ENGR 516 Catholic University of America Partial Differential Equations Find 5 problems in your own discipline that are described by Partial differentials

ENGR 516 Catholic University of America Partial Differential Equations Find 5 problems in your own discipline that are described by Partial differentials

For each problem, provide:

a) The appropriate PDE
b) The type of the PDE using discussed criteria (Show work for credits!)
c) Where and how BC’s and IC’s need to be applied
d) Specify if each PDE has dual nature or not
e) The source of your information.

My professor description: In your undergraduate or graduate studies you took some courses in which you derived or solved problems described by partial differential equations, such as the Navier – Stokes equations you learned with me for Fluid Mechanics problems. Now in your own discipline, try to find problems described by PDEs from the course you took. You can also search the internet for relevant problems of interest in your discipline and answer the questions based on the lecture we discussed last Tuesday.

My important thing to do is to read the two lectures before you start the homework

Note that my major is biomedical engineering so I preferred to do something in that filed but if not you can use other courses. Make sure you write everything in details ENGR 516 – Computational Methods for Graduate Students
Lecture 02:
Basic Concepts for Numerical
Methods
By Dr. Bomela
1
ENGR 516
Lecture Outline
• Learn & understand some basic ideas & concepts
in numerical work:
–
–
–
–
–
Different representations of numbers
Approximation and errors
The estimation of errors
Propagation of errors in numerical Results, and
Algorithm – Stability
2
ENGR 516
General Concepts
• To create a good understanding of the nature of numerical
work, let’s discuss some simple general concepts:
– Floating – point representation of numbers
– Approximation and errors
➢ Roundoff errors,
➢ Truncation errors,
–
–
–
–
Algorithm – Stability,
Propagation of errors in numerical Results,
Basic Error Principle, and
Loss of Significant Digits.
3
ENGR 516
Floating Point Representation
• In decimal notation (based on powers of 10), every real
number is represented by:
– A finite, or
– An infinite sequence of decimal digits (0, 1, 2, …, 9)
• Most computers have 2 ways of representing numbers:
– Fixed point and
– Floating point representations
4
ENGR 516
Fixed – Point System
• All numbers are given with a fixed number of decimals
after the decimal point:
– 62.358
– 0.014
– 1.000
• The above numbers are written with 3 decimals – 3D
• Fixed point representations are impractical in most
scientific computations due to their limited range (0.0004
 0.000), therefore:
– They will not concern us
5
ENGR 516
Floating – Point System
•
6
ENGR 516
Floating – Point System
•
7
Example
The form is:
or
sign  mantissa 10 exponent
  m 10
e
Example: For
− 2.5678102
 = −1
m = 2.5678
e=2
8
ENGR 516
Underflow & Overflow
•
9
ENGR 516
Errors in Numerical Results
• Results of computations of unknown quantities are
generally approximations, i.e., they are not exact but
involve errors:
• Such an error may result from a combination of the
following effects:
– Roundoff errors
– Experimental errors of given data, probably from
measurements,
– Truncation errors from prematurely breaking off, such as in
replacing a Taylor series with the sum of its few terms.
• These errors depend on the computational method used
and must be dealt with individually for each method
10
Approximation and Errors
Why measure/quantify errors?
1) To determine the accuracy of numerical
results.
2) To develop stopping criteria for iterative
algorithms.
http://numericalmethods.eng.usf.
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11
ENGR 516
Roundoff Error
• An error is caused by:
– Chopping, i.e., discarding all decimals from some decimal on, or
– Rounding
• This error is called roundoff error, regardless of whether
we chop or round.
• One of the rounding rules for rounding off a number to k
decimals or to k significant digits is as follows:
– If the number thus discarded is exactly half of a unit, 0.5, round
off to the nearest even decimal: 3.45 = 3.4 and 3.55 = 3.6
– This is supposed to ensure that in discarding exactly half a
decimal, rounding up and rounding down happens about equally
often, on the average: 3.55 = 3.6 & 3.65 = 3.6
12
ENGR 516
Roundoff Error
• Chopping is not recommended!:
– Corresponding error can be larger than that in rounding, and is
systematic, however, it is simpler and faster.
• Rounding errors may ruin a computation completely, even
a small computation:
– In general, these errors become the more dangerous with more
arithmetic operations, perhaps several millions, we perform.
– Therefore, it is important to analyze computational programs for
expected rounding errors & to find arrangement of the
computations such that the effect of rounding errors is as small
as possible.
• The arithmetic in a computer is not exact either and
causes further errors; however, these will not be relevant
13
to our discussion.
ENGR 516
Formulas for Errors
•
14
ENGR 516
Formulas for Errors & Uncertainty
•
15
Truncation Error
• Error caused by truncating or approximating a
mathematical procedure.
Taking only a few terms of a Maclaurin series to
x
e
approximate
2
3
x
x
e x = 1 + x + + + ………………..
2! 3!
If only 3 terms are used,
2


x
x
Truncation Error = e − 1 + x + 
2! 

