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| author | Erik Löffler <erik.loeffler@ost.ch> | 2022-07-28 12:56:49 +0200 |
|---|---|---|
| committer | Erik Löffler <erik.loeffler@ost.ch> | 2022-07-28 12:56:49 +0200 |
| commit | 29fd344738894593ae434a271613815d1aa563ac (patch) | |
| tree | 3e6b47ded088fcb5921d42e7b6ca04c9ef285173 /buch | |
| parent | Added separation for diff. eq. in fourier example. (diff) | |
| download | SeminarSpezielleFunktionen-29fd344738894593ae434a271613815d1aa563ac.tar.gz SeminarSpezielleFunktionen-29fd344738894593ae434a271613815d1aa563ac.zip | |
Added solutions for heat conduction.
Diffstat (limited to '')
| -rw-r--r-- | buch/papers/sturmliouville/waermeleitung_beispiel.tex | 32 |
1 files changed, 32 insertions, 0 deletions
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: +\[ + u(t,x) + = + a_{0} + \sum_{n=1}^{\infty}a_{n}e^{-\frac{n^{2}\pi^{2}\kappa}{l^{2}}t} + \cos\left(\frac{n\pi}{l}x\right) +\] +\[ + a_{0} + = + \frac{1}{l}\int_{0}^{l}u(0,x) dx +\] +\[ + a_{n} + = + \frac{2}{l}\int_{0}^{l}u(0,x)sin\left(\frac{n\pi}{l}x\right) dx +\] |