Policy/grading
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If the door is closed please do not enter to the class. Door will be opened again after ~10 min for you to come in. If you
arrive later, please do not disturb just wait for the next class.
Grading: laboratory work: %20, midterms: 22.5% (each),
final: 35%
computer skills: each students is assumed to be at least familiar with the pc, and mostly used engineering tools/software.
Octave will be used throughout the course.
Attendance: compulsory (80% show-up in attendance records), FA letter will be assigned to those who have less than
80%.
Textbook: by Steven C. Chapra and Raymond Canale: Numerical Methods for Engineers, McGraw-Hill.
Laboratory: Lab session will involve programming of OCTAVE code. The subject of the session will be related with the
theory covered in the classroom. We will try to assign each lab session with a clearly stated task. After completing the lab
session, students will have written a piece of computer code that performs the assigned task.
Lab performance of each student will be evaluated on a weekly basis. The students’ performance will be marked
depending on: the level of attention, the level of effort, and the level of success displayed by the student during the lab
session.
Those activities that result with an UNSATISFACTORY performance are WEB SURFING, CHATTING, or exchanging
EMAILS with other students. (If, the student continues these activities after a warning, he/she will be DISMISSED OFF
that lab session, will be MARKED OFF the attendance sheet.
The lab marks will be added up at the end of the semester, and will contribute to the students' overall cumulative as
indicated on the above table.
Up to four homework assignments will be assigned throughout the semester. Some assignments may receive a higher
credit relative to others depending on the level of complexity of the expected work.
It is strongly recommended to take the lab sessions seriously throughout the semester as they will also be crucial in
understanding the practical aspects of the subject matter and obtaining a good grade.
1
Part 1
Chapter 1
Mathematical Modeling,
Numerical Methods,
and Problem Solving
2
A Simple Mathematical Model
• A mathematical model can be broadly
defined as a formulation or equation that
expresses the essential features of a
physical system or process in mathematical
terms.
• Mathematical Models can be represented by
a functional relationship between dependent
variables, independent variables,
parameters, and forcing functions.
4

Model Function
independent
Dependent
forcing 
 f 
, parameters,

variable
variables
functions


• Dependent variable - a characteristic that usually
reflects the behavior or state of the system
• Independent variables - dimensions, such as time and
space, along which the system’s behavior is being
determined
• Parameters - constants reflective of the system’s
properties or composition
• Forcing functions - external influences acting upon the
system
5
A simple Mathematical Model
Problem of a falling object in air:
FD  cd v 2
Fg= Force of gravity
FD=Drag force
m=mass
v(t)= Velocity of the object
cd= Drag coefficient
m
Fg  mg
Newton’s 2nd law:
Fnet  Fg  FD  ma  m
mg  cd v 2  m
c
dv
 g  d v2
dt
m
dv
dt
dv
dt
(Equation of motion)
v  v(t )
An Ordinary Differential
Equation (ODE)
Need to solve for v(t)
6
Analytical Solution:
cd 2
dv
g v
dt
m
Initial condition: The object is initially at rest
(using Calculus)
v
v(t ) 
t0 ; v0
Dependent variable - velocity v
Independent variables - time t
Parameters - mass m, drag
coefficient cd
Forcing function - gravitational
acceleration g
 gcd 
gm
tanh 
t 
cd
 m 
Terminal velocity
t
t (s)
v (m/s)
0
0
2
18.7292
4
33.1118
6
42.0762
8
46.9575
10
49.4214
12
50.6175

51.6938
7
Numerical Solution:
cd 2
dv
g v
dt
m
 Reformulate the problem so that it can be solved by arithmetic operations
dv
v
 lim
dt t 0 t
dv v v(ti 1 )  v(ti )


dt t
ti 1  ti
v(ti 1 )  v(ti )
c
 g  d v(ti ) 2
ti 1  ti
m
Finite Difference Approximation
(from Calculus)
(Since t is finite, we
put  in the equation)
ti= initial time
v(ti)= velocity at initial time
ti+1= later time
v(ti+1)= velocity at later
time
t=difference in time
v=difference in velocity
8
• To solve the problem using a numerical
method, note that the time rate of change of
velocity can be approximated as:
dv v vti1  vti 


