Chapter 23 ODE (Ordinary Differential Equation)

1
Chapter 23 ODE (Ordinary Differential Equation)

• Speaker Lung-Sheng Chien

Reference 1 Veerle Ledoux, Study of Special
Algorithms for solving Sturm-Liouville and
Schrodinger Equations.
2 ???, university physics,
lecture 5 3 Harris
Benson, university physics, chapter 15

2
Hooks Law (????) 1
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?????????????? (???),??????m??????????????????
??????? (??m)??????????????,????????????, ????????
?????????,??????????????????
????
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3
Hooks Law (????) 2
?????,??????x???
?????m?????????????????,???????????
???????m???????????,??????????????????????????????
,?????????? ( ??? ) ??,??????
??????????m???,??????????,??m???????????,?????????
???? (simple harmonic oscillator)?
?????
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?????
4
Hooks Law (????) 3
dimension analysis
We must match dimension of each term in the
equation
Period (??)
Definition
??????(angular frequency),??? rad/s
guess
or
We need two constraints to determine unknown
constant
1
Initial position
2
Initial velocity
where
5
Hooks Law (????) 4
Simple Harmonic Oscillation
is shifted by phase constant
is phase (??)
the argument
measured in radians (??)
Question 1 why do we guess
in equation
Question 2 where do two constant A, B (or C,D)
come from?
Question 3 is the solution unique?
6
ODE 1
First order linear ODE
1
is first derivative.
First order highest degree of differential
operation
2
, then operator is linear, say
Linear define operator
integration along curve
Observation1 when initial time is determined and
initial condition
is unique
is given, then solution
However when initial condition is specified,
then initial time is determined at once, hence we
say
is one-parameter family, with parameter
First order linear ODE system of dimension two
3
ODE system more than one equation
4
system of dimension two two equations
solution is
since system is de-couple
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ODE 2
has unique solution if two initial conditions
are specified
for square matrix A
we define
implies
let
, then
Observation 2 ODE system of dimension two needs
two initial conditions
, then
is unique solution of
Observation 3 if we write
we may guess that
for any square matrix A
is unique solution of
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ODE 3
Simple Harmonic Oscillation
where
since initial velocity would also affect
Intuition we CANNOT determine
by given initial position
It is well-known when you are in junior high
school, not only free fall experiment but sliding
car experiment.
Transformation between second order ODE and
first order ODE system of dimension two
Define velocity
, then combine Newton second Law
, we have
we need two initial condition to have unique
solution
From Symbolic toolbox in MATLAB, we can
diagonalize matrix
MATLAB code
where
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ODE 4
Formal deduction
with initial condition
Definition fundamental matrix
then solution can be expressed by fundamental
matrix and initial condition
From Symbolic toolbox in MATLAB, we can compute
fundamental matrix easily
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ODE 5
fundamental matrix
is composed of two fundamental solutions
is solution of
with initial condition
is solution of
with initial condition
with initial condition
is linear combination of fundamental solutions
solution of
The space of solutions of
is
The dimension of solution space is two,
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ODE 6
In order to achieve uniqueness, we need to
specify two integration constant
Integral equation
differential equation
Existence and uniqueness (Contraction mapping
principle)
Let
be continuous space equipped with norm
1
is complete under norm
2
by
define a mapping
then
Existence and uniqueness
is a contraction mapping if
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ODE 7
(ignore buoyancy ??)
recover force is opposite to displacement
resistive force is also opposite to displacement
Newton seconds Law
1
what is equivalent ODE system of
2
what is fundamental matrix of this ODE system,
use symbolic toolbox in MATALB
3
what is solution of
with initial condition
4
Can you use contraction mapping principle to
prove existence and uniqueness?
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ODE 7
general solution
We have two choices to determine unknown
constants A and B
1
initial condition
2
boundary condition
period
1
become discrete?
Why does angular frequency
What is physical meaning of discrete angular
frequency?
2
We have still a constant B not be determined,
why?
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Schrodinger equation 1
energy
is angular frequency
photon (??)
is wave number
momentum
http//en.wikipedia.org/wiki/Matter_wave
Louis de Broglie in 1924 in his PhD thesis claims
that matter (object) has the same relation as
photon
de Broglie wavelength
de Broglie frequency of the wave
matter wave (???) is described by wave length
and wave frequency
Fundamental of quantum mechanic
http//en.wikipedia.org/wiki/SchrC3B6dinger_equa
tion
Erwin Schrödinger in 1926 proposed a differential
equation (called Schrodinger equation) to
describe atomic systems.
total energy of a particle
matter wave
and
time-dependent Schrodinger equation
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Schrodinger equation 2
Question physical interpretation of matter wave
probability finding particle on interval
in time
total number of particles in time
Example plane wave
particle has the same probability found in any
position, not physical
time-dependent Schrodinger equation
remove t-dependence, replace
by
time-independent Schrodinger equation
(one-dimensional)
(three-dimensional)
Objective of time-independent Schrodinger
equation find a stationary solution satisfying
with proper boundary conditions
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Schrodinger equation 3
Example Hydrogen atom (???)
parameter
B.C.
http//hyperphysics.phy-astr.gsu.edu/hbase/hyde.ht
ml
discrete energy level
http//www.touchspin.com/chem/SWFs/pt2k61012.swf
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Schrodinger equation 4
Time-independent Schrodinger equation
From dimensional analysis, we extract
dimensionless quantities in this system
1
2
3
4
dimensionless form
Once characteristic length of the system is
determined, for example
Then characteristic energy is
, this is near
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Finite Difference Method 1
First we consider one-dimensional problem
(dimensionless form)
Finite Difference Method (?????) divide
interval
into N1 uniform segments
labeled as
We approximate ODE on finite points, say we want
to find a vector
satisfying
for
We dont need to ask equation on end points,
since equation only holds in interior
1
How to approximate second derivative
on grid points
2
How to achieve solvability though we know
solution indeed exists in continuous sense
3
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Finite Difference Method 2
Definition standard 3-point centered difference
formula
we have
for some
is called local truncation error (LTE) since it
is truncation from Taylors expansion
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Finite Difference Method 3
short-stencil
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Finite Difference Method 4
for
Example n 4
is real symmetric, then A is diagonalizable
since eigenvalue of A is real
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Finite Difference Method 5
model problem no potential ( V 0 )
FDM
Question can we find analytic formula for
eigen-pair in this simple model problem?
solution is
solution is
FDM
for
Conjecture we guess that eigenvector is
satisfies
(boundary condition)
Question How about
for
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Finite Difference Method 6
model problem no potential ( V 0 )
eigen-function
Finite Difference Method on model problem
for
eigen-pair
Question How accurate are numerical eigen-value?
Exact wave number
Numerical wave number
Taylor expansion
where
Question eigenvalue is second order accuracy,
, is this reasonable, why?
Question why does error of eigenvalue increase
as wave number k increases?
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Finite Difference Method 7
model problem no potential ( V 0 )
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Exercise 1 model problem (high order accuracy )
Finite Difference Method on model problem
for
eigen-pair
Question how can we improve accuracy of
eigenvalue of model problem?
Step 1 deduce 4-order centered finite difference
scheme for second order derivative
Step 2 can you transform continuous equation to
discrete equation?
for
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Exercise 2 singular potential
effective potential
finite difference with uniform mesh platform
MATLAB
bound state
There are only six bound state (energy E lt 0 )
Definition bound state means
Exercise plot first 10 eigen-function and
interpret why only first lowest six eigenvalue
corresponds to bound state