From 29fd344738894593ae434a271613815d1aa563ac Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Erik=20L=C3=B6ffler?= <erik.loeffler@ost.ch>
Date: Thu, 28 Jul 2022 12:56:49 +0200
Subject: Added solutions for heat conduction.

---
 .../sturmliouville/waermeleitung_beispiel.tex      | 32 ++++++++++++++++++++++
 1 file changed, 32 insertions(+)

(limited to 'buch/papers/sturmliouville')

diff --git a/buch/papers/sturmliouville/waermeleitung_beispiel.tex b/buch/papers/sturmliouville/waermeleitung_beispiel.tex
index cd7a620..a493749 100644
--- a/buch/papers/sturmliouville/waermeleitung_beispiel.tex
+++ b/buch/papers/sturmliouville/waermeleitung_beispiel.tex
@@ -106,3 +106,35 @@ Differenzialgleichungen aufgeteilt werden:
     =
     0
 \]
+
+% TODO: Rechenweg
+TODO: Rechenweg... Enden auf konstanter Temperatur:
+\[
+    u(t,x)
+    =
+    \sum_{n=1}^{\infty}a_{n}e^{-\frac{n^{2}\pi^{2}\kappa}{l^{2}}t}
+    \sin\left(\frac{n\pi}{l}x\right)
+\]
+\[
+    a_{n}
+    =
+    \frac{2}{l}\int_{0}^{l}u(0,x)sin\left(\frac{n\pi}{l}x\right) dx
+\]
+
+TODO: Rechenweg... Enden isoliert:
+\[
+    u(t,x)
+    =
+    a_{0} + \sum_{n=1}^{\infty}a_{n}e^{-\frac{n^{2}\pi^{2}\kappa}{l^{2}}t}
+    \cos\left(\frac{n\pi}{l}x\right)
+\]
+\[
+    a_{0}
+    =
+    \frac{1}{l}\int_{0}^{l}u(0,x) dx
+\]
+\[
+    a_{n}
+    =
+    \frac{2}{l}\int_{0}^{l}u(0,x)sin\left(\frac{n\pi}{l}x\right) dx
+\]
-- 
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