Numerical Analysis Assignment #8 — Numerical Differentiation  
Question #1
Consider an RC circuit for which the governing equation is: $$ R \frac{dq}{dt}+\frac{q}{C}=0 $$ Show that the exact solution is: $$ q(t)=q_0 \exp\left(-\frac{t}{RC}\right) $$
10.12.16
Question #2
Consider the modified Euler method of numerical differentiation. Do the following:
(a)  Assuming that $$ \Delta t f\left(t_{n+1/2},~\phi_n+\frac{\Delta t}{2}f (t_n,~\phi_n) + O(\Delta t^2) \right) = \Delta t f\left(t_{n+1/2},~\phi_n+\frac{\Delta t}{2}f (t_n,~\phi_n) \right) + O(\Delta t^3) $$ Show that the modified Euler method is second-order accurate.
(b)  For $f(t,\phi)=\frac{1}{\phi}$, show that the assumption in (a) is valid.
Question #3
Using the forward Euler method, solve $q$ at $t=1$ for the RC circuit equation $$ R \frac{dq}{dt}+\frac{q}{C}=0 $$ with $RC=3$ and with the initial condition being $q=2$ at time $t=0$ and with $\Delta t=0.1$. Do so in two different ways:
(a)  By hand
(b)  With a C code that starts as follows:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <assert.h>

double f(
Question #4
Using a second-order Runge-Kutta method, solve $q$ at $t=1$ for the RC circuit equation $$ R \frac{dq}{dt}+\frac{q}{C}=0 $$ with $RC=3$, with the initial condition being $q_0=2$, with $\Delta t=0.1$, and with the constraint $a=0.5$. Do so in two different ways:
(a)  By hand
(b)  With a C code that starts as follows:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <assert.h>

double f(
Recall
The 2nd order Runge-Kutta scheme can be expressed as $$ k_1 = \Delta t f(t_n,~ \phi_n) \\ k_2 = \Delta t f(t_n+\alpha \Delta t, ~ \phi_n + \beta k_1)\\ \phi_{n+1}=\phi_n + a k_1 + b k_2 $$ with the constraints $$ a+b=1\\ b\alpha=\frac{1}{2}\\ b\beta=\frac{1}{2} $$
Answers
1.  
2.  
3.  
4.  
08.29.17
Make PDF
$\pi$