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b/sose2020/ana/uebungen/ana11.tex new file mode 100644 index 0000000..f2a74d3 --- /dev/null +++ b/sose2020/ana/uebungen/ana11.tex @@ -0,0 +1,201 @@ +\documentclass[uebung]{../../../lecture} + +\title{Analysis II: Übungsblatt 11} +\author{Leon Burgard, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Das AWP $(*)$ hat eine globale definierte Lösung $u$. + \begin{proof} + Da $f$ stetig existiert nach Peano eine lokale Lösung $y(t)$ auf $I \coloneqq [t_0, t_0 + T]$. + Es ist also + \[ + y(t) = y_0 + \int_{t_0}^{t} f(s, y(s)) \d s \quad \forall t \in I + .\] + Damit folgt $\forall t \in I$: + \begin{salign*} + \Vert y(t) \Vert &\le \Vert y_0 \Vert + \int_{t_0}^{t} \Vert f(s, y(s)) \d s \\ + &\stackrel{f \text{ linear beschränkt}}{\le} + \Vert y_0 \Vert + \int_{t_0}^{t} (\alpha(s) \Vert y(s) \Vert + \beta(s)) \d s \\ + &= \Vert y_0 \Vert + \int_{t_0}^{t} \alpha(s) \Vert y(s) \Vert \d s + + \int_{t_0}^{t} \beta(s) \d s + \intertext{Da $I$ kompaktes Intervall, nehmen $\alpha(s)$ und $\beta(s)$ ihr + Maximum an. Damit folgt} + \Vert y(t) \Vert &\le \Vert y_0 \Vert + \alpha_{max}\int_{t_0}^{t} \Vert y(s) \Vert \d s + \beta_{\text{max}}(t - t_0) \\ + &\stackrel{t \le T}{\le} \Vert y_0 \Vert + \alpha_{max}\int_{t_0}^{t} \Vert y(s) \Vert \d s + \beta_{\text{max}}T + \intertext{Damit folgt mit dem Lemma von Gronwall} + \Vert y(t) \Vert &\le \underbrace{e^{\alpha_{\text{max}}(t-t_0)} (\Vert y_0 \Vert + T \beta_{max})}_{\eqqcolon \rho(t)} \\ + &\le \rho(t) + .\end{salign*} + Da $\rho(t)$ stetig folgt globale Fortsetzbarkeit von $y(t)$. + \end{proof} + \item Beh.: $f_1$ ist linear beschränkt. + \begin{proof} + Aus der Taylorreihe von $\sin(t)$ folgt $|\sin(t)| \le |t|$. Außerdem ist + $\sqrt{|x|} \le |x| + 1$. Damit folgt direkt + \begin{salign*} + |f_1(t, (x_1, x_2))| &\le |t| |x_1|^{\frac{1}{2}} + |\sin(t)| |x_2| \\ + &\le |t| (|x_1| + 1) + |t| |x_2| \\ + &\le \underbrace{|t|}_{=: \alpha(t)} \left( |x_1| + |x_2| \right) + \underbrace{2 |t|}_{=: \beta(t)} \\ + &= \alpha(t) + \Vert x \Vert_1 + \beta(t) + .\end{salign*} + \end{proof} + Beh.: $f_2$ ist linear beschränkt. + \begin{proof} + Es gilt + \begin{salign*} + |f_2(t, (x_1, x_2))| &\le e^{-t^2 |x_1|} + |x_1| \left| \frac{1}{1+x_2^2} \right| \\ + &\le \frac{1}{e^{t^2}|x_1|} + |x_1| \\ + &\le 1 + |x_1| \\ + &\le \underbrace{1}_{=: \beta(t)} + \underbrace{1}_{=: \alpha(t)} \cdot \Vert x \Vert_1 \\ + &= \alpha(t) \Vert x \Vert_1 + \beta(t) + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Beh.: Das Taylorpolynom $4$-ter Ordnung ist gegeben als + \[ + T_4(u, t, t_0=0) = t - t^2 + \frac{1}{2} t ^{3} - \frac{1}{6} t ^{4} + .