Mathematics 421 Essay 3
Boundary Value Problems
Spring 2008
0.
Introduction
A
second order
differential equation has a general solution containing two
parameters.
Typically these parameters are the values of the solution and its first derivative at a single
point. Under suitable conditions, the theory predicts that such data leads to a unique solution. However,
some natural questions lead to the value of the function at
two different points
being specified. In such
questions, the function restricted to
the interval between those points
is the main object of interest, so
these questions are called
boundary value problems
. If the equation is linear and homogeneous, and the
given boundary values are zero, then a unique solution could only be the zero function. However,
there is
no uniqueness theorem for boundary value problems
. Indeed, certain homogeneous equations with zero
boundary values have nontrivial solutions.
1. The main example
A typical example is
d
2
y
dx
2
C
y
D
0
I
y.0/
D
0
I
y.L/
D
0
for fixed
L
and a parameter
. If
< 0
, write
D
˛
2
. Then, the general solution of the differential
equation is
y
D
ae
˛x
C
be
˛x
. The condition at
x
D
0
requires
b
D
a
, so the solution is a multiple of
sinh
x
. This function is strictly increasing, so the condition at
x
D
L
allows only the zero function. If
D
0
,
the solution is
y
D
a
C
bx
, the condition at
x
D
0
gives
a
D
0
, and again the solution is a multiple of the
increasing function
y
D
x
, and only
b
D
0
allows
y.L/
D
0
. If
> 0
, write
D
˛
2
. Then, the general
solution of the differential equation is
y
D
a
cos
˛x
C
b
sin
˛x
, and the condition at
x
D
0
gives
a
D
0
.
However, if
˛
D
n
=L
, giving
D
n
2
2
=L
2
, all multiples of sin
˛x
satisfy the condition at
x
D
L
.
The functions that appear in these solutions are exactly the
odd functions
on
L
x
L
appearing in
half range Fourier series
.
2. Eigenfunctions
There is something more general than Fourier series involved here. The
boundary condition at each endpoints is allowed to be the requirement that some fixed linear combination
of
y
and
y
0
be zero at that point. For example, the boundary value problem
d
2
y
dx
2
C
y
D
0
I
y.0/
D
0
I
y.L/
C
y
0
.L/
D
0
has a nontrivial solution of the form sin
˛x
when sin
˛L
C
˛
cos
˛L
D
0
. There are still infinitely many such
˛
, but they are characterized by
˛
D
tan
˛L
— an equation that has one root in each interval of the form
.k
1
=
2
/
=L; .k
C
1
=
2
/
=L
. As before,
D
˛
2
, but there is no simple expression for
˛
. It can be shown
by the method used in the first example that only these positive values of
allow nonzero solutions of the
boundary value problem.
In these problems, we are considering the effect of the
linear operator
d
2
=dx
2
on the
linear space
of functions defined on the interval
OE0; LŁ
satisfying certain
homogeneous conditions
at the
boundary
points
x
D
0
and
x
D
L
. We identified functions taken into
multiples of themselves
by the operator. In
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.
 Spring '08
 Bumby
 Derivative, Vector Space, Boundary value problem

Click to edit the document details