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\title[Matematika 4]
{Matematika 4}

\subtitle{5. vaja}

\author[Borut] % 
{B.~Jur\v{c}i\v{c}~Zlobec\inst{1}}

\institute[Univerza v Ljubljani Slovenia] % 
{
  \inst{1}%
    Univerza v Ljubljani,\\Fakulteta za Elektrotehniko\\ 
    1000 Ljubljana, Tr\v{z}a\v{s}ka 25, Slovenija
}


\date[2012] % 
{Matematika FE, Ljubljana, Slovenija \today}

\subject{Matematika}

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\section{Parcialne diferencialne ena\v{c}be, Fouriereva metoda}
%
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\subsection{Nihanje strune}
\begin{frame}{Ena\v{c}ba za nihanje vpete strune 
$\mu \frac{\partial^2 u(x,t)}{\partial t^2}=
\frac{\partial}{\partial x}\left(T(x)\,\frac{\partial u(x,t)}{\partial x}
\right)$}
\formulas{Struna je vpeta v $x=0$ in $x=l$, dol"zina je $l$.\\
Robni pogoji: $u(0,t)=u(l,t)=0$.\\
Napetost strune: $T=konst$.\\
$c^2=\frac{T}{\mu}$, kjer je $\mu$ dol"zinska gostota strune.}
\begin{itemize}
	\item $\displaystyle \frac{1}{c^2}
	\frac{\partial^2 u(x,t)}{\partial t^2}=
	\frac{\partial^2 u(x,t)}{\partial x^2}$
	\item Lo"citev spremenljivk $u(x,t)=X(x)T(t)$.
	\item Lastne funkcije:  \formulas{$\displaystyle -X''
	=\lambda^2 X $ },
	$X(x)=\sin(\lambda x)$. 
	\item Lastne vrednosti:\\ zaradi robnih pogojev je 
	$\lambda l=n \pi$, 
	$n\in\mathbb{N}$, $\lambda_n=\frac{n\pi}{l}$. 
	\item "Casovni del \formulas{$\displaystyle T''+\omega_n^2T=0$}. 
	$\omega_n=c\lambda_n$.
	\item $T_n(t)=A_n\cos(\omega_n t)+B_n\sin(\omega_n t)$
	\item \formulas{$\displaystyle u_n(x,t)=\sin(\lambda_n x)\left(
	A_n\cos(\omega_n t)+B_n\sin(\omega_n t) \right)$}.
\end{itemize}
\end{frame}

\begin{frame}{Ena\v{c}ba za nihanje kro\v{z}e\v{c}e strune
$\mu \frac{\partial^2 u(x,t)}{\partial t^2}=
\frac{\partial}{\partial x}\left(T(x)\,\frac{\partial u(x,t)}{\partial x}
\right)$
}
\formulas{Struna dol"zine $l$ je vpeta v $x=0$.\\
Robni pogoji: $u(0,t)=0$, $|u(l,t)|<\infty$.\\
Napetost strune: $dT=x \omega^2\mu\,dx$, 
$T(x)=\mu\omega^2\int_x^l\xi d\xi$,
$T(x)=\frac{1}{2}\mu\omega^2(l^2-x^2)$, 
$c^2=\frac{1}{2}\omega^2$}. 
\begin{itemize}
	\item Uvedemo novo spremenljivko $\xi=\frac{x}{l}$.
	\item $c^{-2}u_{tt}=\left((1-\xi^2)u_{\xi}\right)_\xi$.
	\item Robni pogoji $u(0,t)=0,\quad |u(1,t)|<\infty$.
	\item Lo"citev spremenljvk $u(\xi,t)=\Xi(\xi)T(t)$.
	\item $-\left((1-\xi^2)\Xi'\right)'=\lambda\Xi$, 
	\formulas{$\displaystyle (1-\xi^2)\Xi''-2\xi\Xi'+\lambda\Xi=0$}.
\end{itemize}
\end{frame}
\begin{frame}{}
\begin{itemize}
	\item Re"sitev je lihi Legendreev polinom, ker je $u(0,t)=0$.  
	\item Lastne vrednosti $\lambda_n=2n(2n-1)$.
	\item Lastne funkcije $\Xi_n(\xi)=P_{2n-1}(\xi)$.
	\item "Casovni del:  
	\formulas{$T''+c^2\lambda T=0$},  
	$\omega_n=c\sqrt{\lambda_n}=\omega\sqrt{n(2n-1)}$.
	$T_n(t)=A_n\cos(\omega_n t)+B_n\sin(\omega_n t)$.
	\item \formulas{$\displaystyle u_n(x,t)=P_{2n-1}(x/l)
	\left(A_n\cos(\omega_n t)+B_n\sin(\omega_n t)\right)$}.
\end{itemize}
\end{frame}

