HOMEWORK PROBLEMS 9
PHYS 516, Fall 2015
Green’s Functions and Integral Equations
Due date: Friday, 12/4/15 — 5.00pm
1. Arfken & Weber Problem 10.5.8. Construct the Green’s function for
Bessel’s inhomogeneous equation
x2
d2
dy
+x
+ (k 2 x2 − 1) y = f (x) ,
dx
dx2
(1)
subject to the boundary conditions y(0) = 0 and y(1) = 0 (i.e., f (0) =
0 = f (1) ). Hint: read Section 11.2 of Arfken & Weber to determine how to
normalize the eigenfunctions upon which the Green’s function is built.
[30 points credit]
2. Arfken & Weber Problem 10.5.11. Construct the one-dimensional Green’s
function for the modified Helmholtz equation
d2 ψ
− k 2 ψ(x) = f (x) .
dx2
(2)
For the boundary conditions, assume that the Green’s function must vanish
for |x| → ∞ .
[20 points credit]
1
3. Arfken & Weber Problem 16.2.6. Use the Fourier transform technique to
solve the Fredholm integral equation
2
e−x =
Z ∞
n
o
exp −(x − t)2 φ(t) dt
(3)
−∞
for φ(t) .
[20 points credit]
4. Arfken & Weber Problem 16.3.9. Solve the Volterra equation
2
φ(x) = 1 + λ
Z x
(x − t) φ(t) dt
(4)
0
by each of the following methods: (i) reduction to an ODE, finding the correct
boundary conditions, (ii) employing a Neumann series, and (iii) using Laplace
transforms.
[30 points credit]
2