latex/assignment_1.tex

@@ -86,6 +86,7 @@
     
 \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 \\
@@ -125,40 +126,7 @@ Using the values that we found for $c_1$ and $c_2$, we get
 
 \section*{Problem 3}
 
-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*}
+% Show how it's derived and where we found the derivation.
 
 \section*{Problem 4}