ReconstructionZvit.tex

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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%òèòóëüíà ñòîð³íêà
\titlepage
\begin{center}
\large {Ëüâ³âñüêèé íàö³îíàëüíèé óí³âåðñèòåò ³ìåí³ ²âàíà Ôðàíêà}\\
 \normalsize{Ôàêóëüòåò ïðèêëàäíî¿ ìàòåìàòèêè òà ³íôîðìàòèêè}\\
  \normalsize{Êàôåäðà îá÷èñëþâàëüíî¿ ìàòåìàòèêè}\\
\end{center}
\vspace*{0.5cm}
\begin{center}
\Large{\textbf{Çâ³ò}}\\
\large{ç êóðñó "Ìåòîäè ðåãóëÿðèçàö³¿ äëÿ ðîçâ'ÿçóâàííÿ îáåðíåíèõ çàäà÷" \\ }
\large{íà òåìó: \\}
\Large{\textbf{×èñåëüíå ðîçâ'ÿçóâàííÿ çàäà÷³ Êîø³ äëÿ ð³âíÿííÿ Ëàïëàñà â 2-âèì³ðí³é äâîçâ'ÿçí³é îáëàñò³ íåïðÿìèì ìåòîäîì ³íòåãðàëüíèõ ð³âíÿíü}}
\end{center}
\normalsize
\vspace*{1cm}\hspace*{11cm}Âèêîíàëè\\
\hspace*{11cm}ñòóäåíòè ãðóïè ÏÌï-51 \\
\hspace*{11cm}Ãëàäèø Ðîìàí\\
\hspace*{11cm}Êàíàôîöüêèé Òàðàñ\\
\\ \\
\hspace*{11cm}Íàóêîâ³ êîíñóëüòàíòè:\\
\hspace*{11cm}ïðîô. Õàïêî Ð.Ñ.\\
\hspace*{11cm}àñ. Áîðà÷îê ².Â.\\
\\
\centerline{Ëüâ³â - 2017}

%!!!!!!!!!!!!!!!!!!!!!!!!ñòîð³íêà çì³ñòó!!!!!!!!!!!!!!!!!!!!!
\tableofcontents
\newpage
\chapter*{Âñòóï}  \addcontentsline{toc}{chapter}{Âñòóï}
\normalsize
\hspace*{\parindent} \hspace*{\parindent} Âïðîäîâæ îñòàíí³õ ðîê³â âñå á³ëüøî¿ ïîïóëÿðíîñò³ íàáóâàþòü íàáëèæåí³, ÷èñåëüí³ òà ìàøèíí³ ìåòîäè ðîçâÿçóâàííÿ ³íòåãðàëüíèõ ð³âíÿíü, îñê³ëüêè ¿¿ çàñòîñóâàííÿ äîçâîëÿº îòðèìàòè åôåêòèâí³ ìàòåìàòè÷í³ îïèñè áàãàòüîõ çàäà÷. Íàêðèêëàä, çà äîïîìîãîþ ³íòåãðàëüíèõ ð³âíÿíü àíàë³çóþòü ïðîöåñè â äèíàì³÷íèõ ñèñòåìàõ, âèçíà÷àþòü ³ìïóëüñíó ôóíêö³þ ë³í³éíî¿ ñèñòåìè, ðîçâ'ÿçóþòü çàäà÷³ îïòèìàëüíî¿ ô³ëüòðàö³¿, çàäà÷³ â³äíîâëåííÿ ñèãíàëó ³ òä.\\
\hspace*{\parindent}Âàæëèâó ðîëü ó ðîçâ'ÿçóâàíí³ ðÿäó çàäà÷ ìàòåìàòè÷íî¿ ô³çèêè â³ä³ãðຠìåòîä ãðàíè÷íèõ ³íòåãðàëüíèõ ð³âíÿíü, çà äîïîìîãîþ ÿêîãî çâîäÿòü â³äïîâ³äíó äèôåðåíö³àëüíó çàäà÷ó äî åêâ³âàëåíòíîãî ³íòåãðàëüíîãî ð³âíÿííÿ. Òîìó ñêëàäຠ³íòåðåñ çàñòîñóâàííÿ åôåêòèâíèõ ìåòîä³â äëÿ ÷èñåëüíîãî ðîçâ'ÿçóâàííÿ  îòðèìàíîãî ³íòåãðàëüíîãî ð³âíÿííÿ. Òîìó çàçíà÷èìî, ùî ìåòîä ãðàíè÷íèõ ³íòåãðàëüíèõ ð³âíÿíü åôåêòèâí³øèé çà ìåòîä ñê³í÷åííèõ ð³çíèöü, ìåòîä ñê³í÷åííèõ åëåìåíò³â òà ³íø³, îñê³ëüêè âîëî䳺 òàêèìè ïåðåâàãàìè ÿê : \begin{itemize}
                                 \item ðîçì³ðí³ñòü çàäà÷³ çìåíøóºòüñÿ íà 1;
                                 \item ìåòîä çàñòîñîâíèé äëÿ îáëàñòåé ç ãðàíèöÿìè äîâ³ëüíî¿ ôîðìè;
                                 \item çàñòîñîâíèé ó âèïàäêó íåîáìåæåíèõ îáëàñòåé.
