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Adding problem description to Project C

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PolishPigeon 2022-01-02 01:22:56 +01:00
parent 91f05a97b0
commit 679ae2635c
9 changed files with 1679 additions and 0 deletions
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ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux Normal file
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{Determine polynomial function fitting experimental data} [1
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Chapter 1.
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Chapter 2.
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\BOOKMARK [0][-]{chapter.1}{Determine polynomial function fitting experimental data}{}% 1
\BOOKMARK [1][-]{section.1.1}{Problem}{chapter.1}% 2
\BOOKMARK [1][-]{section.1.2}{Theoretical introduction}{chapter.1}% 3
\BOOKMARK [0][-]{chapter.2}{Determine trajectory of the motion}{}% 4
\BOOKMARK [1][-]{section.2.1}{a\) Runge-Kutta method of 4th order and Adams PC}{chapter.2}% 5
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\BOOKMARK [2][-]{subsection.2.1.2}{Theoretical Introdution}{section.2.1}% 7
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\BOOKMARK [2][-]{subsection.2.2.1}{Problem}{section.2.2}% 9
\BOOKMARK [2][-]{subsection.2.2.2}{Theoretical Introdution}{section.2.2}% 10

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\documentclass[12pt]{report}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{systeme}
\usepackage{siunitx}
\usepackage[T1]{fontenc}
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\usepackage{bigfoot}
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\title{Numerical Methods, project C, Number 32}
\author{Krzysztof Rudnicki\\ Student number: 307585 \\ Advisor: dr hab. Piotr Marusak}
\date{\today}
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\begin{document}
\maketitle
\tableofcontents
\chapter{Determine polynomial function fitting experimental data}
\section{Problem}
Given following samples:
\begin{center}
\begin{tabular}{| c | c |}
\hline
$x_i$ & $y_i$ \\
\hline
-5 & -6.5743\\
\hline
-4 & 0.9765\\
\hline
-3 & 3.1026\\
\hline
-2 & 1.8572 \\
\hline
-1 & 1.3165 \\
\hline
0 & -0.6144 \\
\hline
1 & 0.1032 \\
\hline
2 & 0.3729 \\
\hline
3 & 2.5327 \\
\hline
4 & 7.3857 \\
\hline
5 & 9.4892 \\
\hline
\end{tabular}
\end{center}
We have to determine polynomial function $ y = f(x) $ that best fits this data.
We will use least-square approximation using system of normal equation with QR factorization.
\section{Theoretical introduction}
\chapter{Determine trajectory of the motion}
\section{a) Runge-Kutta method of $4^{th}$ order and Adams PC}
\subsection{Problem}
We are given following equations:
\[ \frac{dx_1}{dt} = x_2 + x_1(0.5 - x_1^2 - x_2^2) \]
\[ \frac{dx_2}{dt} = -x_1 + x_2(0.5 - x_1^2 - x_2^2) \]
And we have to determine the trajectory of the motion on interval $[0, 15]$ with following initial conditions:
$ x_1(0) = 8; x_2(0) = 9 $
In this section we will use Runge-Kutta method of $4^{th}$ order and Adams PC with different step-sizes until we find an optimal constant step size - when the decrease of the step size does not influence the solution significantly.
\subsection{Theoretical Introdution}
\section{b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}
\subsection{Problem}
We are given following equations:
\[ \frac{dx_1}{dt} = x_2 + x_1(0.5 - x_1^2 - x_2^2) \]
\[ \frac{dx_2}{dt} = -x_1 + x_2(0.5 - x_1^2 - x_2^2) \]
And we have to determine the trajectory of the motion on interval $[0, 15]$ with following initial conditions:
$ x_1(0) = 8; x_2(0) = 9 $
In this section we will use Runge-Kutta method of $4^{th}$ order with step size automatically adjusted by the algorithm, with error estimation made according to the step-doubling rule.
\subsection{Theoretical Introdution}
\begin{thebibliography}{9}
\bibitem{texbook}
Piotr Tatjewski (2014) \emph{Numerical Methods}, Oficyna Wydawnicza Politechniki Warszawskiej
\end{thebibliography}
\end{document}

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\contentsline {chapter}{\numberline {1}Determine polynomial function fitting experimental data}{2}{chapter.1}%
\contentsline {section}{\numberline {1.1}Problem}{2}{section.1.1}%
\contentsline {section}{\numberline {1.2}Theoretical introduction}{2}{section.1.2}%
\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{3}{chapter.2}%
\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{3}{section.2.1}%
\contentsline {subsection}{\numberline {2.1.1}Problem}{3}{subsection.2.1.1}%
\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{4}{subsection.2.1.2}%
\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{4}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}Problem}{4}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{4}{subsection.2.2.2}%