x), \dv{x}{t} = f(t,x)$ (Dynamischer Proz & &&\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{center} + \begin{tabular}{ccc} + succeptible & infected & removed\\ + $S(t)$ & $I(t)$ & $R(t)$ + \end{tabular} + \end{center} \begin{align*} N &= I + S + R\\ \dv{S}{t} &= \nu N - \beta \frac{SI}{N}-\mu S\\ @@ -83,7 +86,7 @@ Notationen: $x' = f(t,x), \dot x = f(t,x), \dv{x}{t} = f(t,x)$ (Dynamischer Proz % \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 + Sei $D = I\times \Omega \subset \R\times \R^n,\ f\colon D\to \R^n$ stetig. Dann heißt \begin{equation} y'=f(t,y)\label{DGLOrd1}\tag{$\star$} \end{equation} @@ -113,7 +116,7 @@ Eine Lösung von \eqref{DGLOrd1} ist eine differenzierbare Funktion $y:I\to \R^n y(t_0) &= y_0&&\text{Anfangsbedingung} \end{align*} Gesucht wird eine differenzierbare Funktion $y\colon I\to \R^n$ derart, dass - \begin{enumerate} + \begin{enumerate}[(a)] \item $\graph(y) \subset D$ \item $y'(t) = f(t,y(t)),\;t\in I$ \item $y(t_0) = y_0$ @@ -128,15 +131,15 @@ Eine Lösung von \eqref{DGLOrd1} ist eine differenzierbare Funktion $y:I\to \R^n \begin{proof} "$\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. + "$\Leftarrow$". 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} +\section{Anfangswertaufgaben: 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 \[T \coloneqq \min_{y(t)}\{\alpha,\frac{\beta}{M}\},\; M\coloneqq \max_{(t,x)\in D}\norm{f(t,x)}\] + Dann existiert eine Lösung $y(t)$ von AWA auf dem Intervall $I \coloneqq [t_0-T,t_0+T]$ mit \[T \coloneqq \min_{y(t)}\left\{\alpha,\frac{\beta}{M}\right\},\; M\coloneqq \max_{(t,x)\in D}\norm{f(t,x)}\] \end{satz} Reminder: \begin{enumerate} @@ -157,7 +160,7 @@ Reminder: \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 $I$. + 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 $y_0^h \coloneqq y_0$. @@ -165,8 +168,8 @@ Reminder: \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 + \begin{enumerate}[1)] + \item \textbf{z.Z.} dass dieses Verfahren durchführbar ist, d.h. $\graph(y^h)\subset D$. Sei $(t,y^h(t))\subset D$ für $t_0 \le t\le t_{k-1}$. Dann gilt \[ \underbrace{(y^h(t))'}_{\coloneqq \dv{y^h(t)}{t}} \equiv f(t_{k-1},y_{k-1}^h),\quad t\in [t_{k-1},t_k] \] @@ -179,12 +182,12 @@ Reminder: &\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\\ + &= \min \left\{\alpha,\frac{\beta}{M}\right\}\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 \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)\\ @@ -195,23 +198,25 @@ Reminder: \item $t_j 0$ $\delta = \frac{\epsilon}{M}$, so gilt $\forall h$ + \end{itemize} + Wählt man für ein beliebiges $\epsilon > 0$ also $\delta = \frac{\epsilon}{M}$, 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}\\ + \norm{y^h(t)} &= \norm{y^h(t) - \smash[b]{\underbrace{y_0}_{\mathclap{y^h(t_0) = y_0}}} + y_0} + \vphantom{\underbrace{y_0}_{\mathclap{y^h(t_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*} @@ -219,7 +224,7 @@ Reminder: \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 $\graph(y)\subset D$. + Offenbar ist $\graph(y)\subset D$. \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} diff --git a/analysisII.pdf b/analysisII.pdf index 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