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Introduction to Algorithms/What is an Algorithm

This is a lesson in the course, Introduction to Algorithms, which is a part of The School of Computer Science

Objective
Our main objective is to learn what an algorithm is and its basic components/concepts. By the end of this you should be able to recall what an algorithm is and what it does.

Definition of Algorithm
Merriam-Webster Online defines "Algorithm" as " a step-by-step procedure for solving a problem or accomplishing some end, especially by a computer". The dictionary definition is missing the component that the procedure must be finite, that is, there has to be a condition for the termination of the algorithm.

Examples:

1. One example would be knitting socks.

The first step is to cast on stitches. Step-by-step the rows are knitted, the heel turned, and finally the last row cast off. The algorithm specifies how to do the sock, but each sock knitted is different, as it uses different wool, but all are similar in structure to each other.

2. Another important example would be sorting a hand of cards.

There are multiple ways of doing this, but one method is to take the lowest card and move it to the leftmost (or rightmost) position. Next, find the second lowest card and move it to the second leftmost position. Repeat until the cards are sorted.

Everyday example:

Here are perhaps slightly whimsical examples of an algorithm.

 * 1) Get a loaf of bread.
 * 2) Cut one slice from end of the loaf of bread.
 * 3) Put slice of bread into toaster.
 * 4) Turn  toaster on.
 * 5) Wait for toaster to finish.
 * 6) Put toasted bread onto a plate.
 * 7) Spread butter on toast.

 * 1) grab tooth paste
 * 2) take off the lid
 * 3) grab tooth brush
 * 4) put a dab of tooth paste onto the tooth brush
 * 5) put lid back onto tooth paste
 * 6) turn on the sink
 * 7) wet the tooth brush
 * 8) turn off the sink
 * 9) brush all tooth surfaces consistently with circular motion
 * 10) when complete, turn on sink
 * 11) fill a small cup with water
 * 12) turn off sink
 * 13) rinse mouth
 * 14) spit into sink
 * 15) turn on sink
 * 16) rinse tooth brush and sink
 * 17) turn off sink
 * 18) put toothbrush and tooth paste away
 * 19) throw away disposable cup
 * 20) shut all drawers/cabinets

As can be seen, the algorithm is a set of steps that can be followed in order to achieve a result.

Basic concepts
This simple example already contains many components commonly found in most algorithms:

 * Instructions. Instructions are the heart and soul of any algorithm. An algorithm will consist of a series of sub-algorithms, each performing a smaller task. Extracting individual digits from the numbers to be added may well be an entire task in itself. Some instructions, like digit addition, can be considered fundamental instructions which cannot be broken up further; all algorithms can eventually be broken down into these instructions, like objects being composed of atoms.

 * Variables. temporary and carry are both temporary values used to store additional information the algorithm needs to function correctly or effectively. Their values can vary as the algorithm progresses, hence their name.

 * Conditionals. Some algorithms may need to make a decision somewhere in it. If the sum of the current two digits is 10 or larger, 1 will be carried over to the next digit position, otherwise nothing (ie. zero) is carried. Conditionals allow an algorithm to selectively execute instructions based on certain conditions which must be satisfied.

 * Looping. The for each...end for instruction means to run certain instructions for each object in a set, in this case for each digit in a number. This is a special case of looping, the act of repeating a set of instructions while (or until) a goal is reached or a condition is satisfied; in this case while there are digits left (or until there are no digits left). Loops can contain any other instructions, including other loops!

 * Subalgorithm. An algorithm may not be able to do all the work on its own. Usually, a large, complex algorithm can be broken up into parts, each perfoming a specific task. Smaller algorithms are easier to reuse, for example the for each ... digit selection algorithm could be reused in subraction, multiplication and division algorithms, and so on. Breaking up algorithms into logical parts also make it easier to analyze their behaviours and properties from a mathematical point of view.

 * Performance. While initial algorithms may seem to function, you may discover that they may not be the best choice in the long term. For example, a simple bubble sort may work for a small amount of items, but if you try it on  a large amount of data, it will take a massive amount of time. An algorithm with better performance would include the Merge Sort.

Assignments
Your assignment is to create a simple algorithm. If you need help then you can go back and look at the example above of an algorithm. How to make a bowl of cereal 1. grab a bowl from the cupboard 2. grab a spoon 3. choose a cereal 4. grab a cereal 5. pour the cereal into the bowl 6. grab the milk 7. pour milk into the bowl 8. put cereal and milk away 9. Enjoy!

