Commit b226c7e9 authored by andre_eikmeier's avatar andre_eikmeier
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Anfang Kapitel 4.4 angepasst

parent 67b1281f
......@@ -970,8 +970,8 @@
\newlabel{eq:inhomogenhomogenRB}{{89}{114}{\textsc {Green}sche Funktion für inhomogene lineare Randwertprobleme}{equation.4.89}{}}
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[]\T1/cmr/m/n/10 Die ver-all-ge-mei-ner-te \T1/cmr/m/sc/10 Green\T1/cmr/m/n/10
sche Funk-ti-on ist ste-tig, da $\OML/cmm/m/it/10 A$ \T1/cmr/m/n/10 und $\OML/c
mm/m/it/10 B$ \T1/cmr/m/n/10 und die \T1/cmr/m/sc/10 Wron-ski\T1/cmr/m/n/10 -De
terminante
[]
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......@@ -12,7 +12,7 @@ Wir können ohne Beschränkung der Allgemeinheit homogene Randbedingungen wähle
Wir suchen eine \hyperref[def:greendef]{\textsc{Green}sche Funktion} für die linke Seite $L$. %(für $c \ge 0 \ge ??$ kennen wir sie schon).
Dafür betrachten wir zunächst die \emph{homogene} Differentialgleichung $L u = 0$.
Diese ist \emph{eindeutig lösbar} und besitzt ein \hyperref[defn:Fundamental]{Fundamentalsystem} $\{u_1, u_2\} \subset \mathfrak{C}([a, b]; \R)$.
Diese besitzt ein \hyperref[defn:Fundamental]{Fundamentalsystem} $\{u_1, u_2\} \subset \mathfrak{C}([a, b]; \R)$.
Wieder sei
\begin{equation*}
W(x)
......@@ -40,12 +40,14 @@ Dann ist $u(x) \coloneqq c_1 u_1(x) + c_2 u_2(x)$ die eindeutige Lösung des zug
0 \\ 0
\end{pmatrix}
\end{equation*}
nur die Lösung hat, das heißt wenn $R \ne 0$.
nur eine Lösung hat, das heißt wenn $R \ne 0$.
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Das homogene Randwertproblem \eqref{eq:homohomo} ist also genau dann eindeutig lösbar, das heißt hat nur die triviale Lösung, wenn $R \ne 0$ ist.
\end{bclogo}
Unter dieser Bedingung definieren wir die neue \textsc{Green}sche Funktion.
\begin{defn}[Verallgemeinerte \textsc{Green}sche Funktion]
Die \emphii{verallgemeinerte \textsc{Green}sche Funktion}\index{Greensche Funktion@\textsc{Green}sche Funktion!verallgemeinert} ist
\begin{equation*}
......@@ -76,7 +78,7 @@ Es gilt $G(a, \xi) = G(b, \xi) = 0$ für alle $\xi \in [a, b]$ (selber nachrechn
%\input{4/ableitung}
\begin{lemma} \label{lemma:homoInhomo}
Die Lösung des Randwertproblems \eqref{eq:inhomogenhomogenRB} ist genau gegeben durch
Unter der Bedingung $R \neq 0$ ist die Lösung des Randwertproblems \eqref{eq:inhomogenhomogenRB} genau gegeben durch
\begin{equation} \label{eq:Greenverallg}
u(x)
= \int_{a}^{b} G(x, \xi) f(\xi) \diff{\xi}.
......
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