\begin{frame}{Ena\v{c}ba za nihanje obe"sene strune 
$\mu \frac{\partial^2 u(x,t)}{\partial t^2}=
\frac{\partial}{\partial x}\left(T(x)\,\frac{\partial u(x,t)}{\partial x}
\right)$}
\formulas{Obe"sena struna, vpeta v $x=l$ in dol"zine $l$.
Os $x$ je navpi"cna.\\ 
Robni pogoji: $|u(0,t)|<\infty$, $|u(l,t)|=0$.\\
Napetost strune:  $T(x)=g\mu x $, $c^2=g$.}
\begin{itemize}
	\item $\displaystyle \frac{1}{c^2}
	\frac{\partial^2 u(x,t)}{\partial t^2}=
	\frac{\partial}{\partial x}\left(x
	\frac{\partial u(x,t)}{\partial x} \right)$.
	\item Uvedemo novo spremenljivko $x=\xi^2$.
	\item $\displaystyle \left(\frac{2}{c}\right)^2u_{tt}=
	\frac{1}{\xi}\left(\xi u_\xi\right)_\xi$.
	\item Lo"cimo spremenljivki $u(\xi,t)=\Xi(\xi)T(t)$, 
	$-\frac{1}{\xi}(\xi\Xi')'=\lambda^2\Xi$.
	\item \formulas{$\Xi''+\frac{1}{\xi}\Xi'+\lambda^2\Xi=0$}.
\end{itemize}
\end{frame}
\begin{frame}{}
\begin{itemize}
	\item Robni pogoji $\Xi(\sqrt{l})=0$, $|\Xi(0)|<\infty$.
	\item Diferencialno ena"cbo re"si Besslova funkcija 
		$\Xi(\xi)=J_0(\lambda\xi)$.
	\item Lastne vrednosti: $\lambda_n^2=\xi_{0,n}^2/l$, 
        kjer je 
	$\xi_{0,n}$ 
	$n$-ta ni"cla Besslove funkcije $J_0(\xi)$.
	\item Lastne funkcije: 
	$\Xi_n(\xi)=J_0(\lambda_n\xi)$.
	\item "Casovni del:  
	\formulas{$\displaystyle 
	T''+\left(\frac{c}{2}\right)^2\lambda^2 T=0$},  
	$\omega_n=\frac{1}{2}c\lambda_n$.
	$T_n(t)=A_n\cos(\omega_n t)+B_n\sin(\omega_n t)$.
	\item \formulas{$u_n(x,t)=J_0(\lambda_n\sqrt{x})
	\left(A_n\cos(\omega_n t)+B_n\sin(\omega_n t) \right)$}.
\end{itemize}
\end{frame}

\begin{frame}{Nihanje obe"sene strune}
\begin{center}
	\includegraphics[scale=0.4]{vrvica1.png}
\end{center}
\end{frame}