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 * Course Home Page


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Introduction to Algorithms/Subalgorithms

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science

Objective
The objective is to teach you all the necessary stuff that is usually taught in a university level course.

Subalgorithm
Algorithms are often broken down into smaller chunks, called Subalgorithm. This is both so that they are easier to read, and also because then parts of the algorithm can be reused. What follows is the above algorithm in a more formal manner. The text in italics below are the names of the Subalgorithm that are being called.

Algorithm for making buttered toast (2.0) get a loaf of bread. cut slice from the loaf of bread. move the slice of bread to the toaster. turn on the toaster. wait for the toaster to finish. move the slice of toast to a plate. spread the slice of toast with butter. turn off the toaster.

Assignments

Subalgorithm for applying mascara

grab the mascara from the cabinet

unscrew the lid

dab any excess product on rim

use the mascara wand and apply to lashes in upward motion

repeat this step until you get the desired results

apply to bottom lashes

screw the lid back on

place mascara back in cabinet

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 * Next Lesson
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license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Introduction to Algorithms/Variables

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science

Variables
In the above example it is fairly simple to follow the piece of toast through the algorithm, but if you imagine that the algorithm could become more complex, and thus the progress of the toast much harder to follow. In order that, at any point in the algorithm, it is clear what is being talked about, names are often given to things. For example

Algorithm for making buttered toast (3.0) LET A = a loaf of bread LET T = a toaster LET P = a plate LET B = some butter LET S = cut slice from A get A move the S to T turn on T wait for T move the S to P spread S with B

Assignments

 * Previous Lesson
 * Next Lesson
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title: Introduction to Algorithms/Variables


text:
Introduction to Algorithms/Flow Control

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science

Flow Control
Our current toast making algorithm works well if you are making toast for yourself, but what if you have a few friends around for breakfast and you need to make them toast as well? The algorithm clearly needs to be improved so that we can make toast for everyone. Also, simply waiting for the toasted is very boring, we could be doing something else while we wait. The third problem with making toast for friends is that they are fussy, what if some of them want butter whilest others prefer margarine, or even just plain toast

Algorithm for making buttered toast (4.0) LET A = a loaf of bread LET T = a toaster LET P = a plate LET B = some butter LET M = some margarine FOR every friend (F) eating toast{ LET S = cut slice from A move S to T turn on T WHILE T is not finished{ talk to F }   move S to P IF F likes B{ LET X = B }ELSEIF F likes M{ LET X = M }ELSE{ LET X = NOTHING }   spread S with X move P to F } As can be seen, to resolve the three problems above, I had to implement the three main types of flow control, a For Loop, a While Loop and an If Statement. It should be obvious from the example what each of these does. I also involved a new variable X, which unlike the variables I have used before changes its value as the algorithm progresses.

Assignments

 * Previous Lesson
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title: Introduction to Algorithms/Flow Control


text:
Introduction to Algorithms/Common Algorithms

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science

Algorithms commonly studied
Many algorithms are currently under intense study for various reasons. They include:

 * Sorting algorithms. A sorting algorithm will take a set of objects and arrange the objects in ascending (or descending) order. While this is a simple problem and there are many existing algorithms to tackle this problem efficiently, new algorithms (and problems with older ones) are published all the time.
 * Examples of sorting algorithms:
 * Bubble sort
 * RadixSort
 * ExtraDix

 * Artificial Intelligence. AI algorithms typically accept an input representing its environment (eg. a chess board) and try to find some legal move to respond favourably to the environment or an opponent (eg. calculating a good chess move). Today modern chess AIs can beat even professionals and regularly challenge world grandmasters to intense, decisive matches. Still, some simple board games remain impervious to attack. AI has also been recently employed in anti-fraud software and email spam filters.

 * Numerical Computation. More and more mathematicians and computer scientists are using computers to do research, eg. to collect data, test new theories or process experimental results. Some researchers have even tried to use computers to prove theorems! (A proof to the four-color theorem was, controversially, completely produced by computer.) Ever faster algorithms need to be invented to cope with this massive increase in demand.

