Homework assignment #10
(due Friday, December 1)
Problem 1. Show that the functions h(x) = 1
2e−|x|/ ( > 0) form a delta family.
Problem 2. Let g(x) = |x|for all x∈R. Find the second derivative g00 in the sense of
distributions.
Problem 3. Consider
d2u
dx2=f(x) with u(0) = 0 and du
dx(L) = 0.
(i) Solve by direct integration.
(ii) Determine G(x, x0) so that
u(x) = ZL
0
f(x0)G(x, x0)dx0.
Problem 4. Consider
d2G
dx2=δ(x−x0) with G(0, x0) = 0 and dG
dx (L, x0) = 0.
(i) Solve directly.
(ii) Check whether G(x, x0) = G(x0, x).
(iii) Compare with Problem 3.
Problem 5. (i) Solve
dG
dx +G=δ(x−x0) with G(0, x0) = 0.
(ii) Show that G(x, x0) is not symmetric.
(iii) Solve
u0+u=fwith u(0) = 0
for any function fon [0, L].