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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.

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Chapter 25 sturm liouville problem ii l.jpg

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] 王信華教授, chapter 8, lecture note of Ordinary Differential equation


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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)


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Sturm’s Comparison [1]

Theorem (Sturm’s 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 (Sturm’s 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)

<proof of (1)>

Simple Prufer


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Sturm’s Comparison [2]

First we consider

suppose

and

on

1

continuity of F, G

2

Suppose

, then


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Sturm’s 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

<proof of (2)>

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

Sturm’s second Comparison theorem

(1)

Between any consecutive two zeros of

, there is at least one zero of

(2)

Hook’s Law:

solution:

Zeros of solution is

with space

Exercise:

Between any consecutive two zeros of

, there is at least one zero of

shows


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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]

where

and

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 (梯形法)


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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

<sol>

is achieved by following 4 conditions

1

2

3

4

where


Slide19 l.jpg

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


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