Assignments

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User:Inyuki~enwikiversity

Hi. This is Inyuki... http://studymaze.org

title: User:Inyuki~enwikiversity


text:
Caregiving and dementia/Participants/Instructions

Instructions

 * 1) Create an account
 * 2) Make sure you are logged in (check in top-right corner)
 * 3) Sign your username on this page by following these steps:
 * 4) Edit the page
 * 5) Add four wiggles ~
 * 6) Save the page - this will leave a link to your userpage with an automatic timestamp!


text:
Talk:Controversies in Science/Was there a mitochondrial Eve?/A critique of The Myth of Eve

The sampling is far too small to be a conclusive scientific explanation. If we consider the vast differences around the world (in Africa alone there are over 2000 language groups) it is not possible that we all originate from a single mother down the lines. Also, the lack of mutations in our lineage suggests that there is far less inter-breeding than the theory of Mitchondrial Eve would have us believe. Tlouc287 (talk) 16:56, 13 April 2012 (UTC)


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User talk:Inyuki~enwikiversity

 Hello and Welcome to Wikiversity Inyuki! You can contact us with questions at the colloquium or me personally when you need help. Please remember to sign and date your finished comments when participating in discussions. The signature icon above the edit window makes it simple. All users are expected to abide by our Privacy, Civility, and the Terms of Use policies while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
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 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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 * Discuss Wikiversity issues or ask questions at the colloquium.
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You do not need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Leutha (talk) 03:30, 14 August 2012 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Inyuki. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Inyuki~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:34, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:11, 19 April 2015 (UTC)

title: User talk:Inyuki~enwikiversity


text:
User:Sohail.alam

 * Let me describe myself with a famous quote by Albert Einstein: " I have no special talent. I am only passionately curious. "... curious about the Physics of the Universe... curious about world of the subatomic particles... curious about how the motion of electrons contribute to all the electronic gadgets that we cannot dream to live without... curious about all that I know not of...

 * Want to reach me: visit -> http://www.robozaa.co.in/

title: User:Sohail.alam


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Introduction to Algorithms/Efficiency

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science

Objective

Algorithm efficiency refers to making the steps of the algorithm concise and eliminating redundancy. This could include eliminating steps altogether by combining or deleting particular ones, or to refining a step to make it run with less code or run time, etc.

Assignments

 * Previous Lesson
 * Next Lesson
 * Course Home Page

title: Introduction to Algorithms/Efficiency


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Sources/First cyan source in Caelum

The first cyan source in Caelum is unknown.

This is a lesson in map reading, coordinate matching, and searching. It is also a project in the history of cyan astronomy looking for the first astronomical cyan source discovered in the constellation of Caelum.

Nearly all the background you need to participate and learn by doing you've probably already been introduced to at a secondary level.

Some of the material and information is at the college or university level, and as you progress in finding cyan sources, you'll run into concepts and experimental tests that are an actual search.

To succeed in finding a cyan source in Caelum is the first step. Next, you'll need to determine the time stamp of its discovery and compare it with any that have already been found. Over the history of cyan astronomy a number of sources have been found, many as point sources in the night sky. These points are located on the celestial sphere using coordinate systems. Familiarity with these coordinate systems is not a prerequisite. Here the challenge is geometrical, astrophysical, and historical.

NGC 1679 in the image at left appears to contain some cyan, probably as a result of a mixture of light blue and yellow.

First step
The first step is to succeed in finding a cyan source in Caelum.

Next, you'll need to determine the time stamp of its discovery and compare it with any that have already been discovered.

Over the history of cyan astronomy a number of sources have been found, many as point sources in the night sky. These points are located on the celestial sphere using coordinate systems. Familiarity with these coordinate systems is not a prerequisite. Here the challenge is geometrical, astrophysical, and historical. The coordinates are usually supplied by the cyan source observers.

Control groups
A control group for this experiment may be a cyan source observed by ancient hominins in their version of the constellation Caelum.

Astronomical cyan sources
The electric blue glow of electricity results from the spectral emission of the excited ionized atoms (or excited molecules) of air (mostly oxygen and nitrogen) falling back to unexcited states, which happens to produce an abundance of electric blue light. This is the reason electrical sparks in air, including lightning, appear electric blue. It is a coincidence that the color of Cherenkov radiation and light emitted by ionized air are a very similar blue despite their very different methods of production.

