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b/sose2020/ana/uebungen/ana10.tex new file mode 100644 index 0000000..d804275 --- /dev/null +++ b/sose2020/ana/uebungen/ana10.tex @@ -0,0 +1,189 @@ +\documentclass[uebung]{../../../lecture} + +\title{Analysis II: Übungsblatt 10} +\author{Leon Burgard, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $T(t) = (T_0 - T_{a})e^{kt}+T_a$ löst das gegebene AWP. + \begin{proof} + Löse zunächst homogene DGL durch Trennung der Variablen: + \begin{align*} + \frac{\mathrm{d}T}{\d t} = kT \implies \frac{\d T}{T} = k \d t + \implies T_h = Ae^{kt} + .\end{align*} + Mit der partikulär Lösung $T_i = T_a$ ($T_a' = 0 = kT_a - kT_a$), folgt: + $T(t) = A e^{kt} + T_a$. Mit $T(0) = T_0$ folgt $A = T_0 - T_a$, also + insgesamt + \[ + T(t) = (T_0 - T_a) e^{kt} + T_a + .\] + \end{proof} + \item Beh.: Es dauert $60$ Zeiteinheiten. + \begin{proof} + Mit $k \coloneqq - \frac{\ln 2}{20}$ folgt + \[ + T(20) = (T_0 - T_a) \exp\left(- \frac{\ln 2}{20}\cdot 20\right) + T_a + = \frac{1}{2} (T_0 - T_a) + T_a = \frac{1}{2} (T_0 + T_a) + .\] Mit $T_0 = 100$ und $T_a = 20$ folgt $T(20) = 60$. + Damit folgt nun direkt + \begin{align*} + T(60) &= (T_0 - T_a) \exp\left( - \frac{\ln 2}{20} \cdot 60 \right) + T_a \\ + &= (T_0 - T_a) \exp\left( - \ln\left( 2^{3} \right) \right) + T_a \\ + &= (T_0 - T_a) \frac{1}{8} + T_a \\ + &= \frac{1}{8} T_0 + \frac{7}{8} T_a + .\end{align*} + Mit $T_0 = 100$ und $T_a = 20$ folgt $T(60) = 30$. + \end{proof} + \item Beh.: $\lim_{t \to \infty} T(t) = 20$. + \begin{proof} + Es gilt da $k < 0$: + \[ + \lim_{t \to \infty} T(t) = \lim_{t \to \infty} ( (T_0 - T_a) \underbrace{e^{kt}}_{\xrightarrow {t \to \infty} 0} + T_a ) = T_a + .\] Mit $T_a = 20$ aus (b) folgt die Behauptung. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Beh.: Falls $y_0 = -1$ ist $y(t) = -1$ Lösung der AWA. + \begin{proof} + Es gilt $y(t_0) = -1 = y_0$ und $y'(t) = 0 = -2t(1 - 1)^2 = -2t(1 + y(t))^2$, $\forall t \in \R$. + \end{proof} + + Beh.: Falls $y_0 \neq -1$ ist $y(t) = - \frac{1}{t_0^2 - t^2 - \frac{1}{1 + y_0}} - 1$ + Lösung der AWA. + \begin{proof} + Falls $y \neq -1$: Trennung der Variablen: + \begin{salign*} + \frac{\d y}{\d t} &= -2t (1+y)^2 \\ + \implies \frac{\d y}{(1+y)^2} &= -2t \d t \\ + \implies \int_{c}^{y} \frac{\d y}{(1+y)^2} \d y &= \int_{t_0}^{t} -2t \d t \\ + \implies - \frac{1}{1 + y} - \frac{1}{1+c} &= -t^2 + t_0^2 \\ + \intertext{Damit folgt} + y(t) &= - \frac{1}{t_0^2 - t^2 + \frac{1}{1+c}} - 1 + \intertext{Durch Einsetzen der Anfangsbedingung folgt} + y(t_0) &= - 1 - c -1 = y_0 \implies c = -2 - y_0 + \intertext{Insgesamt folgt also} + y(t) &= - \frac{1}{t_0^2 - t^2 - \frac{1}{1 + y_0}} - 1 + .\end{salign*} + Dies ist wohldefiniert da $y_0 \neq -1$. + Da $1 \neq 0$ folgt $y(t) \neq -1$ $\forall t \in \R$, d.h. keine weitere Fallunterscheidung notwendig + und $y(t)$ Lösung der AWA. + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Das gegebene AWP hat eine Lösung $y\colon [0, b] \to \R$. + \begin{proof} + Es ist $f(t,y) = \frac{1}{1 + |y|}$ auf $\R \times \R$ stetig. + Bestimme + \[ + M = \max_{(t,y) \in D} |f(t,y) | > 0 + .\] + Wähle $\alpha \coloneqq b > 0$ und $\beta \coloneqq 2 \alpha M$. Dann ist $f$ insbesondere auf + \[ + D = \{(t,y) \in \R \times \R \mid |t| \le \alpha, |y - y_0 | \le \beta \} + \] stetig. + Dann ex. nach dem Satz von Peano auf dem Intervall $[t_0 - T, t_0 + T]$ eine Lösung + der AWA mit + \[ + T = \min \left\{ \alpha, \frac{\beta}{M} \right\} = \min \{\alpha, 2 \alpha \} = \alpha = b + .