\] + \begin{proof} + $u$ ist Lösung des AWP, d.h. es gilt + \begin{align*} + u''(t) &= - \sin(u(t)) - 2u'(t), \quad t \ge 0 + \intertext{Damit folgt} + u^{(3)}(t) &= - \cos(u(t)) u'(t) - 2u''(t) \\ + u^{(4)}(t) &= \sin(u(t)) u'(t)^2 - \cos(u(t)) u''(t) - 2u^{(3)}(t) + \intertext{Mit $u(0) = 0$ und $u'(0) = 1$ folgt} + u^{(3)}(0) &= 3 \\ + u^{(4)}(0) &= -4 + \intertext{Insgesamt folgt dann} + T_4(u, t) &= u(0) + u'(0) t + \frac{u''(0)}{2} t^2 + \frac{u^{(3)}(0)}{3!} t ^{3} + + \frac{u^{(4)}(t)}{4!} t ^{4} \\ + &= t - t^2 + \frac{1}{2} t ^{3} - \frac{1}{6} t ^{4} + .\end{align*} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Beh.: Die Matrix $A(t)$ ist gegeben als + \[ + A(t) = \begin{pmatrix} 1-t & 1 \\ + 2t - t^2 & t-1 + \end{pmatrix} + .\] + \begin{proof} + Durch Nachrechnen folgt + \[ + \phi^{-1} = \begin{pmatrix} t^2 - t + 1 & -t \\ -t^2 & t+1 \end{pmatrix} + .\] Mit $\phi'(t) = A(t) \phi(t)$ folgt also + \[ + A(t) = \phi'(t) \phi^{-1}(t) = \begin{pmatrix} 1 & 1 \\ 2t & 2t-1 \end{pmatrix} + \begin{pmatrix} t^2 - t + 1 & -t \\ -t^2 & t+1 \end{pmatrix} + = + \begin{pmatrix} 1-t & 1 \\ + 2t - t^2 & t-1 + \end{pmatrix} + .\] + \end{proof} + Beh.: Die Lösung $u(t)$ ist gegeben als + \[ + u(t) = \begin{pmatrix} t+1 \\ t^2 \end{pmatrix} + + \begin{pmatrix} \frac{1}{2} t^2 + t \\ \frac{1}{2} t ^{3} + \frac{3}{2} t^2 \end{pmatrix} + .\] + \begin{proof} + Nach VL gilt für die partikuläre Lösung mit $u_b(t_0 = 0) = (0,0)^{T}$ und + $b(t) = (1,t)^{T}$: + \begin{align*} + u_b(t) &= \phi(t) \left( \int_{0}^{t} \phi^{-1}(s) b(s) \d s + \begin{pmatrix} 0 \\ 0 \end{pmatrix} \right) \\ + &= \phi(t) \int_{0}^{t} \begin{pmatrix} 1 - s \\ s \end{pmatrix} \d s \\ + &= \begin{pmatrix} 1 + t & t \\ t^2 & t^2 -t +1 \end{pmatrix} + \begin{pmatrix} + t - \frac{1}{2}t^2 \\ + \frac{1}{2} t^2 + \end{pmatrix} \\ + &= \begin{pmatrix} \frac{1}{2} t^2 + t \\ \frac{1}{2} t ^{3} + \frac{3}{2} t^2 \end{pmatrix} + \intertext{Mit der Anfangsbedingung $u(0) = (1,0)^{T}$ folgt} + u(t) &= c_1 \begin{pmatrix} 1 + t \\ t^2 \end{pmatrix} + c_2 \begin{pmatrix} t \\ t^2 - t + 1 \end{pmatrix} + u_b(t) \\ + u(0) &= c_1 \begin{pmatrix} 1 \\ 0 \end{pmatrix} + c_2 \begin{pmatrix} 0 \\ 1 \end{pmatrix} + + \begin{pmatrix} 0 \\ 0 \end{pmatrix} \stackrel{!}{=} \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \intertext{$c_1 = 0$ also insgesamt} + u(t) &= \begin{pmatrix} t+1 \\ t^2 \end{pmatrix} + + \begin{pmatrix} \frac{1}{2} t^2 + t \\ \frac{1}{2} t ^{3} + \frac{3}{2} t^2 \end{pmatrix} + .