                               \end{itemize}
 Ìåòîþ ö³º¿ ðîáîòè º ÷èñåëüíå ðîçâ'ÿçóâàííÿ çîâí³øíüî¿ çàäà÷³ Êîø³ äëÿ ð³âíÿííÿ Ëàïëàñ ó äâîçâ'ÿçí³é îáëàñò³ íåïðÿìèì ìåòîäîì ãðàíè÷íèõ ³íòåãðàëüíèõ ð³âíÿíü. Òàêèì ÷èíîì ìè çâåäåìî çàäà÷ó ó äâîâèì³ðí³é îáëàñò³ äî çàäà÷³ ç íà îäèíèöþ ìåíøîþ ðîçì³ðíîñòþ. Òàêîæ îäí³ºþ ç ïåðåâàã öüîãî ìåòîäó º ìîæëèâ³ñòü îòðèìàòè ðîçâ'ÿçîê âèñîêî¿ ÿêîñò³ áåç âåëèêèõ ÷èñåëüíèõ çàòðàò.
\newpage
\chapter{Äîñë³äæåííÿ çàäà÷³}
\section{Ïîñòàíîâêà çàäà÷³ Êîø³ äëÿ ð³âíÿííÿ Ëàïëàñà}
 \hspace*{\parindent} Íåõàé â îáìåæåí³é îáëàñò³ $D_2 \subset R^2$ ç ãðàíèöåþ $\Gamma_2$ ì³ñòèòüñÿ îäíîçâ'ÿçíà îáìåæåíà îáëàñòü $D_1 \subset R^2$ ç ãðàíèöåþ $\Gamma_1 \in C^2$, òàêà ùî $\overline{D_1} \subset R^2$. Îáëàñòü $D_1$ íàçèâàòèìåìî âíóòð³øí³ì âêëþ÷åííÿì îáëàñò³ $D_2$. Òàêîæ ïîçíà÷èìî ÷åðåç $D := D_2 \backslash \overline{D_1}$. Ïîòð³áíî çíàéòè ðåãóëÿðíó â îáëàñò³ $D$ ôóíêö³þ $u \in C^2(D)\bigcap C(\overline{D})$ , ÿêà çàäîâîëüíÿº ð³âíÿííÿ Ëàïëàñà
\begin{equation}
\label{Laplace}
        \Delta u = 0 \quad \textup{â } D
\end{equation}
³ â³äïîâ³äí³ ãðàíè÷í³ óìîâè ijð³õëå ³ Íåéìàíà
\begin{equation}
\label{Dirichlet}
        u = f \quad \textup{íà } D
\end{equation}
\begin{equation}
\label{Neuman}
        \frac{\partial u}{\partial \nu} = g \quad \textup{íà } \Gamma_2,
\end{equation}
à òàêîæ $ \frac{\partial u}{\partial \nu}, u \textup{ íà } \Gamma_1$.

\section{Ïîòåíö³àëè ïðîñòîãî ³ ïîäâ³éíîãî øàðó òà ¿õ âëàñòèâîñò³}
\hspace*{\parindent} Íåõàé áëàñòü $D_1$ âíóòð³øíº âêëþ÷åííÿ îáëàñò³ $D_2$ ç ãðàíèöåþ $\Gamma_1$  ç êëàñó $C^p \quad (p\geq2)$. Äëÿ ôóíêö³¿ $\varphi \in C(\Gamma_1)$ âèçíà÷èìî ôóíêö³¿
\begin{equation}
\label{potential1}
u(x) = \int\limits_{\Gamma_1} \, \varphi(y)\Phi(x,y) ds(y) , \quad x \in D_2\setminus\Gamma_1
\end{equation}
òà
\begin{equation}
\label{potential2}
\upsilon(x) = \int\limits_{\Gamma_1} \, \varphi(y)\frac{\Phi(x,y)}{\nu(y)} ds(y) , \quad x \in D_2\setminus\Gamma_1
\end{equation}
ÿê ìîäèô³êîâàí³ ïîòåíö³àëè ïðîñòîãî òà ïîäâ³éíîãî øàðó ç ãóñòèíîþ $\varphi $ â³äïîâ³äíî.\\
\hspace*{\parindent} Òóò $\Phi$ ôóíäàìåíòàëüíèé ðîçâ'ÿçîê ð³âíÿííÿ Ëàïëàñà (\ref{Laplace}) â $\mathbb{R}^2$
\begin{equation}
\begin{split}
\label{fundSol}
    \Phi(x,y) & := -\frac{1}{2\pi}\ln |x-y|, \\
    |x-y| & =\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2},
\end{split}
\end{equation}
 $\nu$ - âíóòð³øíÿ íîðìàëü äî ãðàíèö³ âíóòð³øíüîãî âêëþ÷åííÿ $\Gamma_1$.\\
\hspace*{\parindent} Íà îñíîâ³ êëàñè÷íèõ ðåçóëüòàò³â \cite{Kress} òà ³ç âðàõóâàííÿì âëàñòèâîñòåé ôóíäàìåíòàëüíîãî ðîçâ'ÿçêó (\ref{fundSol}), äëÿ ïîòåíö³àë³â (\ref{potential1}) òà (\ref{potential1}) ñïðàâåäëèâ³ íàñòóïí³ òâåðäæåííÿ.
