Engineering Analysis
(650201)
lec(1)-Introduction
Instructor: Eng. Nada
Email: [email protected]
Philadelphia University Faculty of Engineering
Communication and Electronics Engineering
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Course Outline
Course Title: Engineering Analysis (1) (650201)
Prerequisite: Engineering Mathematics (210206)
Text Book : Advanced Engineering Mathematics By: Erwin Kreyszig 8th edition.1999.
Credit Hours :3
Level :3rd year
Providing Dept. :Communications & Electronics Engineering
Instructor :Nada N Khatib
Office No: 730
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Course Contents
Basic Concepts & Ideas (ch.1)
First Order Differential Equations (ch.1)
Second & Higher Order Differential Equations
(ch. 2)
Laplace transform (ch. 5)
Power Series Method (ch. 4)
Introduction to Partial Differential Equations
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Text Book and References
Advanced Engineering Mathematics By: Erwin Kreyszig
8th edition.1999
Boyce, William E., DiPrima, Richard C., Elementary Differential Equations, Fifth Edition, Wiley, New York, 1992.
Rabenstein, Albert L., Elementary Differential Equations with Linear Algebra, Third Edition, Academic Press, New York, 1982.
Krusemeyer, Mark, Differential Equations, Macmillan Publishing Co., New York, 1994.
Elementary Differential Equations and Boundary Value Problems by W.E. Boyce R.C. Diprima. New York, NY, U.S.A. : John Wiley & Sons, Incorporated, 1986. “
: المعادالت التفاضلية" الدكتور عبد الرحمن القواسمي و المهندسة ندى نبيل الخطيب الطبعة االولى، طبع بدعم من جامعة فيالدلفيا، مطبعة الخط العربي" تطبيقاتها الهندسية
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Course Grading First Exam → (20%)
Second Exam→ (20%)
Quizzes and Homework → (20%) Three homework assignments and 3 quizzes
There is no late homework policy
The grade of each Homework assignment is 8
The grades assigned to Quizzes will be elements of {0, 1, 2, 3, 4}
Missed quizzes cannot be made-up
Final Exam → (40%)
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What is the derivative?
Definition 1
The rate of change of a function
Definition 2
The slope of the tangent line to the graph of a function
Definition 3
The best linear approximation of a function
All of these answers are correct
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What is the Differential Equation?
Differential Equation (DE):is an
equation containing the derivatives or
differentials of one or more dependent
variables, with respect to one or more
independent variables.
DEs are used to describe phenomena that
are in movement or change with time
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What is the Differential Equation?
The DE may contains also the unknown
function itself , as well as the given
function and constants
ttxxt cos32
txdt
dx
dt
xd32943
2
2
cos
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Mathematical Model
DE are of fundamental importance in
engineering mathematics because
many physical laws & relations appear
mathematically in the form of such
equations
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Mathematical Model
DEs describe (almost) everything in the world:
physics
chemistry
engineering
ecology
economy
weather
…
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Modeling
Modeling is the process of describing
a physical situation with a DE
It is the derivation of DEs from physical
or other problems
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Population Models
How does one predict the growth of a
population?
Human populations
Population of bacteria
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Population Models/Human populations
Growth rate is proportional to the
population present
Assume that people die only of natural
causes
Death rate is also proportional to
the population present
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Population Models/Human population
if P(t) represents a population in a
given region at any time t then
= Rate at which
P(t) enters the
region
-
Rate at which
P(t) exits the
region
kppkkdt
dpp )( 21
Rate of
change of
P(t)
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Population Models/Population of bacteria
Reproduce by simple cell division
Parent cell does not die, but
become two new cells
Growth rate is proportional to the
population present pk
dt
dpp 1
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Heating and cooling of Building
How long does it take to change the
building temperature substantially?
How does the building temperature
vary during spring and fall when there
is no furnace heating or air condition?
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Heating and cooling of Building
How does the building temperature
vary in summer when there is air
condition?
How does the building temperature
vary in winter when there is furnace
heating?
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Newton’s law of cooling
Newton’s law of cooling, states that
the temperature of a hot or a cold
object decreases or increases at a rate
proportional to the difference of the
temperature of the object and that of its
surrounding.
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Temperature Problems
T= the temperature of a body at any
time (inside temperature)
)( sTTkdt
dT
Ts= the temperature of the surrounding
medium (outside temperature)
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Temperature Problems
H= additional heating rate
)()()( tUtHTTkdt
dTs
U= furnace heating (+ve)
U= air condition cooling (-ve)
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Pendulums
Determine the
displacement angle,
measured from the
vertical, as a function
of time in pendulums
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Pendulums
The swinging bob in a
grandfather’s clock &
A child’s swing
Are examples of
pendulums
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Mixing problems
The mixing of a
two fluids gives
rise to a DE
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25
Newton’s 2nd law of motion
an object of mass (m) is moving with
acceleration (a) that directly proportional
to the total force (F) acting on it and
inversely proportional to its mass
Fam
atv )( ath )(
we can rewrite the acceleration, a , in one of two ways
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Newton’s 2nd law of motion
Newton’s Second Law can now be written
as a differential equation in terms of either
the velocity, v, or the position
mgtvm )(
mgthm )(
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Newton’s 2nd law of motion
To find the velocity of
the body at any time in
case of free-falling
objects, we need to
solve the DE
mgtvm )(
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Newton’s 2nd law of motion
To find the position of
the body relative to the
ground in case of free-
falling objects, we need
to solve the DE mgthm )(
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Newton’s 2nd law of motion
To find the velocity of the sky driver at
any time
Mass-spring system
The motion of a
basic mechanical
systems (a mass on
an elastic spring)
are governed by
differential
equations.
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Satellite
To determine the
distance r, you
should use Newton’s
2nd law and universal
law of gravitation
then solve DE.
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Suspended Wire
To determine the shape of a wire hanging under its own
weight, such as a wire between two telephone line
poles, we must solve nonlinear DE
Electric circuit
The electric circuits (LR, RC, LRC) models are
differential equations
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Elastic Beam
An important fourth
order Differential
equation governs
the bending of an
elastic beam such
as wooden or iron
girder in a building
or a bridge
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What is the Differential Equation?
We can write any DE in either
following forms:
explicit form
),...,,,,( )1()( nn yyyyxfy