engg2013 unit 1
DESCRIPTION
orde differential equation for engineersTRANSCRIPT
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ENGG2013Unit 1 Overview
Jan, 2011.
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Course info• Textbook: “Advanced Engineering Mathematics” 9th edition, by
Erwin Kreyszig.• Lecturer: Kenneth Shum
– Office: SHB 736 – Ext: 8478– Office hour: Mon, Tue 2:00~3:00
• Tutor: Li Huadong, Lou Wei• Grading:
– Bi-Weekly homework (12%)– Midterm (38%)– Final Exam (50%)
• Before midterm: Linear algebra• After midterm: Differential equations
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Erwin O. Kreyszig (6/1/1922~12/12/2008)
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Academic Honesty
• Attention is drawn to University policy and regulations on honesty in academic work, and to the disciplinary guidelines and procedures applicable to breaches of such policy and regulations. Details may be found at http://www.cuhk.edu.hk/policy/academichonesty/
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System of Linear Equations
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Two variables, two equations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
4
5
6
7
x
y
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System of Linear Equations
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Three variables, three equations
-2 -1.5 -1 -0.5 0 0.5
-2-1
01-8
-6
-4
-2
0
2
4
6
xy
z
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System of Linear Equations
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Multiple variables, multiple equations
How to solve?
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Determinant
• Area of parallelogram
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(a,b)
(c,d)
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3x3 Determinant• Volume of parallelepiped
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(a,b,c)
(d,e,f)
(g,h,i)
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Nutrition problem
• Find a combination of food A, B, C and D in order to satisfy the nutrition requirement exactly.
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Food A Food B Food C Food D Requirement
Protein 9 8 3 3 5
Carbohydrate 15 11 1 4 5
Vitamin A 0.02 0.003 0.01 0.006 0.01
Vitamin C 0.01 0.01 0.005 0.05 0.01
How to solve it using linear algebra?
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Electronic Circuit (Static)
• Find the current through each resistor
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System of linear equations
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Electronic Circuit (dynamic)
• Find the current through each resistor
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System of differential equations
inductor alternatingcurrent
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Spring-mass system
• Before t=0, the two springs and three masses are at rest on a frictionless surface.
• A horizontal force cos(wt) is applied to A for t>0.
• What is the motion of C?
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A B C
Second-order differential equation
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System Modeling
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Physical System
Mathematicaldescription
Physical Laws+
Simplifyingassumptions
Reality
Theory
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How to model a typhoon?
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Lots of partial differential equations are required.
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Example: Simple Pendulum
• L = length of rod• m = mass of the bob• = angle• g = gravitational
constant
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L
m
mg
mg sin
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Example: Simple Pendulum
• arc length = s = L• velocity = v = L d/dt• acceleration = a
= L d2/dt2
• Apply Newton’s law F=ma to the tangential axis:
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L
m
mg
mg sin
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What are the assumptions?
• The bob is a point mass• Mass of the rod is zero• The rod does not stretch• No air friction• The motion occurs in a 2-D plane*• Atmosphere pressure is neglected
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Foucault pendulum @ wiki
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Further simplification
• Small-angle assumption– When is small, (in radian) is very close to sin .
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simplifies to
Solutions are elliptic functions.
Solutions are sinusoidal functions.
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Modeling the pendulum
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modeling
Continuous-time dynamical system
or
for small angle
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Discrete-time dynamical system
• Compound interest– r = interest rate per month– p(t) = money in your account– t = 0,1,2,3,4
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Time is discrete
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Discrete-time dynamical system• Logistic population growth
– n(t) = population in the t-th year– t = 0,1,2,3,4
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Increase in population
Proportionality constant
0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
n
n*(1
-n/K
)
An example for K=1Graph of n(1-n)
Slo
w g
row
th
fast
gro
wth
Slo
w g
row
th
nega
tive
grow
th
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Sample population growth
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0 5 10 15 200
0.2
0.4
0.6
0.8
1
t
n(t)
a=0.8, K=1
Monotonically increasing
Initialized at n(1) = 0.01
a=2, K=1Oscillating
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
1.2
1.4
t
n(t)
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Sample population growth
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0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
1.4
t
n(t)
a=2.8, K=1
Chaotic
Initialized at n(1) = 0.01
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Rough classification
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System
Static Dynamic
Continuous-time Discrete-time
Probabilistic systems are treated in ENGG2040
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Determinism• From wikipedia: “…if you knew all of
the variables and rules you could work out what will happen in the future.”
• There is nothing called randomness.• Even flipping a coin is deterministic.
– We cannot predict the result of coin flipping because we do not know the initial condition precisely.
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