Math 121a homework 8 for -pi lt x lt pi where 0 le a lt pi

Question

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

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

on -∞ < x < ∞. In addition, define h(x) = f(2x). Calculate f ∗ g and h ∗ g, and plot them.

3. Consider the function

Define f_k+1 = f_k ∗ f₀. Explicitly calculate the functions f1, f2, and f3 and plot them.

4. Consider the function

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.