\begin{theorem}
\label{theorem 1}
Íåõàé $\Gamma_0 \in C^p \quad (p\geq2)$ ³ $\varphi \in C(\Gamma_1)$. Òîä³ ïîòåíö³àë ïðîñòîãî øàðó (\ref{potential1}) ç ãóñòèíîþ $\varphi$ º ðåãóëÿðíîþ ôóíêö³ºþ â îáëàñò³ $D$ ³ çàäîâîëüíÿº îö³íêó
$$
\| u\|_{\infty,D_2} \leq C \|\varphi\|_{\infty,\Gamma_1},
$$
äå $\| \cdot\|_{\infty}$ ïîçíà÷ຠíîðìó â ïðîñòîð³ íåïåðåðâíèõ ôóíêö³é, $C > 0$ - äåÿêà êîíñòàíòà.Íà ãðàíèö³ $\Gamma_1$ ìàºìî íàñòóïíå ïðåäñòàâëåííÿ
$$
u(x) = \int\limits_{\Gamma_1} \, \varphi(y)\Phi(x,y) ds(y) , \quad x \in \Gamma_1.
$$
\end{theorem}
\begin{theorem}
\label{theorem 2}
 Ïîòåíö³àë ïîäâ³éíîãî øàðó (\ref{potential2}) ç ãóñòèíîþ $\varphi \in C(\Gamma_1)$ ìîæíà íåïåðåðâíî ïðîäîâæèòè ç îáëàñò³ $D_2$ â $\overline{D_1}$ ³ ç $D_2\backslash \overline{D_1}$ â $\overline{D_1}$ ³ç íàñòóïíèìè çíà÷åííÿìè íà ãðàíèö³ $\Gamma_1$
 $$
\frac{\partial u}{\partial \nu}(x) = \int\limits_{\Gamma_1} \, \varphi(y)\frac{\partial^2\Phi(x,y)}{\partial \nu(x) \partial \nu(x)} ds(y) , \quad x \in \Gamma_1.
$$
Ìàþòü ì³ñöå íàñòóïí³ îö³íêè :
$$
\| \upsilon\|_{\infty,\overline{D_1}} \leq C \|\varphi\|_{\infty,\Gamma_1} ,\quad \| \upsilon\|_{\infty,D_2 \backslash D_1} \leq C \|\varphi\|_{\infty,\Gamma_1}.
$$
\end{theorem}


\section{Çâåäåííÿ äî ñèñòåìè ²Ð }
\hspace*{\parindent} Âèêîðèñòîâóþ÷è ðåçóëüòàòè ïîïåðåäí³õ ï³äðîçä³ë³â, ïîäàìî ðîçâ'ÿçîê çàäà÷³ (\ref{Laplace})-(\ref{Neuman}) ó âèãëÿä³ êîìá³íàö³¿ ïîòåíö³àë³â ïîäâ³éíîãî øàðó
\begin{equation}
\label{solution}
    u(x) = \int_{\Gamma _1} \varphi_1(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y) + \int_{\Gamma _2} \varphi_1(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y), \quad x \in D,
\end{equation}
ç íåâ³äîìèìè ãóñòèíàìè $\varphi_1 \in C(\Gamma_1)$ ³ $\varphi_2 \in C(\Gamma_2)$.