dt t
ti1  ti
9
(rearrange)
cd


v(ti 1 )  v(ti )   g  v(ti ) 2  ti 1  ti 
m


Recursion
relation
 ODE transformed into an algebraic equation.
 can find new v(ti+1) at ti+1 from old v(ti) at ti using arithmetic operations.
 still need to have initial conditions from the physics of the problem
 (t=0 ; v=0) (given).
t (s)
v (m/s)
0
0
2
19.6200
4
36.4137
6
46.2983
8
50.1802
10
51.3123
12
51.6008

51.6938
(for t=2 s)
v
Terminal velocity
(numerical solution)
(analytical solution)
10
t
Bases for Numerical Models
• Conservation laws provide the foundation for many
model functions.
change= increase - decrease
• Different fields of engineering and science apply
these laws to different paradigms within the field.
• Among these laws are:
 Conservation of mass
 Conservation of momentum
 Conservation of charge
 Conservation of energy
11
12
Summary of Numerical Methods
• The book is divided into five categories of
numerical methods:
13
•
•
•
Mathematical models are always needed in solving engineering problems.
Many times, these mathematical models are derived from engineering and
science principles (Newton’s 2nd Law for instance), while at other times the
models may be obtained from experimental data.
Mathematical models generally result in need of using mathematical
procedures that include but are not limited to
(A) differentiation,
(B) nonlinear equations,
(C) simultaneous linear equations,
(D) curve fitting by interpolation or regression,
(E) integration,
(F) differential equations
(G) optimization
These mathematical procedures may be suitable to be solved exactly as you must
have experienced in the series of calculus courses you have taken, but in most
cases, the procedures can be solved approximately using numerical methods.
Since there is an approximation, errors always arise during calculations
14
Lab related
15
Computer Programming
 It is the comprehensive process of
(problem  algorithm  coding executables)
Algorithms:
 An algorithm is the step-by-step procedure for calculations.
 It is a finite list of well-defined instruction for calculting a function.
 Can be expressed in many ways: flowcharts, pseudocodes, etc.
 Programming lanuguages express algorithms so that they can be
executed by a computer.
16
Flowcharts:
 A flowchart is a type of diagram that represents an algorithm.
 Steps are shown by boxes of various kinds.
 The order of the flow is shown by arrows.
Monopoly flowchart
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Flowchart Symbols
Terminal
Start and End of a program
Flowlines
Flow of the logic
Process
Calculations or data manıpulations
Input/Output
Input or output of data and information
Decision
Comparison, question, or decision that
determines alternative paths to be
followed.
On-page
connector
marks where the flow ends at one spot
on a page and continues at another spot.
Off-page
connector
Marks where the algorithms ends on one
page and continues at another 18
A simple addition algorithm
The same algorithm using a flowchart
using natural language
Start
 Step 1: Start the calculation
 Step 2: Input a value for A
 Step 3: Input a value for B
 Step 4: Add A to B and call the
answer C
 Step 5: Output the value for C
 Step 6: End the calculation
Input
A
Input
B
Add A and B
Call the result C
Output
C
end
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Flowchart example for calculating factorial N (N!)
20
Pseudocodes:
 Pseudocode is a higher-level method for describing an algorithm.
 It uses the structural convention of a programming language but is
intended for human reading rather than machine reading.
 It is easier for people to understand than a programming code.
 It is environment-independent description of the key principles of an
algorithm.
 It omits the details not essential for human understanding of the
algorithm (such as variable declarations).
 It can also advantage of natural language.
 It is commonly used in textbooks and scientific publications.
 No standard for pseudocode syntax exists.
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Example 1:
Example 2:
22
Flowchart
Pseudocode
23
Structured programming:
 It is an aim of improving the clarity, quality, and development time of
a computer program by making extensive use of subroutines, block
structures, for and while loops.
 Use of “goto” statements is discouraged, which could lead to a
“spaghetti code” (which is often hard to follow and maintain).
 Programs are composed of simple, hierarchical flow structures.
 Many languages support and encourage structural programming.
Modular programming:
 A kind of structural programming that the act of designing and writing
programs (modules) as interactions among functions that each
perform a single-well defined function.
 Top-down design.
 Coupling among modules are minimal.
24
Octave/Matlab Fundamentals
 Command window: to enter commands and data.
 Graphics window: to plot graphics.
 Edit window: create and edit M-files.
Command window:
 Can be operated just like a calculator
>> 55-16
ans=
39
 Define a scalar value
>> a=4
a=
4
 Assignments can be suppressed by semicolon (;)
>> a=4;
(just stored in memory w/o displaying)
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 Can type several commands on the same line by seperating them
with comas or semicolons. If you use semicolon, they are not
displayed.
>> a=4, A=6; x=1;
a=
4
(MATLAB treats names case-sensitive , i.e. a is not same as A in
above example)
 Can assign complex values as MATLAB handles complex arithmetic
automatically:
>> x=2+i*4
x=
2.0000 + 4.0000i
Predefined variables:
26
>> pi
If you desire more precision
>> format long
>> pi
ans=
3.14159265358979
To return to four decimal version
>> format short
Arrays, vectors, matrices:
 An array is a collection of values represented by a single variable
name. Matrcies are two-dimensional arrays.
 In MATLAB every value is a matrix:
 a scalar: 1x1 matrix
 a row vector: 1xn matrix
 A column vector: nx1 matrix
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 Square brackets ( [ ] ) are used to define an array in command mode.
>> a= [ 1 2 3 4 5]
a=
1
2
3
4
5
is a 1x5 row vector.
 In practive, a vector is usually meant to be a column vector. So, column
vectors are more practical. We can define them by the transpose
operator (‘)
>> b = [ 2 4 6 8 10]’
b=
2
4
6
8
10
28
 A matrix of values can be assigned as follows:
>> A = [1 2 3 ; 4 5 6; 7 8 9]
A=
1
2
3
4
5
6
7
8
9
Alternatively “Enter” key can be used at the end of each row to define
A:
>> A =[ 1 2 3
4 5 6
7 8 9 ];
(strike Enter key after 3 and 6)
 To see the list of all variables at any
session:
>> who
Your variables are:
A
a
ans
b
x
29
 If you want more detail:
>> whos
Name Size Bytes Class
A
3x3
72
double array
a
1x5
40
double array
ans
1x1
8
double array
b
5x1
40
double array
x
1x1
16
double array
(complex)
Grand total is 21 elements using 176 bytes
 To see an individual element in an array:
>> b(4)
ans=
8
>> A(2,3)
ans=
6
30
 Some predefined matrices are very useful.
>> E= zeros(2,3)
E=
0
0
0
0
>> u=ones(1,3)
0
0
u=
1
1
1
 Colon (:) operator is very useful to generating arrays:
>> t=1:5
t=
1
2
3
4
5
If you want increment other than 1, then you type:
>> t= 1:0.5:3
t=
1.0000
1.5000
2.000 2.500
31
3.0000
 Negative increments can also be defined:
>> t= 10: -1: 5
t=
10
9
8
7
6
5
 Colon (:) can also be used to select individual rows:
>> A (2,:)
ans=
4
5
6
returns the second row of the matrix.
 We can also use colon (:) to extract a series of elements from an array
>> t(2:4)
ans=
9
8
7
Second through fourth elements are returned.
32
Mathematical Operations:
 The common operators, in order of priority
^
Exponentiation
-
Negation
* /
Multiplication and Division
\
Left division (in matrix algebra)
+ -
Addition and Subtraction
>> y = pi/4;
>> y ^ 2.45
ans=
0.5533
 To override the priorities use paranthesis:
>> y
=
(-4) ^ 2
y =
16
33
 The real superiority of MATLAB comes in to carry out vector/matrix operation
Inner product (dot product) of two vectors:
>> a*b
ans=
110
Multiply vector with matrices
>> a = [ 1 2 3];
>> a*A
ans=
30
36
42
Multiply matrices
>> A*A
ans=
30
36
66
81
102
126
42
96
150
A^2 will return the same result.
34
 If the inner dimensions are not matched, you get an error message:
>> A*a
