Project-1/latex/assignment_1.tex

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\begin{document}

\title{Project 1}      % self-explanatory
\author{Cory Balaton \& Janita Willumsen}          % self-explanatory
\date{\today}                             % self-explanatory
\noaffiliation                            % ignore this, but keep it.


\maketitle 
    
\textit{https://github.uio.no/FYS3150-G2-2023/Project-1}
    
\section*{Problem 1}

% Do the double integral
\begin{align*}
    u(x) &= \int \int \frac{d^2 u}{dx^2} dx^2\\
         &= \int \int -100 e^{-10x} dx^2 \\
         &= \int \frac{-100 e^{-10x}}{-10} + c_1 dx \\
         &= \int 10 e^{-10x} + c_1 dx \\
         &= \frac{10 e^{-10x}}{-10} + c_1 x + c_2 \\
         &= -e^{-10x} + c_1 x + c_2
\end{align*}

Using the boundary conditions, we can find $c_1$ and $c_2$ as shown below:

\begin{align*}
    u(0) &= 0 \\
    -e^{-10 \cdot 0} + c_1 \cdot 0 + c_2 &= 0 \\
    -1 + c_2 &= 0 \\
    c_2 &= 1
\end{align*}

\begin{align*}
    u(1) &= 0 \\
    -e^{-10 \cdot 1} + c_1 \cdot 1 + c_2 &= 0 \\
    -e^{-10} + c_1 + c_2 &= 0 \\
    c_1 &= e^{-10} - c_2\\
    c_1 &= e^{-10} - 1\\
\end{align*}

Using the values that we found for $c_1$ and $c_2$, we get

\begin{align*}
    u(x) &= -e^{-10x} + (e^{-10} - 1) x + 1 \\
         &= 1 - (1 - e^{-10}) - e^{-10x}
\end{align*}

\section*{Problem 2}

% Write which .cpp/.hpp/.py (using a link?) files are relevant for this and show the plot generated.

\section*{Problem 3}

% Show how it's derived and where we found the derivation.

\section*{Problem 4}

% Show that each iteration of the discretized version naturally creates a matrix equation.

\section*{Problem 5}

\subsection*{a)}

% Phrase it better
$n = m - 2$, since $\textbf{A}$ is used to solve for all of the points in 
between the end-points $0$ and $1$. For the complete solution, we need to add
$u(0)$ and $u(1)$.

\subsection*{b)}

When solving for $\vec{v}$, we find the approximate solutions for $u(x)$
that are in between the end-points, but not the end-points themselves.

\section*{Problem 6}

\subsection*{a)}

% Use Gaussian elimination, and then use backwards substitution to solve the equation

\subsection*{b)}

% Figure it out

% Linelevant files on gh and possibly add some comments

\section*{Problem 8}

%link to relvant files and show plots

\section*{Problem 9}

% Shon*{Proalgorithm, then calculate FLOPs, then link to relevant files

\section*{Problem 10}

% Time and show result, and link to relevant files

\end{document}