16
Example 1 —Maclaurin series
Calculate the value of e1.2 with an absolute
relative approximate error of less than 1%.
e
1.2
1.2 2 1.2 3
= 1 + 1.2 +
+
+ ……………….
2!
3!
a %
terms
e1.2
1
1
__
___
2
2.2
1.2
54.545
3
2.92
0.72
24.658
4
3.208
0.288
8.9776
5
3.2944
0.0864
2.6226
6
3.3151
0.020736
0.62550
n
Ea
6 terms are required. How many are required to get
at least 1 significant digit17correct in your answer?
ENGR 516
Another Example of Truncation Error Differentiation
f (x + Δx ) − f (x )
f (x ) =
Δx → 0
Δx
lim
For a finite ‘ Δx’
f (x + x ) − f (x )
f (x ) 
x
18
Dividing a small difference of large
quantities by a small quantity
http://numericalmethods.eng.usf.edu
Another Example of Truncation Error Differentiation
Using a finite x
to approximate the derivative of a function ( ), f (x) :
f ( x) 
f ( x + x) − f ( x)
x
( )
( + ∆ )
( )
secant line
P
+ ∆ − ( )
( )
Graphical Representation
of Forward Difference
Approximation of 1st
Derivative
tangent line
Q

∆
+ ∆
Figure 1. Approximate derivative using finite Δx
19

Example 2 —Differentiation
f (3)
Find
for
and x = 0.2
f ( x) = x 2
using
f (3 + 0.2) − f (3)
0.2
3 .2 2 − 3 2
f (3.2) − f (3)
=
=
0.2
0 .2
f ( x + x) − f ( x)
f ( x) 
x
f ‘ (3) =
=
1.24
10 .24 − 9
= 6.2
=
0.2
0.2
The actual value is
f ‘ ( x) = 2 x,
f ‘ (3) = 2  3 = 6
Truncation error is then, 6 − 6.2 = −0.2
Can you find the truncation error with x = 0.1 ?
20
Another Example of Truncation Error Integration
Using finite rectangles to approximate an integral,
since the integral can be represented by the surface
area below the curve.
y
90
y = x2
60
30
x
0
0
1.5
3
4.5
6
21
7.5
9
10.5
12
Example 3 — Integration
Use two rectangles of equal width to
approximate the area under the curve for
f ( x) = x 2 over the interval [3,9]
y
9
90
60
x
dx

30
3
2
y = x2
2 | =6
2 | =3
x
0
0
3
6
9
22
12
Integration example (cont.)
Choosing a width of 3, we have
9
2
2
x
dx
=
(
x
)

3
x =3
(6 − 3) + ( x 2 )
x =6
(9 − 6)
= (32 )3 + (6 2 )3
= 27 + 108 = 135
Actual value is given by
9
9
 x 3   93 − 33 
2
3 x dx =  3  =  3  = 234