Traveling cyan sources
Many cyan sources do not remain in a constellation for lengthy periods. Some of these are the Sun and sources apparently in orbit around the Sun. The Sun travels through the 13 constellations along the ecliptic (the plane of the Earth's orbit around the Sun): the 12 of the Zodiac and the constellation Ophiuchus. These are described in source astronomy. Caelum is not one of the constellations of the Zodiac.

First sources
The "earliest known astronomy anywhere in the world [is] that of the Australian Aborigines, whose culture has existed for some 40,000 years".

"The Aranda tribes of Central Australia, for example, distinguish red stars from white, blue and yellow stars."

Backgrounds
To introduce yourself to some aspects of the challenge may I suggest reading the highlighted links mentioned above, and if you're curious, those listed under the section "See also" below.

Cyan rays are a form of radiation that is currently part of the electromagnetic radiation intersecting the Earth. More information about radiation is in radiation astronomy.

Constellations
The Wikipedia article about the constellation Caelum contains a high school level description. The figure at right shows the sky map of Caelum. Around the edges of the map are coordinates related to longitude and latitude, but with the Earth rotating on its axis every 24 hours the celestial coordinates must remain fixed. How has this been accomplished?

Also, in the Wikipedia article is a list of stars in Caelum. What's the difference between a star and an astronomical cyan source?

From the Wikipedia article on the Zodiac, is Caelum a zodiacal constellation?

Caelum is a faint constellation in the southern sky, introduced in the 1750s by Nicolas Louis de Lacaille.

"Caelum is depicted as a sculptor’s chisel. It first appeared in Lacaille’s map of the southern stars published in 1756, as “les Burins,” a pair of crossed burins connected by a ribbon. (Burins are sharp engraving tools.)"

"In Johann Bode’s star atlas Uranographia, the constellation still had the longer name, Caela Scalptoris."

Testing a source
There are many web sites that may have a cyan source listed for the constellation Caelum.

Wikipedia sources
A. Constellation article

Under "Notable features" in the Wikipedia article on the constellation Caelum is the list of stars in Caelum. Click on this link. In the table of this Wikipedia article is α Cae. To the right are coordinates:
 * 1) Right ascension (RA): 04h 40m 32.62s and
 * 2) Declination (Dec): -41° 51' 48.9".

Find these coordinates on the Caelum map at the right. Is alpha Caeli really inside the boundaries of the constellation?

To evaluate the star as a cyan source, skip ahead to the section "Cyan source".

B. Wikipedia search

Another way to look for cyan sources in the constellation is to perform a search on Wikipedia. Try "Caelum cyan" without the quotes. This yields no possible astronomical cyan source candidates. "Caelum blue-green" yields only Eridanus (constellation) where a blue-green star is described. This may be a good candidate for the first cyan source in Eridanus.

To evaluate a cyan source, skip ahead to section "Cyan source".

SIMBAD sources
Another way to find possible cyan sources in Caelum is to use search queries on SIMBAD.

Click on the SIMBAD link under "External links" below, then click on "Criteria query" or "by criteria".

In the tan box, type in "region(04 40 32.62 -41 51 48.9, 10m)", without the quotes. This tells the SIMBAD computer you are interested in a circular region of the celestial sphere centered on the coordinates for alpha Caeli, with a radius of 10 arcminutes (m).

Notice on the page over at the right from the tan colored box: "Return". The default is "object count". Click on "submit query". In a few moments a result something like "Number of objects:	   	 9" should appear. Click "Back" to see the tan box again.

To see if you have found at least one object, change "Return" to "display" by clicking on the circle to its left, then "submit query".

SIMBAD should display a list of objects. Read through the resource cyan astronomy for clues that may indicate whether a particular spectral type (Sp type) is a cyan source. If none of the objects listed seems to be described as a cyan source, try going "Back" and increasing the arcminutes from "10m" to "20m", and repeat until a cyan source is found.

Once you believe you have discovered a cyan source, proceed to the section "Cyan source".

Cyan source
There are several ways to evaluate a cyan source for the constellation Caelum.

Wikipedia source
Click on the link to the Wikipedia article. After you've enjoyed reading about the source, use the 'find' command of your browser to see if this Wikipedia page mentions anything about "cyan", "blue-green", "green-blue", or "cyan rays". Does the article mention whether or not the source is a cyan source?

What is the current time stamp for the Wikipedia article on the source? [Hint]: look for something like "This page was last modified on 12 January 2012 at 06:47." very near the bottom of the page. For now this is an adequate time stamp.