\begin{frame}{Nihanje vpete okrogle membrane 
$\Delta u(r,t)=\frac{1}{c^2}\frac{\partial^2 u(r,t)}{\partial t^2} $}
\formulas{Polmer membrane je $a$. Gledamo lastna nihanja, 
	ki niso odvisna od kota. 
$\displaystyle \Delta u= u_{rr}+\frac{1}{r}u_r$. Robna pogoja $u(a,t)=0$ in $|u(0,t)|<\infty$.}
\begin{itemize}
	\item Ena"cba $\displaystyle u_{rr}+\frac{1}{r}u_r=\frac{1}{c^2}u_{tt}$.
	\item Lo"cimo spremenljivke $u(r,t)=R(r)T(t)$. 
	\item $-(R''+\frac{1}{r}R')=\lambda^2 R$, $T''+c^2\lambda^2 T=0$.
	\item Vpeljemo novo neznanko $\xi=r\lambda$, 
	\formulas{$\displaystyle \frac{d^2R(\xi)}{d\xi^2}+
	\frac{1}{\xi}\frac{dR(\xi)}{d\xi}+R(\xi)=0$}.
	\item Ker je $|R(\xi)|<\infty$, je 
        $R(\xi)=J_0(\xi)$, $R(r)=J_0(\lambda r)$.
	\item Upo"stevamo "se drugi robni pogoj 
	$R(\lambda a)=0$, $\lambda a=\xi_{0,n}$,\\ $n$-ta ni"cla 
	Besslove funkcije $J_0(\xi)$, 
        $\displaystyle\lambda_n = \frac{\xi_{0,n}}{a}$.
	\item $R_n(r)=J_0(\lambda_n r)$, 
	\formulas{$T''+c^2\lambda_n^2T=0$, $\omega_n=c\lambda_n$}. 
	\item \formulas{$u_n(r,t)=R_n(r)T_n(t)=J_0(\lambda_n r)
	\left(A_n\cos(\omega_n t)+B_n\sin(\omega_n t)\right)$}.
\end{itemize}
\end{frame}

\begin{frame}{Nihanja pravokotne membrane
$\Delta u(x,y,t)=\frac{1}{c^2}\frac{\partial^2 u(r,t)}{\partial t^2} $}
\formulas{Stranici sta  $a$ in $b$.
$\displaystyle \Delta u= u_{xx}+u_{yy}$. Robni pogoji $u(0,y,t)=u(a,y,t)=u(x,0,t)=u(x,b,t)=0$.}
\begin{itemize}
	\item Ena"cba $\displaystyle u_{xx}+u_{yy}=\frac{1}{c^2}u_{tt}$. 
	\item Lo"cimo spremenljivke $u(x,y,t)=X(x)Y(y)T(t)$.
	\item Ena"cba $\displaystyle 
	\frac{X''}{X}+\frac{Y''}{Y}=\frac{1}{c^2}\frac{T''}{T}$, robni pogoji
	$X(0)=X(a)=Y(0)=Y(b)=0$. 
	\item Prostorski del:  
	\formulas{$X''+\lambda_m^2 X=0$, $Y''+\mu_n^2 Y=0$}. 
	Lastni vrednosti $\lambda_m=\frac{m\pi}{a}$ in 
	$\mu_n=\frac{n\pi}{b}$, $m,n\in \mathbb{N}$. 
	\item "Casovni del 
	\formulas{$T''+\omega_{mn}^2T=0$, $\omega_{mn}=
	\pi c\sqrt{\left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2}$}. 
	\item Lastne funkcije \formulas{$\displaystyle u_{mn}(x,y,t)=
		\sin(\lambda_m x)\sin(\mu_n y)
		\left(A_{mn}\cos(\omega_{mn}t)+
		B_{mn}\sin(\omega_{mn}t)\right)$}.
\end{itemize}
\end{frame}