3
Truncation error is then
234 − 135 = 99
Can you find the truncation error with 4 rectangles?
23
ENGR 516
Accuracy in Tables
•
24
How is Absolute Relative Error used as a
stopping criterion?
If |a |  s where s is a pre-specified tolerance, then
no further iterations are necessary, and the process
is stopped.
If at least m significant digits are required to be correct
in the final answer, then
|a | 0.5  10 2−m %
http://numericalmethods.eng.usf.
edu
25
ENGR 516
Algorithms
• Numerical methods can be formulated as algorithms:
– An algorithm is a step – by – step procedure that states a
numerical method in a form, a “pseudocode”, understandable to
humans
– The algorithm is then used to write a program in a
programming language that the computer can understand so that
it can execute the numerical method.
– For routine tasks, your CAS or some other software system may
contain programs that you can use or include as parts of larger
programs of your own.
26
ENGR 516
Algorithm – Stability
• To be useful, an algorithm should be stable, i.e., small
changes in the initial data should cause only small
changes in the results, otherwise it is unstable.
• This numerical instability should be avoided by choosing
a better algorithm & must be distinguished from
mathematical instability of a problem called “ill –
conditioning”.
Some algorithms are stable only for certain initial data,
so that one must be careful in such a case & monitor the
residuals in running the program.
27
Example 4 – True Error
Forward Difference Approximation
The derivative, f (x) of a function f (x) can be
approximated by the equation,
f ‘ ( x) 
f ( x + h) − f ( x )
h
Note: = ∆
0.5 x
If f ( x) = 7e and h = 0.3
a) Find the approximate value of f ‘ (2)
b) True value of f ‘ (2)
c) True error for part (a)
http://numericalmethods.eng.usf.
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28
Example 4 (cont.)
Solution:
a) For x = 2 and h = 0.3
f (2 + 0.3) − f (2)
0.3
f (2.3) − f (2)
=
0.3
f ‘ (2) 
7e 0.5( 2.3) − 7e 0.5( 2 )
=
0.3
22 .107 − 19 .028
= 10.263
=
0.3
http://numericalmethods.eng.usf.
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29
Example 4 (cont.)
b) The exact value of f ‘ (2) can be found by using
our knowledge of differential calculus.
f ( x) = 7e 0.5 x
f ‘ ( x ) = 7  0.5  e 0.5 x
= 3.5e0.5 x
So the true value of
f ‘ (2)
f ‘ (2) = 3.5e 0.5( 2 )
= 9.5140
is
True error is calculated as
Et = True Value – Approximate Value
= 9.5140 − 10.263 = −0.722
http://numericalmethods.eng.usf.
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30
Example 5 — Relative True
Error
Following from the previous example 4 for true error,
find the relative true error for f ( x) = 7e 0.5 x at f ‘ (2)
with h = 0.3
From the previous example,
Et = −0.722
Relative True Error is defined as
True Error
True Value
− 0.722
=
= −0.075888
9.5140
t =
as a percentage,
t = −0.075888  100 % = −7.5888 %
http://numericalmethods.eng.usf.
edu
31
ENGR 516
Propagation of Errors
• In numerical methods, calculations are NOT made with
exact numbers. How do these inaccuracies propagate
through the calculations?
– Error propagation refers to how error bounds at the beginning
& in later steps, roundoff for example, propagate into the
computation & affect accuracy, sometimes very drastically:
• Error Propagation theorem:
(a) In addition & subtraction, an error bound (uncertainty) for
the results is given by the sum of the error bounds for the
terms.
(b) In multiplication & division, an error bound for the relative
error of the results is given (approximately) by the sum of the
bounds for the relative errors of the given numbers.
∈ ෤ × ෤
=
− ෤ ෤
1
2
≤
+
= ∈ 1 + ∈ 2 ≤ 1 + 2

32
Propagation of Errors In Formulas
If f is a function of several variables X 1 , X 2 , X 3 ,……., X n−1 , X n
then the maximum possible value of the error in f is
f
f
f
f
f 
X 1 +
X 2 + ……. +
X n−1 +
X n
X 1
X 2
X n−1
X n
http://numericalmethods.eng.usf.
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33
Example 6:
Find the bounds for the propagation in adding two numbers. For
example if one is calculating X +Y where:
X = 1.5 ± 0.05 and Y = 3.4 ± 0.04
Solution
Maximum possible value of X = 1.55 and Y = 3.44
Maximum possible value of X + Y = 1.55 + 3.44 = 4.99
Minimum possible value of X = 1.45 and Y = 3.36.
Minimum possible value of X + Y = 1.45 + 3.36 = 4.81
Hence
4.81 ≤ X + Y ≤4.99.
http://numericalmethods.eng.usf.
edu
34
Example 7:
The strain, , in an axial member of a square cross-section
is given by
Given:
F
= 2
h E
F = 72  0.9 N
h = 4  0.1 mm
E = 70  1.5 GPa
Find the maximum possible error in the measured strain, .
http://numericalmethods.eng.usf.
edu
35
Example 7:
Solution
72
=
−3 2
9
(4  10 ) (70  10 )
−6
= 64.286 10
= 64.286



 =
F +
h +
E
F
h
E
http://numericalmethods.eng.usf.
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36
Example 7:

1
= 2
F h E
Thus
E =

2F
=− 3
h
hE

F
=− 2 2
E
h E
1
2F
F

F
+

h
+
E
2
3
2 2
h E
hE
h E
1
2  72
=
 0.9 +
 0.0001
(4 10 −3 ) 2 (70 10 9 )
(4 10 −3 )3 (70 10 9 )
+
Hence
72
9

