Math 054 partial differential equations - hw assignment 9

Question

Math 054 Partial Differential Equations - HW Assignment 9

1. Is it possible to find two functions in L²(-∞, ∞) neither one of which is identically zero but whose convolution product is zero?

2. Consider a function f(x) in L²(-∞, ∞) with Fourier transform F(α) and let

g(x) = F^-1[F(α)I_A(α)], for some A > 0

In what way, if any, is g(x) related to f(x).

3. Solve the following by taking the Fourier Transform with respect to x.

u_t = c²u_xx, t > 0

u(x, 0) = f(x), -∞ < x < ∞

4. Let k denote a positive integer and let

Compute the Fourier transform F_k(α) for k = 1, 2, ... and sketch the graphs of f_k(x) and F_k(α) for k = 1, 2, 3. Relate the changes in f_k(x) as k varies to the corresponding changes in F_k(α).

5. Solve the following by taking the Fourier Transform with respect to x.

u_tt(x, t) = a²u_xx, t > 0

u(x, 0) = 0, u_t(x, 0) = sin πx, -∞ < x < ∞