diff --git a/latex/assignment_1.pdf b/latex/assignment_1.pdf index 80e2ee3..c29ca0d 100644 Binary files a/latex/assignment_1.pdf and b/latex/assignment_1.pdf differ diff --git a/latex/assignment_1.tex b/latex/assignment_1.tex index 211f82a..87de2c0 100644 --- a/latex/assignment_1.tex +++ b/latex/assignment_1.tex @@ -84,84 +84,22 @@ \textit{https://github.uio.no/FYS3150-G2-2023/Project-1} -\section*{Problem 1} +\input{problems/problem1} -% 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*} +\input{problems/problem2} -Using the boundary conditions, we can find $c_1$ and $c_2$ as shown below: +\input{problems/problem3} -\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*} +\input{problems/problem4} -\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*} +\input{problems/problem5} -Using the values that we found for $c_1$ and $c_2$, we get +\input{problems/problem6} -\begin{align*} - u(x) &= -e^{-10x} + (e^{-10} - 1) x + 1 \\ - &= 1 - (1 - e^{-10}) - e^{-10x} -\end{align*} +\input{problems/problem7} -\section*{Problem 2} +\input{problems/problem8} -% 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)} - -\subsection*{b)} - -\section*{Problem 6} - -\subsection*{a)} - -% Use Gaussian elimination, and then use backwards substitution to solve the equation - -\subsection*{b)} - -% Figure it out - -\section*{Problem 7} - -% Link to relevant files on gh and possibly add some comments - -\section*{Problem 8} - -%link to relvant files and show plots - -\section*{Problem 9} - -% Show the algorithm, then calculate FLOPs, then link to relevant files - -\section*{Problem 10} - -% Time and show result, and link to relevant files +\input{problems/problem9} \end{document} diff --git a/latex/problems/problem1.tex b/latex/problems/problem1.tex new file mode 100644 index 0000000..779a58f --- /dev/null +++ b/latex/problems/problem1.tex @@ -0,0 +1,35 @@ +\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*} diff --git a/latex/problems/problem10.tex b/latex/problems/problem10.tex new file mode 100644 index 0000000..72de7f2 --- /dev/null +++ b/latex/problems/problem10.tex @@ -0,0 +1,3 @@ +\section*{Problem 10} + +% Time and show result, and link to relevant files diff --git a/latex/problems/problem2.tex b/latex/problems/problem2.tex new file mode 100644 index 0000000..b90aa58 --- /dev/null +++ b/latex/problems/problem2.tex @@ -0,0 +1,3 @@ +\section*{Problem 2} + +% Write which .cpp/.hpp/.py (using a link?) files are relevant for this and show the plot generated. diff --git a/latex/problems/problem3.tex b/latex/problems/problem3.tex new file mode 100644 index 0000000..03fc02f --- /dev/null +++ b/latex/problems/problem3.tex @@ -0,0 +1,4 @@ + +\section*{Problem 3} + +% Show how it's derived and where we found the derivation. diff --git a/latex/problems/problem4.tex b/latex/problems/problem4.tex new file mode 100644 index 0000000..2f690f3 --- /dev/null +++ b/latex/problems/problem4.tex @@ -0,0 +1,3 @@ +\section*{Problem 4} + +% Show that each iteration of the discretized version naturally creates a matrix equation. diff --git a/latex/problems/problem5.tex b/latex/problems/problem5.tex new file mode 100644 index 0000000..5d32e6f --- /dev/null +++ b/latex/problems/problem5.tex @@ -0,0 +1,6 @@ + +\section*{Problem 5} + +\subsection*{a)} + +\subsection*{b)} diff --git a/latex/problems/problem6.tex b/latex/problems/problem6.tex new file mode 100644 index 0000000..d3798e5 --- /dev/null +++ b/latex/problems/problem6.tex @@ -0,0 +1,9 @@ +\section*{Problem 6} + +\subsection*{a)} + +% Use Gaussian elimination, and then use backwards substitution to solve the equation + +\subsection*{b)} + +% Figure it out diff --git a/latex/problems/problem7.tex b/latex/problems/problem7.tex new file mode 100644 index 0000000..a39867c --- /dev/null +++ b/latex/problems/problem7.tex @@ -0,0 +1,3 @@ +\section*{Problem 7} + +% Link to relevant files on gh and possibly add some comments diff --git a/latex/problems/problem8.tex b/latex/problems/problem8.tex new file mode 100644 index 0000000..4d20743 --- /dev/null +++ b/latex/problems/problem8.tex @@ -0,0 +1,3 @@ +\section*{Problem 8} + +%link to relvant files and show plots diff --git a/latex/problems/problem9.tex b/latex/problems/problem9.tex new file mode 100644 index 0000000..e5bd862 --- /dev/null +++ b/latex/problems/problem9.tex @@ -0,0 +1,3 @@ +\section*{Problem 9} + +% Show the algorithm, then calculate FLOPs, then link to relevant files