\begin{frame}{Dirichletov robni problem: $\Delta u =0$ 
za polpas, $0<x<1$, $0<y<\infty$.}
\formulas{Robni pogoji $u(0,y)=0$, $u(1,y)=0$, $u(x,0)=1$ 
$\lim_{y\to\infty}u(x,y)=0$}.
\begin{itemize}
 	\item Ena"cba $\displaystyle u_{xx}+u_{yy}=0$.
        \item Lo"cimo spremenljivke $u(x,y)=X(x)Y(y)$.
        \item Ena"cba $\displaystyle 
        \frac{X''}{X}+\frac{Y''}{Y}=0$, robni pogoji
        $X(0)=X(1)=0$, $|Y(y)|$ je omejena funkcija $0<y<\infty$.
 	\item $X''+\lambda_n^2 X=0$, $\lambda_n=n \pi$ 
	$n\in\mathbb{N}$.
	\item $X_n(x)=\sin(n\pi x)$ in $Y_n(y)=A_n e^{-n\pi y}+B_n e^{n\pi y}$.
	\item Koeficienti $B_n=0$, ker je funkcija $Y(y)$ omejena. 
	\item \formulas{$u(x,y)=\sum_{n=1}^\infty A_n\sin(n\pi x)e^{-n\pi y}$}.
	\item $u(x,0)=\sum_{n=1}^\infty A_n\sin(n\pi x)=1$, 
	$A_n=2\frac{1-(-1)^2}{n\pi}$. 	
\end{itemize}
\end{frame}
\begin{frame}{Graf k prej"snji nalogi $z=\sum_{n=1}^{10}u_n(x,y)$}
	\begin{center}
		\includegraphics[scale=0.5]{dirichlet.png}
	\end{center}
\end{frame}

\begin{frame}{Dirichletov robni problem, $\Delta u=0$ za krog, $0\le r\le 1$.}
\formulas{Robni pogoji: $|u(r,\phi)|<\infty$, $u(1,\phi)=f(\phi)$}.
\begin{itemize}
 	\item Ena"cba: 
	$\displaystyle u_{rr}+\frac{1}{r}u_{r}+
	\frac{1}{r^2}u_{\phi\phi}=0$.
        \item Lo"cimo spremenljivke: $u(r,\phi)=R(r)\Phi(\phi)$.
        \item Ena"cba: $\displaystyle 
        \frac{R''}{R}+\frac{1}{r}\frac{R'}{R}+
	\frac{1}{r^2}\frac{\Phi''}{\Phi}=0$. 
	\item Robni pogoji:
        $|R(r)|$ je omejena funkcija,\\ $\Phi(\phi)$ je periodi"cna funkcija 
	s periodo $2\pi$. 
 	\item Ena"cbi:
	\begin{itemize} 
		\item \formulas{$\Phi''+\lambda_n \Phi=0$, 
		$\lambda_n=n^2$, $n\in\mathbb{N}$} in 
	\item \formulas{$r^2R''(r)+rR'(r)-n^2R(r)=0$}.
	\end{itemize}
	\item $\Phi_n(\phi)= A_n\sin(n \phi)+B_n\cos(n \phi)$ in $R_n(r)=r^n$.
	\item \formulas{$u(r,\phi)=A_0+
	\sum_{n=1}^\infty\left(A_n\cos(n \phi)+B_n\sin(n \phi)\right)r^n$}.
	\item $u(1,\phi)=
	f(\phi)=A_0 +\sum_{n=1}^\infty 
	\left(A_n\cos(n \phi)+B_n\sin(n\phi)\right)$. 	
\end{itemize}
\end{frame}
\begin{frame}{Graf k prej"snji nalogi $z=\sum_{n=1}^{10}u_n(x,y),\quad 
f(\phi)=\mbox{sign}\,(\phi), \quad -\pi\le\phi\le\pi$.}
	\begin{center}
		\includegraphics[scale=0.4]{dirichlet1.png}
	\end{center}
\end{frame}

