## Homework Assignment #4

(due: May 6 in class)

### Theory Part

Problem 1 (20%):
Starting from equation 5.30 in the textbook, derive the expression for the fourth order Adams-Bashforth technique (equation 5.24).

Problem 2 (30%):

Consider the following initial value problem:

dy/dt = -20*y + 20*t^2 + 2*t
0 <= t <= 1
y(0) = 1/3

Assume you want to solve this problem with the Trapezoidal Method. Show that applying Newton's Method leads to the equation

wj+1 = wj *(1-10h)/(1+10h) + h/(1+10h) * [10*(tj)2 + 10*(tj+1)2 + tj + tj+1]

### Numerics Part

Problem 3 (50%):

Consider the initial value problem in problem 2.

The exact solution to this problem is y(t) = t^2 + 1/3*exp(-20*t)

Solve this problem numerically using
a) the Trapezoidal Method (cf. problem 2)
b) Euler's Method
c) the standard Runge-Kutta Method of order 4
Use the timesteps h=0.2, h=0.12, h=0.1, and h=0.02 for all methods. Compare the results; in particular, compare which methods become unstable, and make a statement about the regions of absolute stabilty based on the values of h that were chosen.

Hint: You should "recycle" your old scripts for the Euler Method and RK4.

You should turn in the following:

- Plots that show the different solutions (and the exact result) as a function of t for all h;