??? Error using ==> mtimes
Inner matrix dimensions must agree.
 MATLAB normall treat simple arithmetic operations in vector/matrix
operations. You can also do an element-by-element operation. To do
that you put a dot (.) in front of the arithmetic operator.
>> A .^ 2
ans=
1
16
49
4
25
64
9
36
81
35
Built-in Functions:
 MATLAB is very rich in predefined functions (e.g., sqrt, abs,
sin, cos, acos, round, ceil, floor, sum, sort, min,
max, mean,…)
For a list of elementary functions:
>> help elfun
 They operate directly on matrix quantities.
>> log (A)
ans=
0
1.3863
1.9456
0.6931
1.6094
2.0794
1.0986
1.7918
2.1972
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Graphics:
 Consider plotting time-versus-velocity for the falling object in air. The
solution to the problem was:
v(t ) 
 gcd 
gm
tanh 
t 
cd
 m 
 First need to define the time array:
>> t=[0:2:20]’;
This will define time (t) as 0,2,4..20. Total number of elements:
>> length(t)
ans=
11
 Assign values to the parameters:
>> g = 9.81 ; m = 68.1
;
cd=0.25 ;
 Now calculate (v):
>> v = sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t);
37
 To plot t versus t:
>> plot(t,v)
A graph appears in a graphics window.
 You can add many properties to the graph. For example
>> title ‘A falling object in air’
>> xlabel ‘t in second’
>> ylabel ‘v in meter per second’
>> grid
 If you want to see data points on the plot:
>> plot (t,v,’o’)
 If you want to see both lines and points on the same plot
>> plot(t,v)
>> hold on
>> plot (t,v,’o’)
>> hold off
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Programming with Matlab
M-files:
 We use the editor window to generate M-files.
 M-files contain a series of statements that can be run all at once.
 Files are stored with extension .m
 M-files can be script files or function files.
 Many built-in functions are all given in the form of M-files.
39
Script files:
 merely a series of Matlab commands that are saved on a file.
 They can be executed by typing the filename in the command
window.
Consider plotting the v(t) of the falling object problem. A script file can
be
typed in the editor window as follows:
t=[0:2:20]’;
g = 9.81 ; m = 68.1 ;
cd=0.25 ;
v = sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t);
plot(t,v)
title ‘A falling object in air’
Xlabel ‘t in second’
Ylabel ‘v in meter per second’
grid
40
This script can be saved as an “.m” file and be executed at once in
Function files:
 M-files that start with the word function.
 Unlike script files they can accept input arguments and return outputs.
 They must be stored as “functionname.m”.
Consider writing a function freefall.m for the falling body problem:
function v = freefall(t,m,cd)
% Inputs
% t(nx1) = time (nx1) in second
% m = mass of the object (kg)
% cd = drag coefficient (kg/m)
% Output
% v(nx1) = downward velocity (m/s)
g = 9.81 ; % acceleration of gravity
v = sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t);
return
41
Structured programming in Matlab:
 Simplest M-file instructions are performed sequentially. Program
statements are executed line by line at the top of the file and
moving down to the end. To allow non-sequential paths we use
- Decisions (for branching of flow)
Relational operators
- if structure
- if…else structure
- if…elseif structure
- switch structure
- Loops (repetitions)
- for … end structure
==
Equal
~=
Not equal
<
Less than
>
Greater than
<=
Less than or equal to
>=
Greater than or equal to
- while structure
- while … break structure
42
Excel
 It is a spreadsheet application.
 Uses a grid of cells arranged in numbered rows and letter-named
columns for data manipulations.
 Maximum of 1048K rows , and 16K columns. (version 12.0).
 It features calculations, graphing, pivot tables, etc.
 A wide variety of graphing tools, e.g. plotting, charting, histograms, etc.
 A wide variety of simple built-in functions for statistical, financial, and
engineering calcualtions.
 It is embeded programming language (Visual Basic for Applications,
VBA).
 Easy way to generate VBA files by macro recording.
 File formats:
archive)
.xlsx: Excel workbbok w/o macro (a ZIP compressed
43