 Äàë³ âðàõóâàâøè ïîäàííÿ ðîçâ'ÿçêó (\ref{solution}) ³ çàñòîñóâàâøè ôîðìóëè ñòðèáê³â ïîòåíö³àë³â ïîäâ³éíîãî øàðó, ëåãêî áà÷èòè, íàêëàâøè óìîâè ijð³õëå ³ Íåéìàíà, çàäà÷ó (\ref{Laplace})-(\ref{Neuman}) ìîæíà çâåñòè äî íàñòóïíî¿ ñèñòåìè ãðàíè÷íèõ ³íòåãðàëüíèõ ð³âíÿíü.
\begin{equation}
\label{initEquation}
    \begin{cases}
        \int_{\Gamma _1} \varphi_1(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y) + \frac{1}{2}\varphi_2(x) + \int_{\Gamma _2}\varphi(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y) =f(x), \quad x \in \Gamma_2,\\
       \int_{\Gamma _1} \varphi_1(y)\frac{\partial \Phi(x,y)}{\partial\nu(x) \nu(y)}ds(y) + \frac{\partial}{\partial\nu(x)}\int_{\Gamma _2}\varphi(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y) =g(x), \quad x \in \Gamma_2,\\
    \end{cases}
\end{equation}
\hspace*{\parindent}Òàêîæ ç ìåòîþ çìåíøåííÿ ñòåïåíÿ ñèíãóëÿðíîñò³ â ã³ïåðñèíãóëÿðíîìó ³íòåãðàë³ â ñèñòåì³ (\ref{initEquation}) ñêîðèñòàºìîñü ð³âí³ñòþ
\begin{equation}
     \frac{\partial}{\partial\nu(x)}\int_{\Gamma _2}\varphi(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y)=\int_{\Gamma_2}\frac{\partial\varphi_2}{\partial \Theta}(y) \frac{\Phi(x,y)}{\partial \Theta(x)}ds(y),\quad x \in \Gamma_2,
\end{equation}
äîâåäåííÿ ÿêî¿ ìîæíà çíàéòè ó \cite{Kress}. Òóò $\Theta$ - öå îäèíè÷íèé âåêòîð äîòè÷íî¿ äî $\Gamma_2$.\\
\hspace*{\parindent}Òîìó ñèñòåìà ñèñòåìà ³íòåãðàëüíèõ ð³âíÿíü (\ref{initEquation}) íàáóäå âèãëÿäó
\begin{equation}
\label{system}
    \begin{cases}
        \int_{\Gamma _1} \varphi_1(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y) + \frac{1}{2}\varphi_2(x) + \int_{\Gamma _2}\varphi(y)\frac{\partial \Phi(x,y)}{\partial \nu(y)}ds(y) =f(x), \quad x \in \Gamma_2,\\
       \int_{\Gamma _1} \varphi_1(y)\frac{\partial \Phi(x,y)}{\partial\nu(x) \nu(y)}ds(y) + \int_{\Gamma_2}\frac{\partial\varphi_2}{\partial \Theta}(y) \frac{\Phi(x,y)}{\partial \Theta(x)}ds(y) =g(x), \quad x \in \Gamma_2,\\
    \end{cases}
\end{equation}

\section{Ïàðàìåòðèçàö³ÿ}
\hspace*{\parindent}Íåõàé ãðàíèö³ $\Gamma_1$ ³ $\Gamma_2$ çàäàþòüñÿ ïàðàìåòðè÷íî ÿê
\begin{equation}
\label{Gamma1}
    \Gamma_1 := \big\{\eta(t)+ d := (\eta_1(t) + d_1, \eta_2(t) + d_2), \quad t \in [0, 2\pi]\big\},
\end{equation}
\begin{equation}
\label{Gamma2}
    \Gamma_2 := \big\{\sigma(t) + d := (\sigma_1(t) + d_1, \sigma_2(t) + d_2), \quad t \in [0, 2\pi]\big\},
\end{equation}
äå òî÷êà $d=(d_1, d_2)$ âêàçóº íà â³äñòàíü â³ä ïî÷àòêó êîîðäèíàò. Ôóíêö³¿ $\sigma : [0, 2\pi] \rightarrow \mathbb{R}^2$ òà $\eta : [0, 2\pi] \rightarrow \mathbb{R}^2$ - $2\pi$-ïåð³îäè÷í³, íàëåæàòü êëàñó $C^2[0,\pi]$, º ³í'ºêòèâíèìè ³ òàê³, ùî $\sigma'(t) \neq 0$ òà $\eta'(t) \neq 0$ $ \forall t\in [0,\pi]$.
