Physics 498/MMA
Handout 10
Oct 10th 2002
Mathematical Methods in Physics I
http://w3.physics.uiuc.edu/
∼
mstone5
Prof. M. Stone
305 Loomis Laboratory
University of Illinois
Here are some optional problems on integral equations.
They are taken
verbatim
from
Paul Goldbart’s homework sets.
1) Integral equations
:
a) Solve the inhomogeneous type II Fredholm integral equation
u
(
x
) =
e
x
+
λ
Z
1
0
xy u
(
y
)
dy .
b) Solve the homogeneous type II Fredholm integral equation
u
(
x
) =
λ
Z
π
0
sin(
x

y
)
u
(
y
)
dy .
c) Solve the inhomogeneous type II Fredholm integral equation
u
(
x
) =
x
+
λ
Z
1
0
y
(
x
+
y
)
u
(
y
)
dy
to second order in
λ
using
i) the LiouvilleNeumannBorn series; and
ii) the Fredholm series.
d) By di±erentiating, solve the integral equation:
u
(
x
) =
x
+
R
x
0
u
(
y
)
dy
.
e) Solve the integral equation:
u
(
x
) =
x
2
+
R
1
0
xy u
(
y
)
dy
.
f) Find the eigenfunction(s) and eigenvalue(s) of the integral equation
u
(
x
) =
λ
Z
1
0
e
x

y
u
(
y
)
dy .
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Stone
 Work, integral equation, Fredholm integral equation

Click to edit the document details