\end{align*} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Bedingung $(**)$ für die eindeutige Lösbarkeit von (RWP-2.Ord) ist äquivalent + zur Bedingung $(*)$, wenn (RWP-2.Ord.) als System 1. Ordnung umformuliert wird. + \begin{proof} + Es sei ein RWP-2.Ord. wie beschrieben gegeben. Definiere + $y_1(t) \coloneqq u(t)$ und $y_2(t) \coloneqq y_1'(t)$. Dann + mit + \[ + B_a = \begin{pmatrix} \alpha_0 & \alpha_1 \\ 0 & 0 \end{pmatrix} + \quad + \text{und} + \quad + B_b = \begin{pmatrix} 0 & 0 \\ \beta_0 & \beta_1 \end{pmatrix} + \] als äquivalente Randwertbedingung + \[ + B_a y(a) + B_b y(b) = g := \begin{pmatrix} \eta_0 \\ \eta_1 \end{pmatrix} + .\] Sei nun $\{y, z\} $ ein beliebiges Fundamentalsystem der homogenen Gleichung + mit $\phi(a) = \mathbb{I}$. Dann gilt + $y_1(a) = z_2(a) = 1$ und $y_2(a) = z_1(a) = 0$. Damit folgt mit + \[ + A \coloneqq \begin{pmatrix} \alpha_0 & \alpha_1 \\ \beta_0 y_1(b) + \beta_1 y_2(b) + & \beta_0 z_1(b) + \beta_1 z_2(b) \end{pmatrix} + \] die zu $(**)$ äquivalente Bedingung $\text{det}(A) \neq 0$. + + Damit g.z.z. $A = B_a + B_b \phi(b)$. + \begin{align*} + B_a + B_b \phi(b) &= \begin{pmatrix} \alpha & \alpha_1 \\ 0 & 0 \end{pmatrix} + + \begin{pmatrix} 0 & 0 \\ \beta_0 & \beta_1 \end{pmatrix} + \begin{pmatrix} y_1(b) & z_1(b) \\ y_2(b) & z_2(b) \end{pmatrix} \\ + &= \begin{pmatrix} \alpha_0 & \alpha_1 \\ \beta_0 y_1(b) + \beta_1 y_2(b) + & \beta_0 z_1(b) + \beta_1 z_2(b) \end{pmatrix} = A + .\end{align*} + \end{proof} + \item Hier gilt + in Analogie zu (a) für (i) bis (iii): $\alpha_0 = 1$, $\alpha_1 = 0$, $\beta_0 = 1$ und + $\beta_1 = 0$. Sei außerdem $u_1 = \sin(t)$ und $u_2 = \cos(t)$. + \begin{enumerate}[(i)] + \item Beh.: Es existiert eine eindeutige Lösung. + \begin{proof} + Mit + \[ + \text{det}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = -1 \neq 0 + \] und (a) folgt die Behauptung. + \end{proof} + \item Beh.: Es existiert keine Lösung. + \begin{proof} + Das Kriterium aus (a) liefert + \[ + \text{det}\begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix} = 0 + .\] Ang. es ex. eine Lösung $u(t)$ des RWP. Dann hat diese die + Form $u(t) = c_1 \sin(t) + c_2 \cos(t)$. + Es gilt weiter $u(0) = 0 \implies c_2 = -1$, aber + $u(\pi) = 0 \implies c_2 = 1$ $\contr$. + \end{proof} + \item Beh.: Es existieren unendlich viele Lösungen. + \begin{proof} + Das Kriterium aus (a) liefert + \[ + \text{det}\begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix} = 0 + .\] Z.z.: $\forall c_1 \in \R$ ist $u(t) = c_1 \sin(t) + 1$ eine Lösung des + RWP. Es ist $u(0) = u(\pi) = 1$ und $u(t)$ ist mit $c_1$ und $c_2 = 0$ Lösung + des RWP. + \end{proof} + \end{enumerate} + \end{enumerate} +\end{aufgabe} + +\end{document}