1
.
5

10
(4 10 −3 ) 2 (70 10 9 ) 2
= 5.3955
= (64.286  5.3955 )
http://numericalmethods.eng.usf.
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37
Example 8:
Subtraction of numbers that are nearly equal can create
unwanted inaccuracies. Using the formula for error
propagation, show that this is true.
Solution
Let
z = x− y
Then
z =
z
z
x +
y
x
y
= (1)x + (−1)y
= x + y
So the relative change is
x + y
z
=
z
x− y
http://numericalmethods.eng.usf.
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38
Example 8:
For example if
x = 2  0.001
y = 2.003  0.001
0.001 + 0.001
z
=
z
| 2 − 2.003 |
∆

= 0.6667
= 66.67%
http://numericalmethods.eng.usf.
edu
39
ENGR 516
Basic Error Principle For
Approximate Error
• Every numerical method should be accompanied by an
error estimate.
• If such a formula is not known, is extremely complicated
to obtain, or is impractical because it involves information
not available, estimate error by comparison:
Approximate Error ( E a ) = Present Approximation – Previous Approximation
40
Example 9 — Approximate Error
For f ( x) = 7e 0.5 x at x = 2 find the following,
a) f (2) using h = 0.3
See solution on Example on True Error
f (2 + 0.3) − f (2)
22 .107 − 19 .028
=
= 10.263
0.3
0.3
f (2) using h = 0.15
f ‘ (2) 
b)
c) approximate error for the value of f (2) for part b)
Solution:
b) For x = 2 and h = 0.15
7e 0.5( 2.15) − 7e 0.5( 2) = 20.50 − 19.028 = 9.8800
f (2.15) − f (2)
f (2 + 0.15) − f (2)
=
f (2) 
=
0.15
0.15
0.15
0.15
‘
c) So the approximate error, E a is
Ea = Present Approximation – Previous Approximation
= 9.8800 − 10.263 = −0.38300
http://numericalmethods.eng.usf.
edu
41
Relative “Approximate Error”
• Defined as the ratio between the approximate
error & the present approximation.
Relative Approximate Error ( a) =
Approximate Error
Present Approximation
From the previous example, the approximate
value of was found to be = −0.38300
http://numericalmethods.eng.usf.
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42
Relative “Approximate Error”
Let’s find the relative approximate error
a =
=
Approximate Error
Present Approximation
− 0.38300
= −0.038765
9.8800
as a percentage,
a = −0.038765 100 % = −3.8765 %
Absolute relative approximate errors may also need to
be calculated,
a =| −0.038765 | = 0.038765 or 3.8765%
http://numericalmethods.eng.usf.
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43
Table of Values
0.5 x
For f ( x) = 7e at x = 2 with varying step size, h
h
f (2)
a
m
0.3
10.263
N/A
0
0.15
9.8800
3.877%
1
0.10
9.7558
1.273%
1
0.01
9.5378
2.285%
1
0.001
9.5164
0.2249%
2
http://numericalmethods.eng.usf.
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44
ENGR 516
Loss of Significant Digits
•
45
Effect of Carrying Significant
Digits in Calculations
http://numericalmethods.eng.usf.edu
46
Find the contraction in the
diameter
Tc
D = D   (T )dT
Ta
Ta=80oF; Tc=-108oF; D=12.363”
α = a0+ a1T + a2
2
T
47
Thermal Expansion
Coefficient vs Temperature
D = D  T
T(oF)
-340
-300
-220
-160
-80
0
40
80
α (μin/in/oF)
2.45
3.07
4.08
4.72
5.43
6.00
6.24
6.47
Plot Data from the table
48
o
Alpha ( in/in/ F)
Regressing Data in Excel
(general format)
8.00
4.00
0.00
-400
Fit a curve through
Data & find its
Equation using a
general format
-200
0
200
T (oF)
α = -1E-05T2 + 0.0062T + 6.0234
49
Observed and Predicted Values
α = -1E-05T2 + 0.0062T + 6.0234
-340
α (μin/in/oF)
Given
2.45
α (μin/in/oF)
Predicted
2.76
-300
3.07
3.26
-220
4.08
4.18
-160
4.72
4.78
-80
5.43
5.46
0
6.00
6.02
40
6.24
6.26
80
6.47
6.46
T(oF)
50
Alpha ( in/in/ F)
Regressing Data in Excel
(scientific format)
o
8.00
4.00
0.00
-400
-200
0
200
Fit a curve through
T (oF)
Data & find its
Equation using a
2 + 6.2714E-03T + 6.0234
α
=
