Math 121A: Homework 8-
1. Consider the differential equation
Y'' = f(x)
on the range -1 ≤ x ≤ 1 subject to y(-1) = 0 and y(1) = 0.
(a) Calculate a Green function solution of the form
y(x) = -1∫1 G(x, x')f(x')dx'.
(b) Plot G(x, x') on the range -1 ≤ x ≤ 1 for the cases of x' = -2/3, -1/3, 0, 1/3, 2/3. Explain in words what the plotted functions represent.
(c) Explicitly calculate the solution y(x) for the case when
![2365_Figure.png](https://secure.tutorsglobe.com/CMSImages/2365_Figure.png)
Plot the solution, and explain how its form is related to the plots in part (b).
(d) Explicitly calculate the solution for the case of f(x) = x, plot the solution, and check that the solution satisfies the differential equation and the boundary conditions.
2. Define
f(x) = δ(x - 2) + δ(x) + δ(x + 2)
and
![1323_Figure1.png](https://secure.tutorsglobe.com/CMSImages/1323_Figure1.png)
on -∞ < x < ∞. In addition, define h(x) = f(2x). Calculate f ∗ g and h ∗ g, and plot them.
3. Consider the function
![34_Figure2.png](https://secure.tutorsglobe.com/CMSImages/34_Figure2.png)
Define fk+1 = fk ∗ f0. Explicitly calculate the functions f1, f2, and f3 and plot them.
4. Consider the function
![1611_Figure3.png](https://secure.tutorsglobe.com/CMSImages/1611_Figure3.png)
for -π < x < π, where 0 ≤ a < π. Plot the function and determine whether it is odd, even, or neither. Calculate the Fourier series of f and plot it for the cases of a = 1 and a = 2 using the first ten non-zero terms.