\hspace*{\parindent}Òîä³ ñèñòåìó (\ref{system}) ìîæíà ïîäàòè ó ïàðàìåòðèçîâàíîìó âèãëÿä³
\begin{equation}
\label{systemParamtrized}
    \begin{cases}
        \displaystyle\int_0^{2\pi}\mu_1(\tau)H_{11}(t,\tau)d\tau + \frac{\mu_2(t)}{2|\sigma'(t)|} + \int_0^{2\pi}\mu_2(\tau)H_{12}(t,\tau)d\tau = f(\sigma(t)),\, t \in [0,2\pi]\\
         \displaystyle\int_0^{2\pi}\mu_1(\tau)H_{21}(t,\tau)d\tau + \int_0^{2\pi}\mu_2'(\tau)K_{22}(t,\tau)d\tau = g(\sigma(t)),\, t \in [0,2\pi]
    \end{cases}
\end{equation}
$$\mu_1(\tau) = \varphi_1(\eta(\tau))|\eta'(\tau)| \quad \mu_2(\tau) = \varphi_2(\sigma(\tau))|\sigma'(\tau)|$$
ç ãëàäêèìè $2\pi$-ïåð³îäè÷íèìè ôóíêö³ÿìè
\begin{equation}
\label{H11}
   H_{11}(t,\tau)=\frac{1}{2\pi}\frac{\langle\sigma(t)-\eta(\tau),\nu(\eta(\tau))\rangle}{|\sigma(t)-\eta(\tau)|^2}
\end{equation}
\begin{equation}
\label{H21}
    H_{21}(t,\tau) = \frac{1}{2\pi}\Bigg(\frac{\langle\nu(\eta(\tau)), \nu(\sigma(t))\rangle}{|\sigma(t)-(\eta(\tau))|^2} - \frac {2\langle\sigma(t)-\eta(\tau), \nu(\eta(\tau))\rangle\langle\sigma(t)-\eta(\tau),\nu(\sigma(t))\rangle}{|\sigma(t)-\eta(\tau)|^4}\Bigg)
\end{equation}
\begin{equation}
\label{H12}
H_{12}(t,\tau) =
        \frac{1}{2\pi}\frac{\langle\sigma(t)-\sigma(\tau),\nu(\eta(\tau))\rangle}{|\sigma(t)-\eta(\tau)|^2}, \quad t \neq \tau
\end{equation}
ç ä³àãîíàëüíèìè åëåìåíòàìè
\begin{equation}
\label{H12diagon}
\lim_{\tau \to t}H_{12}(t,\tau) = \frac{1}{2\pi}\frac{\langle\sigma^{''}(t), \nu(\sigma(t))\rangle}{|\sigma^{'}(t)|^2}.
\end{equation}
³ âåêòîðîì íîðìàë³ òà äîòè÷íî¿ â³äïîâ³äíî
\begin{equation}
\label{normalVector}
   \nu(\sigma(t))=\frac{1}{|\sigma(t)|}\Big(\sigma_2^{'}(t),
   -\sigma_1^{'}(t)\Big),
\end{equation}
\begin{equation}
\label{Dotichna}
   \Theta(\sigma(t))=\frac{1}{|\sigma(t)|}\Big(\sigma_1^{'}(t),\sigma_2^{'}(t)\Big).
\end{equation}
\hspace*{\parindent}ϳä³íòåãðàëüíà ôóíêö³ÿ
\begin{equation}
\label{K22}
K_{22}(t,\tau) = -\frac{1}{2\pi}\frac{\langle\sigma(t)-\sigma(\tau),\Theta(\sigma(\tau))\rangle}{|\sigma(t)-\sigma(\tau)|^2},
\end{equation}
ì³ñòèòü ã³ïåðñèíãóëÿðíó îñîáëèâ³ñòü, äëÿ âèä³ëåííÿ ÿêî¿ çä³éñíèìî íàñòóïí³ ïåðåòâîðåííÿ, ñêîðèñòàâøèñü ôîðìóëîþ ³íòåãðóâàííÿ ÷àñòèíàìè ³ âðàõóâàâøè $2\pi$-ïåð³îäè÷í³ñòü ôóíêö³é $K_{22}$ òà $\mu_2$,
\begin{equation}
\int_0^{2\pi}\mu_2'(\tau)K_{22}(t,\tau)d\tau = -\int_0^{2\pi}\mu_2(\tau)\frac{1}{2\sin^2(\frac{t-\tau}{2})}\{{K_{22}}_t'(t,\tau)2\sin^2(\frac{t-\tau}{2})\}d\tau.
