this is for holding javascript data
Chris Spencer edited Problem 1.tex
about 10 years ago
Commit id: da8670bacefdeb8760e0014afd655358ab459066
deletions | additions
diff --git a/Problem 1.tex b/Problem 1.tex
index 7e3c6df..e69de29 100644
--- a/Problem 1.tex
+++ b/Problem 1.tex
...
\section{Problem 1}
Define the function we will be using for the forward differencing as \[u=ge^{ikx_j}\] and the forward differencing equation as \[\Delta_x^{'}f_j=\frac{f_{j+1}-f_j}{\Delta}\] where are function $f$ will be our functuon $u$.\\
Putting this is we get \[\Delta_x^{'}u=\frac{ge^{ikx_{j+1}}-ge^{ikx_j}}{\Delta}\] define $x_{j+1}=x_j+\Delta$ then the above equation becomes \[\Delta_x^{'}u=\frac{ge^{ikx_j+\Delta}-ge^{ikx_j}}{\Delta}\]\\
Factoring we get \[\Delta_x^{'}u=\frac{ge^{ikx_j}(e^{ik\Delta}-1)}{\Delta}\] and taylor expanding we get
\[\approx\frac{ge^{ikx_j}(1+ik\Delta+\frac{(ik\Delta)^2}{2}-1)}{\Delta}\]
