Remove comment for problem 3

latex/assignment_1.tex

@@ -86,7 +86,6 @@
     
 \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 \\
@@ -126,7 +125,40 @@ Using the values that we found for $c_1$ and $c_2$, we get
 
 \section*{Problem 3}
 
-% Show how it's derived and where we found the derivation.
+To derive the discretized version of the Poisson equation, we first need
+the taylor expansion for $u(x)$ around $x + h$ and $x - h$.
+
+\begin{align*}
+    u(x+h) &= u(x) + u'(x) h + \frac{1}{2} u''(x) h^2 + \frac{1}{6} u'''(x) h^3 + \mathcal{O}(h^4)
+\end{align*}
+
+\begin{align*}
+    u(x-h) &= u(x) - u'(x) h + \frac{1}{2} u''(x) h^2 - \frac{1}{6} u'''(x) h^3 + \mathcal{O}(h^4)
+\end{align*}
+
+If we add the equations above, we get this new equation:
+
+\begin{align*}
+    u(x+h) + u(x-h) &= 2 u(x) + u''(x) h^2 + \mathcal{O}(h^4) \\
+    u(x+h) - 2 u(x) + u(x-h) + \mathcal{O}(h^4) &= u''(x) h^2 \\
+    u''(x) &= \frac{u(x+h) - 2 u(x) + u(x-h)}{h^2} + \mathcal{O}(h^2) \\
+    u_i''(x) &= \frac{u_{i+1} - 2 u_i + u_{i-1}}{h^2} + \mathcal{O}(h^2) \\
+\end{align*}
+
+We can then replace $\frac{d^2u}{dx^2}$ with the RHS (right-hand side) of the equation:
+
+\begin{align*}
+    - \frac{d^2u}{dx^2} &= 100 e^{-10x} \\
+    \frac{ - u_{i+1} + 2 u_i - u_{i-1}}{h^2} + \mathcal{O}(h^2) &= 100 e^{-10x} \\
+\end{align*}
+
+And lastly, we leave out $\mathcal{O}(h^2)$ and change $u_i$ to $v_i$ to 
+differentiate between the exact solution and the approximate solution, 
+and get the discretized version of the equation:
+
+\begin{align*}
+align*    \frac{ - u_{i+1} + 2 u_i - u_{i-1}}{h^2} &= 100 e^{-10x} \\
+\end{align*}
 
 \section*{Problem 4}
 
@@ -136,16 +168,8 @@ Using the values that we found for $c_1$ and $c_2$, we get
 
 \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)}
@@ -156,7 +180,9 @@ that are in between the end-points, but not the end-points themselves.
 
 % Figure it out
 
-% Linelevant files on gh and possibly add some comments
+\section*{Problem 7}
+
+% Link to relevant files on gh and possibly add some comments
 
 \section*{Problem 8}
 
@@ -164,7 +190,7 @@ that are in between the end-points, but not the end-points themselves.
 
 \section*{Problem 9}
 
-% Shon*{Proalgorithm, then calculate FLOPs, then link to relevant files
+% Show the algorithm, then calculate FLOPs, then link to relevant files
 
 \section*{Problem 10}