\end{equation}
Äàë³ ñêîðèñòàºìîñü ïðàâèëîì Ëîï³òàëÿ, îá÷èñëèìî íåîáõ³äí³ ïîõ³äí³ ³ îòðèìàºìî òîòîæí³ñòü
\begin{equation}
\label{H22inEq}
\int_0^{2\pi}\mu_2'(\tau)K_{22}(t,\tau)d\tau = \int_0^{2\pi}\mu_2(\tau)\frac{1}{2\sin^2(\frac{t-\tau}{2})}H_{22}(t,\tau)d\tau,
\end{equation}
äå $H_{22}$ - ãëàäêà $2\pi$-ïåð³îäè÷íà ôóíêö³ÿ
\begin{multline}
    H_{22}(t,\tau)=\frac{1}{\pi}( \frac{2\langle\sigma(t)-\sigma(\tau),\sigma^{'}(\tau)\rangle\langle\sigma(t)-\sigma(\tau),\sigma^{'}(t)\rangle}{|\sigma(t)-\sigma(\tau)|^4}\\ -\frac{\langle\sigma^{'}(t),\sigma^{'}(\tau)\rangle}{|\sigma(t)-\sigma(\tau)|^2})\sin^2(\frac{t-\tau}{2})
\end{multline}
ç ä³àãîíàëüíèìè åëåìåíòàìè
\begin{equation}
    \lim_{\tau \to t}H_{22}(t,\tau)=-\frac{1}{2\pi|\sigma(t)|}.
\end{equation}
\hspace*{\parindent}Òàêèì ÷èíîì ñèñòåìó íåêîðåêòíèõ ³íòåãðàëüíèõ ð³âíÿíü áóëî çâåäåíî äî íàñòóïíîãî âèãëÿäó
\begin{equation}
\label{systemParamFinal}
    \begin{cases}
        \displaystyle\frac{1}{2}\mu_2(t) + \int_0^{2\pi}\Big\{\mu_1(\tau) H_{11}(t,\tau)+
        \mu_2(\tau)H_{12}(t,\tau)\Big\}d\tau = f(\sigma(t)),\\
         \displaystyle\int_0^{2\pi}\Big\{ \mu_1(\tau)H_{21}(t,\tau) + \mu(\tau)\frac{1}{2\sin^2\big(\frac{t-\tau}{2})}H_{22}(t,\tau)\Big\}d\tau=g(\sigma(t))
    \end{cases}
\end{equation}
\section{Ìåòîä òðèãîíîìåòðè÷íèõ êâàäðàòóð}
\hspace*{\parindent} ×èñåëüíå ðîçâ’ÿçóâàííÿ ñèñòåìè ³íòåãðàëüíèõ ð³âíÿíü(\ref{systemParamFinal}) çä³éñíèìî ìåòîäîì êâàäðàòóð \cite{Chapko_Kress}. Äëÿ öüîãî íà ðîçáèòò³ $t_i := \frac{i\pi}{n},\quad i =0,..,2n-1, n \in \mathbb{N}$  ðîçãëÿíåìî òàê³ êâàäðàòóðí³ ôîðìóëè:
\begin{equation}
\label{quadrFormul1}
\frac{1}{2\pi}\int_0^{2\pi} \, f(\tau) d\tau \approx\frac{1}{2n}\sum_{j=0}^{2n-1}f(t_j)
\end{equation}
\begin{equation}
\label{quadrFormul3}
\frac{1}{2\pi}\int_0^{2\pi} \, \frac{f(\tau)}{2\sin^2{\frac{t-\tau}{2}}} d\tau \approx\sum_{j=0}^{2n-1}f(t_j)T^n_j(t)
\end{equation}
ç âàãîâèìè ôóíêö³ÿìè
$$
T^n_j(t) := -\frac{1}{n}\sum_{m=1}^{n-1}{m\cos{m(t-t_j)}} + \frac{\cos{n(t-t_j)}}{2}.
