VL 11 PP Folien

1
Universidad de La Habana
Lectures 5 6 Difference Equations Kurt Helmes
22nd September  - 2nd October, 2008
2
Part 1 Introduction Part 2 First-Order
Difference Equations Part 3 First-Order
Linear Difference Equations
3
Introduction
4
Part 1.1
An Example
5
Dagobert- Example
6
Starting Point
Given
K0
initial capital ( in Euro )
p
interest rate ( in )
r
7
Objective
Find ....

• 1. The amount of capital after 1 year.

• 2. The amount of capital after 2 years.

• n. The amount of capital after n years.

8
Solution
After one year the amount of capital is
How much capital do we have after 2 years?
9
Solution
After one year the amount of capital is
After two years the amount of capital is
10
Solution
After one year the amount of capital is
After two years the amount of capital is
11
Solution
After n years the amount of capital is
12
Observation
special difference equation
recursion formula
13
Part 1.2
Difference Equations
14
Illustration
A difference equation is a special system of
equations, with

• (countably) infinite many equations,

• (countably) infinite many unknowns.

15
Hint
The solution of a difference equation is a
sequence (countably infinite many numbers).
16
How do we recognize a difference equation?
17
Definition Difference Equation
Explicit form
Implicit form
18
First-Order Difference Equations
19
Part 2.1
A Model for theHog Cycle
20
Hog Cycle (Example)
21
Starting Point
Given Hog-corn price ratio in Chicago in the
period 1901-1935
22
Starting Point
Stylized
23
Starting Point

• Find
• A (first) model, which explains /
  describes the cyclical fluctuations of the
  prices (ratio of prices).

24
Model (Part 1) Supply and Demand
The suppply of hogs
The demand of hogs
25
Model (Part 2) Supply and Price
Assumption
26
Model (Part 2)
Nature of the dependance
Assumption
The supply function is linear
27
Figure 1 Graphical representation of the
supply function
28
Model (Part 3) Demand and Price
Assumption
For the demand we assume If the hog price
increases, the demand will decrease, thus
29
Figure 2 Graphical representation of the
demand function
30
Model (Part 4) Equilibrium
Postulate
Supply equals demand at any time
31
Model (Part 4) Equilibrium
The equilibrium relation yields a defining
equation for the price function
32
Solution (Part 4) Equilibrium
Thus we obtain the following difference equation
33
Model (Part 4) Equilibrium
This difference equation is

• first-order
• linear
• inhomogeneous

34
Model
(Part 5) Analysis
solution formula
35
Deriving the Solution Formula
....
36
Figure 2
37
Model (Part 5) Analysis
Results
The equation / solution is stable.
The equation / solution is unstable.
38
Figure 3 Price development for
39
Figure 4 Price development for
40
Figure 5 Price development for
41
Summary
The given difference equation has a unique
solution it can be solved explicitly.
The price is the sum of a constant and a power
function.
42
CONCLUSION
We can model and analyze dynamic processes with
difference equations.
43
Part 2.2
Definitions und Concepts for First-Order
Difference Equations
44
Definition
A (general) first-order nonlinear difference
equation has the form
(F is defined for all values of the variables.)
45
Important Questions

• Does at least one solution exist?

• Is there a unique solution?
• How many solutions do exist?
• How does the solution change, if parameters
  of the system of equations are changed
  (sensitivity analysis)?

46
Important Questions

• Do explicit formulae for the solution exist?

• How do we calculate the solution?

• Does the system of equations has a special
  structure ?

