Scientific Computing - Quiz 1 - Question 7

Problem:

Reconsider the fundamental theorem of calculus as expressed in (1.1.16 found in the lecture note packet). But now, integrate from $t-\Delta t$ to $t+\Delta t$. Using (1.1.19), what is the new version of (1.1.20):

Solution:

We start with $y(t + \Delta t) - y(t - \Delta t) = \int\limits_{-t - \Delta t}^{t + \Delta t}{f(t, y)dt}$.

Next we approximate the integral by constant and put terms around, we get

$y_{n+1} = y_{n-1} + 2\Delta t f(t, y_n)$.