$$
\hspace*{\parindent}Ö³ ôîðìóëè îòðèìàí³ ç âèêîðèñòàííÿì òðèãîíîìåòðè÷íî¿ ³íòåðïîëÿö³¿ äëÿ ãëàäêî¿ ÷àñòèíè $f$ ï³ä³íòåãðàëüíî¿ ôóíêö³¿ ³ ïîäàëüøîãî òî÷íîãî ³íòåãðóâàííÿ \cite{Kress}. ϳñëÿ çàñòîñóâàííÿ âèïèñàíèõ êâàäðàòóðíèõ ïðàâèë (\ref{quadrFormul1})-(\ref{quadrFormul3}) äî ³íòåãðàë³â ó ñèñòåì³ (\ref{systemParamFinal}) ³ êîëîêàö³¿ ó âóçëàõ êâàäðàòóðíèõ ôîðìóë îòðèìàºìî òàêó ñèñòåìó ë³í³éíèõ ð³âíÿíü ðîçì³ðí³ñòþ $4n\times4n$:
\begin{equation}
\label{systemDiscret}
  \begin{cases}
      \displaystyle \frac{\pi}{n}\sum_{j=0}^{2n-1}\mu_1(t_j)H_{11}(t_i,t_j)+\frac{\mu_2(t_i)}{2|\sigma^{'}(t_i)|} \\+ \displaystyle\frac{\pi}{n}\sum_{j=1}^{2n-1}\mu_2(t_j)H_{22}(t_i,t_j)=f(\sigma(t_i))\\
        \displaystyle \frac{\pi}{n}\sum_{j=0}^{2n-1}\mu_1(t_j)H_{21}(t_i)(t_j)\\
         \displaystyle+2\pi\sum_{j=0}^{2n-1}\mu_2(t_j)T_j^n(t_i)H_{22}^{(1)}(t_i,t_j)=q(\sigma(t_i))
    \end{cases}
\end{equation}
ùîäî íåâ³äîìèõ çíà÷åíü $\mu_{1j} \approx \mu_1(t_j)$ ³ $\mu_{2j} \approx \mu_2(t_j)$.\\
\hspace*{\parindent}Ó ïðàö³ \cite{Chapko_Kress} íà ï³äñòàâ³ òåî𳿠êîëåêòèâíî êîìïàêòíèõ îïåðàòîð³â äîñë³äæåíî çá³æí³ñòü ³ ïîõèáêó ïðîïîíîâàíîãî ìåòîäó. Çîêðåìà, ÿêùî \\$\Gamma_2 \in C^\infty$ ³ $F \in C^{p+1}$,  òî ìຠì³ñöå îö³íêà $\|\tilde{\mu_n} - \mu\|_{0,\alpha}\leq Cn^{-p}$, äå $\tilde{\mu_n}$ - òðèãîíîìåòðè÷íèé ïîë³íîì ïîáóäîâàíèé çà äîïîìîãîþ çíà÷åíü $\mu_j$, çíàéäåíèõ ç ñèñòåìè (\ref{systemDiscret}). Îòæå öåé ìåòîä íàëåæèòü äî àëãîðèòì³â áåç íàñè÷åííÿ,òîáòî éîãî òî÷í³ñòü ïîâ’ÿçàíà ³ç ãëàäê³ñòþ âõ³äíèõ äàíèõ. Çàóâàæèìî, ùî çà àíàë³òè÷íîñò³ $\Gamma_2$  ³ $F$ îòðèìóºìî åêñïîíåíö³éíó çá³æí³ñòü.

\section{Ðåãóëÿðèçàö³ÿ Ò³õàíîâà}
\hspace*{\parindent}Ìåòîä ðåãóëÿðèçàö³¿ Ò³õîíîâà - àëãîðèòì, ùî äîçâîëÿº çíàõîäèòè íàáëèæåíèé ðîçâ'ÿçîê íåêîðåêòíèõ îïåðàòîðíèõ ð³âíÿíü âèäó $Ma=b$. Áóâ ðîçðîáëåíèé À.Í. Ò³õîíîâèì â 1965 ð .Îñíîâíà ³äåÿ ïîëÿãຠâ çíàõîäæåíí³ íàáëèæåíîãî ðîçâ'ÿçêó ð³âíÿííÿ $Ma=b$ ó âèãëÿä³ $x_{\delta} = R(b_{\delta}, \lambda)$, äå $R(b_{\delta},\lambda)$ - ðåãóëÿðèçóþ÷èé îïåðàòîð, äå $\lambda$ - ïàðàìåòð ðåãóëÿðèçàö³¿. ³í ïîâèíåí ãàðàíòóâàòè, ùî ïðè íàáëèæåíí³ $b_{\delta}$ äî òî÷íîãî çíà÷åííÿ $b_{T}$ ïðè $\delta \rightarrow 0$ íàáëèæåíèé ðîçâ'ÿçîê $a_{\delta}$ ïðÿìóº äî áàæàíîãî òî÷íîãî ðîçâ'ÿçêó $a_{T}$ ð³âíÿííÿ $Ma = b_{T}$.\\
\hspace*{\parindent} Ó íàøîìó âèïàäêó ó ÿêîñò³ ðåãóëÿðèçîâàíîãî ðîçâ'ÿçêó ð³âíÿííÿ $Ma=b$, äå $M$ - ìàòðèöÿ $4n\times4n$ ³ $b \in \mathbb{R}^{4n}$, áðàòèìåìî ðîçâ'ÿçîê, ÿêèé ì³í³ì³çóº íàñòóïíèé âèðàç:
$$
  a_{\lambda} = \arg\min{\|Ma_{\lambda} -b \|^{2}_2 + \lambda\|a_\lambda\|^{2}_2},
$$
ùî åêâ³âàëåòíî ðîçâ'ÿçêó òàêî¿ ñèñòåìè:
 \begin{equation}
  (M^TM + \lambda I)a_\lambda = M^Tb
 \end{equation}
 äå $M^T$- òðàíñïîíîâàíà ìàòðèöÿ $M$.