e.g. a) linear or nonlinear,
b) one- or multidimensional ?
47
Remark
If the initial value of the solution (sequence)
of a difference equation is given, i.e.
then we call our problem an
initial value problem
related to a first-order difference equation.
48
Remark
The initial value problem of a first-order
difference equation has a unique solution.
49
Remark
50
Definition Invariant Points
invariant points.
F right-hand side.
51
Invariant Points
52
Invariant Points
53
(No Transcript)
54
Iteration rule
55
(No Transcript)
56
(No Transcript)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
Newtons-Method (Example)
63
Starting Point
Finding the roots of a nonlinear function
analytically is rarely possible. Therefore we
have to use numerical methods.
64
Starting Point
For differentiable functions a numerical
root-finding algorithm exists. It goes back to
Isaac Newton (1643 1727).
65
Goal
66
Idea
67
Idea
68
Idea
Determine the intersection of the tangent with
the x-axis.
69
Idea
70
Idea
Repeat this operation many times.
71
Figure 12 Schematic representation of
Newtons Method
72
Solution
73
Solution
74
Solution
By the same idea we compute x2, x3, ... as
(difference equation)
75
Solution
This is a (nonlinear) first-order difference
equation, and
76
Numerical Example
Consider the problem of finding the root of
.
The difference equation according to Newtons
Method is
77
First-Order Linear Difference Equations
78
Part 3.1
First-Order Linear Difference Equations
with a Constant a-Term
79
Definition
Time-dependent, inhomogeneous linear difference
equations of first order with constant a-term
take the form
Equation
80
Definition
Time-dependent, inhomogeneous linear difference
equations of first order with constant a-term
take the form
Equation
Iteration Rule
81
Lösungsformel
Time-dependent, inhomogeneous linear difference
equations of first order with constant a-term
have the solution
Solution formula
82
Deriving the Solution Formula
....
83
Special Case
For first-order linear difference equations with
constant coefficients it holds
84
Example of an Exam Exercise
1
Solution
Backwards iteration yields
85
Example of an Exam Exercise
1
Solution
86
Example of an Exam Exercise
1
Solution
87
Example of an Exam Exercise
1
Intermediate Calculation
Expanding the equation
88
Example of an Exam Exercise
1
Solution
Expanding the equation
89
Example of an Exam Exercise
1
Solution
General condensation of the terms
90
Example of an Exam Exercise
2
91
Example of an Exam Exercise
2
92
Example of an Exam Exercise
2
The initial value problem can be solved either
directly by using the solution formula, i.e.
93
Example of an Exam Exercise
2
94
Example of an Exam Exercise
2
or by forward iteration
95
Example of an Exam Exercise
2
Continuing with forward iteration
and in general
96
Example of an Exam Exercise
97
Example of an Exam Exercise
and it holds
98
Dagobert- Example
(with deposits and payments)
99
Starting Point
interest factor
100
Find Formula for the account balance
Solution formula
101
Formula for the account balance
The discounted capital flow is
102
Summary
The discounted capital stock at time t equals the
capital stock at time t0 plus the sum of the
discounted deposits minus the sum of the
discounted payments up to time t .
103
Part 3.2
First-Order Linear Difference Equations with
Variable Coefficients
104
Definition
First-order linear difference equations with
variable coefficients take the form
105
Solution formula
The solution of first-order linear difference
equations with variable coefficients is given by
106
Dagobert- Example
with variable interest rate andproportional
deposits and payments
107
Starting Point
Consider a capital model with time-dependent
interest factor
interest factor
deposits
payments
108
Starting Point
Special Case Capital model with proportional
deposits and payments
109
Proportional In- and Outpayments
110
Numerical Example
Capital stock
1000 Euro
111
Numerical Example
Capital stock
1000 Euro
Interest factor
112
Numerical Example
Capital stock
1000 Euro
Interest factor
Rate of deposits
113
Numerical Example
Capital stock
1000 Euro
Interest factor
Rate of deposits
Rate of payments
114
Numerical Example
115
Numerical Example
116
Numerical Example
117
Part 3.3
Stability of First-Order Linear Difference
Equations
118
Definition Stability
A first-order difference equation is called
stable, if
the solution of the homogeneous equation
converges for any initial value to zero.
cf.. 1) unstable 2) chaotic
119
Stability Conditions
120
Stability Conditions
Remark 1
If
holds for one time point s,
121
Stability Conditions
Remark 2
Stability comes along in different forms
Example
1
122
Figure 13 Schematic representaion of
stability - Case A
123
Stability Conditions
Remark 2
Stability comes along in different forms
Example
2
124
Figure 14 Schematic representaion of
stability - Case B
125
Stability Conditions
Remark 3
126
Figure 15 Schematic representaion of
stability - Case C