From reading the Wikipedia article on the source, if you believe the text demonstrates that the source is not a cyan source in Caelum edit the "Non-cyan source in Caelum" section near the bottom of the page with an entry similar to "# Alpha Caeli 12 January 2012 at 06:47 Wikipedia article "Alpha Caeli".", without the outer quotes, and finish the entry with four "~"s without the quotes after the period. The date included with your designation or username is a time stamp for the entry. The last portion of the entry is the source of your information.

On the other hand, if there are one or more sentences in the article that you believe demonstrates that the source is a cyan source in Caelum edit the section below "Cyan source in Caelum" with a similar entry.

Go to the section entitle, "Challenging an entry".

SIMBAD source
To check any source (even one from Wikipedia) on SIMBAD, click of the "External link" to the "SIMBAD Astronomical Database".

At the lower right side of the SIMBAD Astronomical Database page is a "Basic search" box. There are several ways to try your target:
 * 1) source name: without the quotes or
 * 2) source coordinates: without the quotes, for example, "04 40 32.62 -41 51 48.9".

If you are looking at a SIMBAD generated table which lists possible targets, click on one.

On its SIMBAD page move over to the right side until you see an Aladin visual photograph of the object. Is it a cyan source?

Even if the source does not look cyan, look down the left hand side of the page for "Spectral type:". From your reading of cyan astronomy, do you believe the source is a cyan source, or not? Noting that SIMBAD does, or does not consider the source to be a cyan source is important, so skip down to the "SIMBAD time stamp" section.

If you have already found a cyan source (or a table of them) using SIMBAD, click on the blue link identifier for the first.

SIMBAD time stamp
Peruse the SIMBAD page for a time stamp or date of last revision. [Hint: it may look something like "2012.01.09CET20:10:02" and be in the upper right.]

If the entry at SIMBAD convinces you that the source is not a cyan source, edit the "Non-cyan source in Caelum" section near the bottom of this page and type in an entry similar to "# Source Name 2012.01.09CET20:10:02 SIMBAD article "SIMBAD source name".", without the first set of quotes, followed by four ~s.

If your SIMBAD analysis convinces you that you have found a cyan source in Caelum (did you check the coordinates vs. the map of Caelum?), make an entry something like the ones in the section "Cyan source in Caelum".

Challenging an entry
Any entry in either the section "Cyan source in Caelum" or "Non-cyan source in Caelum" can be challenged. The time stamp may be challenged to see if there is an earlier one. The source may be challenged by an earlier source.

Wikipedia challenge
Is Wikipedia a 'primary source', or does the Wikipedia article cite a source?

Even though Wikipedia has an article on the source, is it a good place to stop in testing whether the source has been detected as an astronomical cyan source?

If the Wikipedia article cites a primary source, skip down to the section on "Primary sources".

SIMBAD challenge
Is SIMBAD a 'primary source'?

SIMBAD is an astronomical database provided by the Centre de Données astronomiques de Strasbourg. It is an authoritative source, but they do occasionally make a mistake.

If you find a cyan source within the constellation on SIMBAD, the next step is to find the earliest time stamp of discovery.

Wikipedia test source
For a Wikipedia article that cites a primary source, scroll down to the reference and open the reference. Read through the article looking for where the source mentioned in the Wikipedia article occurs. Some primary source authors may use source designations that are not mentioned in the Wikipedia article. To look for other designations, click on the link to SIMBAD in the "External links" on this page, enter the source name from the Wikipedia article, and see if other names are mentioned in the article.

When none of the names are mentioned, click on the link for "Google Advanced Search" in the list of "External links", enter the source name or designation(s) such as "Gliese 866", with "cyan" to see if the source has a reference indicating it is a cyan source. And, look for the earliest one. Compose an entry using the primary source.

SIMBAD test source
Further down the SIMBAD page is a list of "Identifiers". Click on the blue bold portion.

On the page that appears, there should be a primary source listed after Ref:. Click on the blue link with the oldest year. This yields an earlier time stamp and entry citation like the current one in the section "Cyan source in Caelum". If you find another source or an earlier time stamp, compose a similar entry and edit the section. Additional information to add into the reference can be found by clicking on "ADS services" from the SIMBAD page.

Changing an entry
From your analysis of the source so far, is it a cyan source?