\begin{frame}{Splo"sna nihanja vpete okrogle membrane:\\ 
	$\Delta u=\frac{1}{c^2}u_{tt}$ $0\le r\le a$.}
\formulas{Robni pogoji: $|u(0,\phi,t)|<\infty$, $u(a,\phi,t)=0$, 
	$u(r,\phi+2\pi,t)=u(r,\phi,t)$, periodi"cnost}.\\
\formulas{Za"cetni pogoji: $u(r,\phi,0)=f(r,\phi)$ in 
$u_t(r,\phi,0)=g(r,\phi)$}
\begin{itemize}
 	\item Ena"cba: 
	$\displaystyle u_{rr}+\frac{1}{r}u_{r}+
	\frac{1}{r^2}u_{\phi\phi}=\frac{1}{c^2}u_{tt}$.
        \item Lo"cimo spremenljivke: $u(r,\phi,t)=R(r)\Phi(\phi)T(t)$.
        \item Ena"cba: $\displaystyle 
        \frac{R''}{R}+\frac{1}{r}\frac{R'}{R}+
	\frac{1}{r^2}\frac{\Phi''}{\Phi}=\frac{1}{c^2}\frac{T''}{T}$. 
	\item Robni pogoji:
        $|R(0)|<\infty$, $R(a)=0$ in \\ $\Phi(\phi)$ je periodi"cna funkcija 
	s periodo $2\pi$. 
\end{itemize}
\end{frame}
\begin{frame}
\begin{itemize}
 	\item Ena"cbe:
	\begin{itemize} 
		\item \formulas{$\Phi''+m^2 \Phi=0,\quad m\in\mathbb{N}$}, 
		\item \formulas{$T''+c^2\lambda^2 T=0$},
		\item \formulas{$r^2R''(r)+rR'(r)+(\lambda^2 r^2-m^2)R(r)=0$},
		\item Vpeljemo novo spremenljivko $\xi=\lambda r$ 
			dobimo ena"cbo
		\item \formulas{$\displaystyle
			\xi^2R''(\xi)+\xi R'(\xi)+(\xi^2-m^2)R(\xi)=0$.}\\
			Re"sitev je \formulas{$R(\xi)=AJ_m(\xi)+BN_m(\xi)$}.
		\item {$\lambda_{mn}=\xi_{mn}/a$, 
		$\omega_{mn}=c\lambda_{mn}$}\\
		$\xi_{mn}$ je $n$-ta ni"cla Besslove funkcije $J_m(\xi)$. 
	\end{itemize}
	\item Re"sitve:
	\begin{itemize} 
		\item $\Phi_m(\phi)= \sin(m \phi)$, 
		\item $R_{mn}(r)=J_m(\lambda_{mn}r)$
		\item $T_{mn}=
		A_{mn}\cos(\omega_{mn}t)+B_{mn}\sin(\omega_{mn}t)$.
	\end{itemize}
	\item \formulas{$\displaystyle 
	u(r,\phi,t)=\sum_{m,n}\sin(m \phi)J_m(\lambda_{mn}r)
	\left(A_{mn}\cos(\omega_{mn}t)+B_{mn}\sin(\omega_{mn}t)\right)$}. 
\end{itemize}
\end{frame}

\begin{frame}{Graf lastne funkcije $u_{2,3}(r,\phi,0)$.}
\begin{center}
	\includegraphics[scale=0.5]{membrana.png}
\end{center}
\end{frame}


