// Main VL template
#import "../preamble.typ": *

// Fix theorems to be shown the right way in this document
#import "@preview/ctheorems:1.1.3": *
#show: thmrules

// Main settings call
#show: conf.with(
	// May add more flags here in the future
	num: 5,
	type: 0, // 0 normal, 1 exercise
	date: datetime.today().display(),
	//date: datetime(
	//	year: 2025,
	//	month: 5,
	//	day: 1,
	//).display(),
)

= Uebersicht

= Partielle Differentialgleichungen

ODE
$
	arrow(x) (t) => m (dif ^2 arrow(x)) / (dif t^2 )  =  F (arrow(x), dot(arrow(x))).
$
PDE
$
	(partial ^2 psi) / (partial t^2 ) =  u^2 (partial ^2 psi) / (partial x^2 ) , space psi (x,t) \
	(partial T) / (partial t) =  D (partial ^2 T) / (partial x^2 ) , space "Fourier Gesetz" arrow(j) =  kappa arrow(nabla) T.
$
Poisson-Gleichung
$
	Delta phi =  - (rho (arrow(x))) / (epsilon_0 ).
$
Lineare partielle DGL
$
	L [phi (arrow(x))] =  rho (arrow(x)) \
	phi (arrow(x)) =  phi_0 (arrow(x)) "auf Rand" partial Omega \
	"gesucht ist die Loesung von Gebiet" Omega.
$

#example[
	Poisson-Gleichung.

	Fuehre eine Diffferenzdiskretisierung durch
	$
		phi (x,y) tilde.equiv phi (I_(x) , I_(y) ) \
		Delta phi =  (partial ^2 phi) / (partial x^2 ) +  (partial ^2 phi) / (partial y^2 ) =  (phi (i_(x) + 1 , i_(y) ) - 2 phi (i_(x) , i_() )) / () 
	$
]

Doolittle Verfahren
Gauss-Elimination Verfahren