If you have found an earlier time stamp for the source than the one listed in the section below "Non-cyan source in Caelum" and the answer to the above question is "no", you can edit the section with your result. Or, you can leave the entries as is and try another star or object.

If you have found an earlier time stamp for the source than the one listed in the section below "Cyan source in Caelum", edit the section with your result. Or, if you found another cyan source with a comparable or earlier time stamp, edit the section with your result.

Cyan sources in Caelum

 * 1) The list of stars in Caelum contains the entry "3 G. Cae" at right ascension (RA) 04h 27m 05.92s declination (Dec) -46° 56' 49.6", spectral type F8V. Inputting the RA and Dec into SIMBAD using "region(04 27 05.92 -46 56 49.6,10m)", clicking on "display", then "submit query" yields a list of objects containing LTT 1989 approximately at the coordinates for "3 G. Cae" but with a visual spectral type of F5.5V. Using 40m, yields HD 28079 an F9V and HD 28155 an F7V. --Marshallsumter (talk) 10:59, 15 August 2012 (UTC)
 * 2) The list of stars in Caelum contains the entry "27 G. Cae" at RA 05h 03m 53.94s Dec -41° 44' 43.2", F8V. Using a "Query by criteria" on SIMBAD with region(05 03 53.94 -41 44 43.2,10m) yields HR 1651 an F8V. --Marshallsumter (talk) 11:45, 15 August 2012 (UTC)

Non-cyan sources in Caelum

 * 1) Alpha Caeli A is an F-type main sequence star with a visual stellar classification of F2V. Temperature 6,991 K. Cyan stellar sources are likely between stellar classifications F7V and F9V. --Marshallsumter (talk) 10:40, 15 August 2012 (UTC)
 * 2) Beta Caeli "is categorized as an F-type main sequence star ... F2V" . --Marshallsumter (talk) 10:40, 15 August 2012 (UTC)
 * 3) Gamma Caeli, according to SIMBAD, is a K3III. --Marshallsumter (talk) 10:40, 15 August 2012 (UTC)
 * 4) Delta Caeli, according to SIMBAD, is a B2IV-V. --Marshallsumter (talk) 10:40, 15 August 2012 (UTC)

Hypotheses

 * 1) The first cyan source in Caelum may have been observed some 42,000 b2k.

title: Sources/First cyan source in Caelum


text:
Beaver pipes

Beaver Pipes are a non-destructive way of controlling beaver activity in an ecosystem. The process of building beaver pipes is quite simple, and often serves as a permanent way to prevent beavers from damming water.

Many articles exist on different ways to build beaver pipes, however, all have one common theme: divert water downstream while allowing beavers to continue to build their dams to whatever size they choose. This photo shows beaver pipes constructed at Enzo Creek Nature Sanctuary, a private nature sanctuary near Big Rapids, Michigan.

Managers of this watershed experienced the all too common problem with beavers: their tenacious desire to dam water. The water level at Enzo Creek Nature Sanctuary had been growing year after year, impacting the fauna in Hunt Marsh, an 18-acre (73,000 m2) wetland which serves as both a waterfowl nesting area and refuge.

Managers physically removed the dams from Enzo Creek, the only outlet to the watershed; however, beavers would quickly work to rebuild them.

Lethal removal of beavers from the marsh was contrary to the mission of the sanctuary, so this non-lethal method was approved and adopted.

Construction
The construction of these beaver pipes was a two day project for one person and involved the following steps:

 * 1) Slowly lower the water level behind the dam. Physically cut a notch in the position on the dam where beaver pipes are to be installed. In this case, a shovel and pitchfork were used. It is important to note that this process may take a significant amount of time depending on the volume of water already in reserve behind the dam. In the case at Cielo, this dam controlled the water level in a very small pond and was drained to the desired level in about two days. It is suggested that prior to beginning this project, a marker be installed that shows the target water depth so installers know when the correct water depth has been achieved.
 * 2) Insert pipes across the notch. In this case, as shown, the upstream pipes are made from 10-foot (3.0 m) sections of perforated 4" PVC pipe. The perforations allow water to enter the pipes. The upstream ends of the pipes have a 90 degree elbow pointing downward into the water, and are anchored to steel fence post supports at the upstream end of the pipes as well as in the notch. The perforated pipe is joined to non-perforated pipe at the notch, and then appropriate elbows are used to drop the pipes to the downstream position for water to exit. Threaded caps are placed on the final section, and then plugged with end caps. Steel fence post supports are used on the downstream portion as well to hold the pipes in position. Flexible galvanized wire is best to attache the pipes to the posts.
 * 3) Beavers complete the project. With the downstream pipes capped, water will continue to run through the notch. As shown in the photo, beavers will quickly move to dam the flowing water over and around the pipes. As they do this, the water level will rise behind the pipes. After the beavers have a good seal established over and around the pipes, remove the end caps and the water level upstream will reach equilibrium with the level of the upstream pipes. Remember it is the sound of running water that will attract beavers to build a dam, so the further downstream you divert the water, the better.

