Math 228A
Homework 1
Due Tuesday, 10/12/10
1. Let
L
be the linear operator
Lu
=
u
xx
,
u
x
(0) =
u
x
(1) = 0.
(a) Find the eigenfunctions and corresponding eigenvalues of
L
.
(b) Show that the eigenfunctions are orthogonal in the
L
2
[0
,
1] inner product:
(
u, v
)
=
integraldisplay
1
0
uv dx.
(c) It can be shown that the eigenfunctions,
φ
j
(
x
), form a complete set in
L
2
[0
,
1].
This
means that for any
f
∈
L
2
[0
,
1],
f
(
x
) =
∑
j
α
j
φ
j
(
x
). Express the solution to
u
xx
=
f,
u
x
(0) =
u
x
(1) = 0
,
(1)
as a series solution of the eigenfunctions.
(d) Note that equation (1) does not have a solution for all
f
.
Express the condition for
existence of a solution in terms of the eigenfunctions of
L
.
2. Define the functional
F
:
X
→ ℜ
by
F
(
u
) =
integraldisplay
1
0
1
2
(
u
x
)
2
+
fu dx,
where
X
is the space of real valued functions on [0
,
1] that have at least one continuous
derivative and are zero at
x
= 0 and
x
= 1.
The Frechet derivative of
F
at a point
u
is
defined to be the linear operator
F
′
(
u
) for which
F
(
u
+
v
) =
F
(
u
) +
F
′
(
u
)
v
+
R
(
v
)
,
where
lim

v
→
0

R
(
v
)


v

= 0
.
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 Fall '11
 Wiley
 Math, Differential Equations, Numerical Analysis, Equations, Derivative, uxx, finite difference formula

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