\section{Ïîäàííÿ Ðîçâ'ÿçêó}
$u$  íà $\Gamma_1$
\begin{equation}
u(x)=\frac{1}{2}\varphi_1(x) + \int_{\Gamma_1}\varphi_1(y)\frac{\partial\Phi(x,y)}{\partial\nu(y)}ds(y) +\int_{\Gamma_2}\varphi_2(y)\frac{\partial\Phi(x,y)}{\partial\nu(y)}ds(y)
\end{equation}
\begin{equation}
u(t_i)=\frac{\mu_1(t_i)}{2|\eta^{'}(t)|} + \frac{\pi}{n}\sum_{j=0}^{2n-1} \mu_1(t_j)H_{11}^{(1)}(t_i,t_j)+\frac{\pi}{n}\sum_{j=0}^{2n-1} \mu_2(t_j)H_{12}^{(1)}(t_i,t_j)
\end{equation}
\begin{equation}
    H_{11}^{(1)}(t,\tau)=
    \begin{cases}
       \displaystyle \frac{1}{2\pi} \frac{\langle\eta(t)-\eta(\tau),\nu(\eta(\tau))\rangle}{|\eta(t)-\eta(\tau)|^2} \quad , t\neq\tau\\
       \displaystyle \frac{1}{2\pi} \frac{\langle\eta(t)^{''},\nu(\eta(\tau))\rangle}{2|\eta(t)^{'}|^2} \quad , t\doteq\tau
    \end{cases}
\end{equation}
\begin{equation}
H_{12}^{(1)}(t,\tau)=\frac{1}{2\pi}\frac{\langle\eta(t)-\sigma(\tau), \nu(\sigma(\tau))\rangle}{|\eta(t)-\sigma(\tau)|^2}
\end{equation}
\newpage
\chapter*{Âèñíîâîê} \addcontentsline{toc}{chapter}{Âèñíîâîê}
\hspace*{\parindent}Ó äàí³é ðîáîò³ ðîçãëÿíóòî ìåòîä ãðàíè÷íèõ ³íòåãðàëüíèõ ð³âíÿíü äëÿ íàáëèæåíîãî ðîçâ'ÿçóâàííÿ çàäà÷³ Êîø³ äëÿ ð³âíÿííÿ Ëàïëàñà. Ïîêàçàíî ¿¿ íåêîðåêòí³ñòü. Ç âèêîðèñòàííÿì âëàñòèâîñòåé ïîòåíö³àë³â ïðîñòîãî øàðó îòðèìàíî ñèñòåìó ²Ð ç ãëàäêèìè ³ ñèíãóëÿðíèìè ï³ä³íòåãðàëüíèìè ôóíêö³ÿìè. Äàë³ çàäà÷ó Êîø³ ïåðåòâîðåíî ó ñèñòåìó ë³í³éíèõ àëãåáðè÷íèõ ð³âíÿíü, øëÿõîì ïàðàìåòðèçàö³¿ òà íàáëèæåíîãî îá÷èñëåííÿ ³íòåãðàë³â çà äîïîìîãîþ â³äïîâ³äíèõ òðèãîíîìåòðè÷íèõ êâàäðàòóðíèõ ôîðìóë. Äî îòðèìàíî¿ íåêîðåêòíî¿ ñèñòåìè ë³í³éíèõ àëãåáðà¿÷íèõ ð³âíÿíü çàñòîñîâàíî ìåòîä ðåãóëÿðèçàö³¿ Ò³õàíîâà.\\
\hspace*{\parindent}Ïðîâåäåíî ÷èñåëüíå ðîçâ'ÿçóâàííÿ çàäà÷³ ó äâîçâ'ÿçí³é îáëàñò³ ç âèêîðèñòàííÿì ìîâè ïðîãðàìóâàííÿ R.
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\renewcommand{\bibname}{Ñïèñîê ë³òåðàòóðè}
\begin{thebibliography}{99}
\addcontentsline{toc}{chapter}{Ñïèñîê ë³òåðàòóðè}
 \bibitem{Chapko_Kress}
\emph{Chapko R.,Kress R.} On a quadrature method for a logarithmic integral equation of the first kind// World of Scientific Series in Applicable Analisis -Vol.2 Contributions in Numerical Mathematics (Agarwal, ed.) World Scientific, Singapore, 1993. P.127-140.

\bibitem{Kress}
\emph{Kress R.} Linear Integral Equations, 2nd. ed. /R. Kress - New York: Springer-Verlag ,Heidelberg, 1999. - 367 p.

\end{thebibliography}
\end{document}