title: Beaver pipes


title: File:Wave.3.A.FKdomain.20120815.pdf


text: File:Convergence plot.svg

title: File:Convergence plot.svg


text:
User:Egm6322.s12.team2.steele.m2/Mtg1

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg1.djvu|Page 1-1]]
… the “big three” technologies for the numerical solution of PDEs, which came into their own in successive decades: 1950s: Finite difference methods 1960s: Finite element methods 1970s: Spectral methods Trefethen 2000, Spectral methods, p.ix. It’s all about solving differential equations (Des) (ODEs and PDEs): Deterministic or stoachastic (for uncertainty quantification). 3 tightly-connected courses: (Disclaimer:  My course organization) EGM 6321 Principles of Engineering Analysis 1 (PEA1): Analytical methods for solving 1st-order and 2nd-order ODEs (Ordinary Differential Equations), orthogonal polynomials, tools for spectral method. EGM 6341 Numerical methods 1 (NM1): Numerical integration of functions (quadratures) and ODEs, finite difference method, tools for spectral method. EML 5526 Finite Element Method (FE1): Solving linear PDEs, steady-state and transient problems.

[[media:Pea1.f11.mtg1.djvu|Page 1-2]]
A little more details … PEA1: Legendre differential equation (DE) Legendre polynomials (orthogonal) Self-adjoint DE 				Fourier-Legendre series Analytical solution Approximation for spectral method More … NM1: Gauss-Legendre quadrature Integrating ODEs in time Clenshaw-Curtis quadrature for spectral method More… FE1: Self-adjoint (transient) PDEs Evaluating FE matrices Solving semi-discrete equations in time More …

[[media:Pea1.f11.mtg1.djvu|Page 1-3]]
Course wiki: http://en.wikiversity.org/wiki/User:Egm6321.f11 (en = English, Egm6321=course number, f11=fall 2011, my wiki username) Course info: Click at course title Syllabus: From course info Other important info: Policy, report guidelines, … Also access from Sakai 	https://lss.at.ufl.edu/ Web search for “vu-quoc wikiversity”

Important links Lecture plan: Including lecture notes, references, inspiring quotations, and other info related to the course. See Fall 2010, Fall 2009. Report table: Reports, for 30% course grade, are “Exam 3” in 7 installments (“take home”, done in team, open book, open notes); formal project reports, NOT the usual HW for 10% course grade. See examples in Fall 2010, Fall 2009. Writing tools: Important docs for efficient wiki writing; read Writing tools, pros and cons Almost-WYSIWYG wiki writing Other courses

[[media:Pea1.f11.mtg1.djvu|Page 1-4]]
HW: Read course info, syllabus, policy Browse Lecture Plan Fall 2011 (to be updated frequently), Fall 2010, Fall 2009 Browse Report Table Fall 2010, Fall 2009 Read Writing tools, pros and cons Almost-WYSIWYG wiki writing Report guidelines Open skype account, and fill team-formation form (to be sent out by e-mail) (end HW) /// Note: HW = assignments, no written report required Rm.n = Report number “m”, problem number “n “. 3 slashes (///) mark the end of a section that begins with a blue, underlined title. Page number 1-4 = Meeting number 1, page 4. (end Note) /// Enroll = Agree with teaching and learning style (“Open-source” teaching and learning)

[[media:Pea1.f11.mtg1.djvu|Page 1-5]]
Note: Class mailing lists 3 class mailing lists = 1 for on-campus students + 2 for off-campus students

Important course info, announcements, etc. are conveyed by e-mail via the class mailing lists.

It is important that you are on the class mailing list. I sent out a test e-mail to the class mailing lists; if you did not receive this test e-mail, that means your e-mail address was not on one of the lists.