-1.2360E-05T
scientific format
51
Observed and Predicted Values
α = -1.2360E-05T2 + 6.2714E-03T + 6.0234
-340
α (μin/in/oF)
Given
2.45
α (μin/in/oF)
Predicted
2.46
-300
3.07
3.03
-220
4.08
4.05
-160
4.72
4.70
-80
5.43
5.44
0
6.00
6.02
40
6.24
6.25
80
6.47
6.45
T(oF)
52
Observed and Predicted Values
α = -1.2360E-05T2 + 6.2714E-03T + 6.0234
α = -1E-05T2 + 0.0062T + 6.0234
T(oF)
α (μin/in/oF)
Given
α (μin/in/oF)
Predicted
α (μin/in/oF)
Predicted
-340
2.45
2.46
2.76
-300
3.07
3.03
3.26
-220
4.08
4.05
4.18
-160
4.72
4.70
4.78
-80
5.43
5.44
5.46
0
6.00
6.02
6.02
40
6.24
6.25
6.26
80
6.47
6.45
6.46
53
ENGR 516 – Computational Methods for Graduate Students
Lecture 03:
Partial Differential Equations
(PDEs)
&
Their Classification
By Dr. Bomela
1
ENGR 516
Course Description from Cardinal Station
(http://cardinalstation.cua.edu)
• Discretization methods:
– finite differences,
– finite volumes,
– finite elements,
• Stability and convergence;
• Parabolic, hyperbolic, and elliptic partial differential
equations (PDEs)
• Model equations & numerical solution methods.
• Numerous programming exercises will be assigned.
2
ENGR 516
Lecture Outline
• Introduce various types of PDEs & their
classification
• Discuss several examples from Physics &
Engineering disciplines to highlight the
classification procedure that:
– Helps elucidate where & how Boundary Conditions
BC’s & Initial Conditions IC’s need to be applied
– Enables the development of the appropriate
numerical methods to solve each type of the PDE.
• Introduce Homework 02
3
ENGR 516
Roles of Analysis & Modeling
System (Matter)
– Domain
Surroundings
Interactions
(Causes) Q & W
Mathematical Model
• Analysis is the use of certain physical & mathematical
principles to establish the relationship (mathematical
model) between cause (input parameter) & effect
4
(output) as applied to the system
Steps in Analysis – Develop the
Mathematical Model
ENGR 516
Physical
Model
Mathematical
formulation
Model
formulation
Real – life problem
Mathematical
Model
Numerical
method
Analytical
Solution
Numerical
Solution
Verification
Measurements
Validation
Validation
Experimental observations (data)
5
ENGR 516
What is a PDE ?
• Partial Differential Equations have 2 or more
independent variables
 2u  2u
3 2 + 2 = x2 + y2
x
y
subject to certain conditions: where u is the dependent variable, and
and y are the independent variables.
• Ordinary Differential Equations have only one
independent variable
dy
3 + 5 y 2 = 3e − x , y(0) = 5
dx
6
ENGR 516
Example of an Ordinary Differential
Equation
( )
Hot Water
, ∀,
Spherical
Ball
= : Specific heat
∞

An overall Energy Balance gives: ℎ ∞ − = ∀

• Assumption: Ball is a lumped system
– Dependent variable of interest function of time alone
• Number of Independent variables: One ( )
7
ENGR 516
Example of a Partial Differential
Equation
Spherical
Ball
Hot Water
( , , , )
Spherical Coordinates
k   2 T 
k
 
T 
k
 2T
T
r
+
sin

+
=

C
, t  0, T (r , ,  ,0) = Ta




2
2
2
2
2
  r sin  
t
r r  r  r sin   
• Assumption: Ball is not a lumped system.
• Number of Independent variables: Four ( , , , )
• Dependent variable: = , , ,
8
ENGR 516
Introducing PDEs
• PDEs are mathematical models of various physical &
geometrical problems arising when:
– The unknown functions (solutions) depend on 2 or more variables,
usually:
➢ Time, , and
➢ One or several space variables , , or .
• PDEs describe the behavior of systems under consideration
• Most problems in dynamics, elasticity, heat transfer,
electromagnetic theory & quantum mechanics require PDEs
– Only the simplest physical systems can be modeled by ODEs, and
– The range of applications of…
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