\] Mit $t_0 = 0$ existiert also inbesondere eine Lösung $y(t)$ auf $[0 + b] \subseteq [-T, T]$. + \end{proof} + \item Beh.: Diese Lösung $y$ ist für festes $y_0 \in \R$ eindeutig. + \begin{proof} + Nach VL g.z.z., dass $f(t,y)$ Lipschitz-stetig bezüglich $y$ ist. Seien dazu + $x, y \in \R$. Dann ist + \begin{salign*} + |f(t,x) - f(t,y)| &= \left| \frac{1}{1 + |x|} - \frac{1}{1+|y|} \right| \\ + &= \left| \frac{|y|-|x|}{(1 + |x|)(1+|y|)} \right| \\ + &\le \frac{|x-y|}{1 + |x|+|y|+|xy|} \\ + &\le 1 \cdot |x-y| + .\end{salign*} + Mit $L \coloneqq 1$ folgt die Behauptung. + \end{proof} + \item Sei nun zusätzlich $v\colon [0,b] \to \R$ Lösung der AWA mit $v(0) = v_0$. + + Beh.: + \[ + |y(t) - v(t)| \le e^{t} |y_0 - v_0| \quad \forall t \in [0,b] + .\] + \begin{proof} + Definiere + \[ + w(t) \coloneqq |y(t) - v(t)| + .\] Da $y$ und $v$ die AWA lösen, erfüllen sie die Integralgleichung. Damit folgt + \begin{salign*} + w(t) &= \left| y_0 - v_0 + \int_{t_0}^{t} f(s, y(s)) \d s - \int_{t_0}^{t} f(s, v(s)) \d s \right| \\ + &\le |y_0 - v_0| + \int_{t_0}^{t} |f(s, y(s)) - f(s, v(s))| \d s \\ + &\stackrel{\text{(b)}}{\le} |y_0 - v_0| + \int_{t_0}^{t} |y(s) - v(s)| \d s \\ + &= |y_0 - v_0| + \int_{t_0}^{t} w(s) \d s \\ + &\stackrel{\text{Lemma v. Gronwall}}{\le } e^{t-t_0} |y_0 - v_0| + .\end{salign*} + Mit $t_0 = 0$ folgt die Behauptung. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{align*} + y'(t) &= 3 (y(t))^{\frac{2}{3}}, t \in I \\ + y(0) &= 0 + .\end{align*} + \begin{enumerate}[(a)] + \item Beh.: Es ex. unendlich viele Lösungen für diese AWA. + \begin{proof} + Die Funktion $y(t) = (t-t_0)^{3}$ löst die AWA mit $y(t_0) = 0$. Denn + \[ + y'(t) = 3 (t-t_0)^{2} = 3 (t-t_0)^{2 \cdot 3 \cdot \frac{1}{3}} = 3 \left((t-t_0)^{3}\right)^{\frac{2}{3}} = 3 (y(t))^{\frac{2}{3}} \quad (*) + .\] + Da $I \subseteq \R$ abgeschlossenes Intervall mit $0 \in I$, ex. $a, b \in \R$ + mit $a < 0$, $b > 0$, s.d. $I = [a,b]$. Definiere nun + \[ + y_d(t) \coloneqq \begin{cases} + 0 & a \le t \le d \\ + (t-d)^{3} & \text{sonst} + \end{cases} + .\] g.z.z.: $\forall d \in R$ mit $0 \le d \le b$ ist $y_d(t)$ eine Lösung der AWA. + + Es gilt $y_d(t_0) = y_d(0) = 0$. Weiter ist mit für $t \neq d$: + \[ + y'_d(t) = \begin{cases} + 0 = 3 (y(t))^{\frac{2}{3}} & a \le t < d \\ + 3 (y(t))^{\frac{2}{3}} & \text{sonst} + \end{cases} + .\] Für $t = d$ gilt + \begin{align*} + \lim_{h \searrow 0} \frac{y_d(d + h) - y_d(d)}{h} + &= \lim_{h \searrow 0} \frac{(d+h-d)^{3} - (d-d)^{3}}{h} \\ + &= \lim_{h \searrow 0} \frac{h^{3}}{h} \\ + &= \lim_{h \searrow 0} h^2 \\ + &= 0 \\ + &= \lim_{h \nearrow 0} \frac{y_d(d+h) - y_d(d)}{h} + .\end{align*} + Also gilt für $t \in I$: $y_d'(t) = 3 (y_d(t))^{\frac{2}{3}}$. Damit + löst $y_d(t)$ die AWA. + \end{proof} + \item Beh.: $f(t,y) \coloneqq 3 y^{\frac{2}{3}}$ ist nicht Lipschitz-stetig. + \begin{proof} + Sei $L > 0$ beliebig. Dann wähle $x = 0$ und $y < \left(\frac{3}{L} \right)^{3}$. Dann folgt + \begin{salign*} + \left| f(x) - f(y) \right| &= \left|3 y^{\frac{2}{3}}\right| \\ + &= \left|3 y^{-\frac{1}{3}}\right| |y| \\ + &> \left| 3 \frac{L}{3}\right| |y| \\ + &= L |y| + .\end{salign*} + \end{proof} + Die Lipschitz-stetigkeit von $f$ bezüglich $y$ + ist Voraussetzung für den Eindeutigkeitssatz aus der VL. + \item siehe (a). + \end{enumerate} +\end{aufgabe} + +\end{document}