Chapter 25 Sturm-Liouville problem (II)

1
Chapter 25 Sturm-Liouville problem (II)

• Speaker Lung-Sheng Chien

Reference 1 Veerle Ledoux, Study of Special
Algorithms for solving Sturm-Liouville and
Schrodinger Equations.
2 ?????, chapter 8, lecture
note of Ordinary Differential equation

2
Prufer method
Sturm-Liouville Dirichlet eigenvalue problem
Scaled Prufer transformation
Simple Prufer transformation
scaling function
where
So far we have shown that Sturm-Liouville
Dirichlet problem has following properties
1
Eigenvalues are real and simple, ordered as
Eigen-functions are orthogonal in
with inner-product
2
Eigen-functions are real and twice differentiable

3
Moreover we have implemented (Scaled) Prufer
equation
with Forward Euler Method (not stable, but it
can be used so far)
3
Sturms Comparison 1
Theorem (Sturms first Comparison theorem) let
be eigen-pair of Sturm-Liouville problem.
. Precisely speaking
suppose
, then
is more oscillatory than
Between any consecutive two zeros of
, there is at least one zero of
Theorem (Sturms second Comparison theorem) let
be solutions of Sturm-Liouville problem.
suppose
and
on
(1)
Between any consecutive two zeros of
, there is at least one zero of
(2)
ltproof of (1)gt
Simple Prufer
4
Sturms Comparison 2
First we consider
suppose
and
on
1
continuity of F, G
2
Suppose
, then
5
Sturms Comparison 3
and
1
2
Question How to deal with the case
Between any consecutive two zeros of
, there is at least one zero of
ltproof of (2)gt
Suppose
has consecutive zeros at
Without loss of generality, we assume
Moreover
, we may assume
and
1
2
Apply result of (1), set
, then
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Pitfall 1
Recall Sturm-Liouville Dirichlet eigenvalue
problem
1
Eigenvalues are real and simple, ordered as
Question How about asymptotic behavior of
eigenvalue, say
Eigen-functions are orthogonal in
with inner-product
2
Question are eigen-functions complete in
is eigen-pair of
Eigen-functions are real and twice differentiable

3
The more important question is
Question is operator
diagonalizable in
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Pitfall 2
Question How about asymptotic behavior of
eigenvalue, say
General Sturm-Liouville problem
Model problem
Sturms second Comparison theorem
(1)
Between any consecutive two zeros of
, there is at least one zero of
(2)
Hooks Law
solution
Zeros of solution is
with space
Exercise
Between any consecutive two zeros of
, there is at least one zero of
shows
8
Pitfall 3
Question are eigen-functions complete in
General Sturm-Liouville problem
Model problem
Question solution of modal problem is
Is such eigenspace
complete in
Consider space
with inner-product
1
is orthogonal in
is a closed subspace
2
is unique
decomposition
where
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Pitfall 4
Informally,
for some
to be determined
Formally speaking, when we write
, in mathematical sense we construct partial sum
such that
in L2 sense.
in
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Pitfall 5
Exercise
is the solution of
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Pitfall 6
and
where
Theorem trigonometric basis is complete in
in L2 sense, where
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Pitfall 7
Exercise we have shown
where
We abbreviate f as
1
If function f is even, say
, then
2
If function f is odd, say
, then
Modal problem
has eigen-pair
From above exercise, for any
, we can do odd extension
then
. Hence
Question How about if we do even extension
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Pitfall 8
Question is operator
diagonalizable in
From Prufer transformation, we can show
and
1
Eigenvalues are real and simple, ordered as
Eigen-functions are orthogonal in
with inner-product
2
Define domain of operator L with Dirichlet
boundary condition as
Clearly we have
,but we can not say
is diagonalizable in
Finite dimensional matrix computation
infinite dimensional functional analysis
Jordan form
Question does such
exists?
Idea if we can show that
, then even such
exists,
, why?
Then operator L is diagonalizable in
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Scaled Prufer Transformation 1
Scaled Prufer transformation
Time-independent Schrodinger equation
where
Suppose we choose
Question function f is continuous but not
differentiable at x 1. How can we obtain
has jump discontinuity at
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Scaled Prufer Transformation 2
and
Observation
does not exist, we ignore it.
Then fundamental Theorem of Calculus also holds,
say
, fundamental Theorem of Calculus holds,
1
f is continuous
2
, fundamental Theorem of Calculus holds,
3
Question although fundamental theorem of
calculus holds for function f , but if
is given,
How can we find f(x) numerically and have
better accuracy?
Reason to discussion of fundamental theorem of
calculus
depends on S(x), accuracy of
is equivalent to accuracy of obtaining S(x)
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Numerical integration 1
Ignore odd power since it does not contribute to
integral
general form
Trapzoid rule (???)
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Numerical integration 2
Example given a partition
and grid function
We use Trapezoid rule to find
1
Exercise 1 let
2
Try number of grids 10, 20, 40, 80, 160,
compute
and measure maximum error
Plot error versus grid number, what is order of
accuracy ?
Exercise 2 let
3
1
If x 1 is in the grid partition, what is
order of accuracy
If x 1 is NOT in the grid partition, what is
order of accuracy
2
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Scaled Prufer Transformation 3
Question can we modify function f slightly such
that it is continuously differentiable
, say
and
where
is polynomial of degree 3,
are chosen such that
ltsolgt
is achieved by following 4 conditions
1
2
3
4
where
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Scaled Prufer Transformation 4
and
but
has jump discontinuity at
Exercise 3 try to construct
where
is polynomial of degree 5
1
use Symbolic toolbox to determine coefficients
2
plot
3
use Trapezoid method to compute
,what is order of accuracy ?
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Review Finite Difference Method
Model problem
for
FDM
eigen-pair
solution is
Question why does error of eigenvalue increase
as wave number k increases?
Substitute
Exercise 4 find analytic solution of
where
Then use FDM to solve
What is order of accuracy? measure