\begin{frame}{Prevajanje toplote po palici z izoliranima kraji"s"cema}
\formulas{$\displaystyle u_{xx}=\frac{1}{\sigma^2}u_t$, $u_x(0,t)=u_x(a,t)=0$, 
$u(x,0)=f(x)$.}
\begin{itemize}
	\item $\displaystyle u_{xx}(x,t)=\frac{1}{\sigma^2}u_t(x,t)$. 
		$u(x,t)=X(x)T(t)$. 
	\item $\displaystyle \frac{X''}{X}=\frac{1}{\sigma^2}\frac{T'}{T}$.
	\item Krajevni del: $\frac{X''}{X}=-\lambda^2$
		\begin{itemize}
			\item Ena"cba: \formulas{$X''+\lambda^2 X=0$}.
			\item $X(x)=A \cos(\lambda x)+B \sin(\lambda x)$.
			\item Robni pogoji: 
				$X'(x)=-A\lambda\sin(\lambda x)+
				B\lambda\cos(\lambda x)$.
			\item $X(0)=0\rightarrow B=0$, 
				$X(a)=0\rightarrow \lambda a= n\pi$, 
			$n\in \mathbb{N}$.
			\item Lastne funkcije $X_n(x)=\cos(\frac{n\pi}{a}x)$.
		\end{itemize}
	\end{itemize}
\end{frame}
\begin{frame}
	\begin{itemize}
	\item "Casovni del: 
	$\frac{1}{\sigma^2}\frac{T'}{T}=-\left(\frac{n\pi}{a}\right)^2$. 
	$\mu_n=\sigma\frac{n\pi}{a}$. 
		\begin{itemize}
			\item Ena"cba \formulas{$\displaystyle 
			T'+\mu_n^2T=0$}. 
			\item $\displaystyle T_n(t)=A_n e^{-\mu_n^2 t}$.
		\end{itemize}
	\item Lastne funkcije: $u_n(x,t)=A_n\cos(\frac{n\pi}{a}x)e^{-\mu_n^2 t}$.
	\item Za"cetni pogoj: 
		$f(x)=\sum_{n=0}^\infty A_n\cos(\frac{n\pi}{a}x)$.
	\item $\displaystyle A_n=\frac{2}{a}\int_0^a 
		f(x)\cos(\frac{n\pi}{a} x) dx$, $n>0$. 
	\item Re"sitev: 
	\formulas{$\displaystyle u(x,t)=A_0+\sum_{n=1}^\infty 
		A_n\cos(\frac{n\pi}{a}x)e^{-\mu_n^2 t}$}.
	\item Stacionarna re"sitev 
	$\displaystyle \lim_{t\to\infty}u(x,t)=A_0=\frac{1}{a}\int_0^af(x) dx$.  

\end{itemize}
\end{frame}

\begin{frame}{Temperatura na kraji"s"cih je konstantna.}
\formulas{$\displaystyle 
u_{xx}=\frac{1}{\sigma^2}u_t$, $u(0,t)=\tau_0$, 
$u(a,t)=\tau_a$ in $u(x,0)=f(x)$.}
\begin{itemize}
	\item Pi"s"cimo staconarno re"sitev, $\overline{u}_t=0$, 
	to je re"sitev neodvisna od "casa, $\overline{u}_{xx}=0$.
	\item  Re"sitev je linearna funkcija: 
	$\overline{u}(x)=\tau_0+\frac{\tau_a-\tau_0}{a}x$.
	\item $u(x,t)=\overline{u}(x)+v(x,t)$, $v(0,t)=v(a,t)=0$.  
	\item $v(x,t)=X(x)T(t)$. 
	$\frac{X''}{X}=\frac{1}{\sigma^2}\frac{T'}{T}=-\lambda^2$. 
	\item $X(x)=A\cos(\lambda x)+B\sin(\lambda x)$, $X(0)=X(a)=0$.
	\item $v_n(x,t)=A_n\sin(\frac{n\pi}{a}x)
	e^{-\left(\sigma\frac{n\pi}{a}\right)^2t}$.
	\item Re"sitev: \formulas{$\displaystyle 
	u(x,t)=\tau_0+\frac{\tau_a-\tau_0}{a}x +
	\sum_{n=1}^\infty A_n\sin(\frac{n\pi}{a}x)
        e^{-\left(\sigma\frac{n\pi}{a}\right)^2t}$}.
	\item $\displaystyle A_n=\frac{2}{a}\int_0^a
	\left(f(x)-\left(\tau_0+\frac{\tau_a-\tau_0}{a}x\right)\right)
	\sin(\frac{n\pi}{a}x)dx$.
\end{itemize}
\end{frame}
\end{document}