If you are registered for the course, and did NOT receive my test e-mail, please send me an e-mail with the subject header “PEA1: Add e-mail to class mailing list” Subject tag for ALL e-mails related to the course And go see the registrar to have your gatorlink e-mail in the classroll.

[[media:Pea1.f11.mtg1.djvu|Page 1-6]]
Motivation: High-speed trains German Transrapid Emsland 500 km/h: youtube video, Uploaded by TransrapidSupporter on Feb 14, 2007

title: User:Egm6322.s12.team2.steele.m2/Mtg1


text:
User:Egm6322.s12.team2.steele.m2/Mtg2

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg2.djvu|Page 2-1]]
Motivation: High-speed trains (cont’d) German Transrapid (electromagnetic attraction) Japanese Maglev (electrodynamic repulsion) French TGV (wheel on rail) Vu-Quoc & Olsson 1989 CMAME: Vehicle/structure interaction, where vehicle is the high-speed maglev and the structure is the flexible guideway.

=Equations of Motion (EOM) of Wheel/Magnet=

[[media:Pea1.f11.mtg2.djvu|Page 2-2]]
$$Y^1(t) =$$ nominal position of wheel (without guideway deformation) $$S = x^1 =$$ horizontal coordinate $$u^1(S, t) =$$ axial deformation (displacement) of guideway, where t is the time parameter $$u^2(S, t) =$$ transverse deformation (displacement) of guideway $$\displaystyle u^2_(,s) := \frac{\partial u^2(S, t)}{\partial S} =$$ guideway slope (small deformation)
 * = -> “equal by definition (non symmetric”	Note latex code for reuse and for training. Demo with online latex equation editor.

Note: latex code Use \display style to prevent fraction from being squished down, i.e., preserve the font size in the numerator and in the denominator.

[[media:Pea1.f11.mtg2.djvu|Page 2-3]]
Note: Equal-by-definition symbols
 * = 	non-symmetric symbol, good to use, flexible.

The quantity on the side of the colon is defined by the quantity on the side of the equal sign. e.g.,		A := B		A is defined by B 			A =: B		B is defined by A $$\overset{=}$$ symmetric symbol, not clear $$A \overset{=} B$$	A is defined by B? OR B is defined by A? Need a convention: left-hand side defined by right-hand side (read “def” from left to right) $$\overset{\Delta}{=}$$ similar to $$\overset{=}$$, but worse, since at least “def” has an implicit reading direction from left to right, whereas  is completely symmetric. ///

[[media:Pea1.f11.mtg2.djvu|Page 2-4]]
Axial displacement of guideway under moving wheel/magnet

$$\displaystyle \left. u^1(S,t) \right|_{S=Y^1(t)}$$ is  evaluated at $$S = Y^1(t)$$ Note: latex code “\left.” Is used to balance “\right|” to get nothing on the left and the vertical bar | on the right. Note the dot “.” (after “left”).

General setting: First total time derivative:

[[media:Pea1.f11.mtg2.djvu|Page 2-5]]
With $$\displaystyle \dot Y^1 := \frac{dY^1(t)}{dt}$$ R*1.1: Second total time derivative Show that Similarly for the other partial derivatives.

title: User:Egm6322.s12.team2.steele.m2/Mtg2


text:
User:Egm6322.s12.team2.steele.m2/Mtg3

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg3.djvu|Page 3-1]]
Note: R*m.n = Problem either (i) already explained in lectures in previous versions, or (ii) already assigned with solution in reports in previous years. You can do one of the following: Solve the problem without looking at existing solutions (lecture notes or reports). Put a boxed comment at the top of your solution saying that you did the solution on your own, without looking at existing solution. Example: EGM 6341 Spring 2011 (NM1 S11) Team 5 HW5.1 Look at existing solutions to get help for the solution. Need to (i) refer to the relevant documents (lecture notes, reports), with proper medawiki links, and (ii) put a boxed comment at the top of your solution indicating the improvements that you brought to the existing solutions (e.g., provide clearer explanation for “dummies”, deeper theoretical background or applications, correct misprints/errors, etc., or if your solution is completely different, then point out the differences). Example: PEA1 F10 Team 6 HW1.1 Ref[7].NM1 S11 Team 5 HW1.3. (need a link to “existing solution”)

[[media:Pea1.f11.mtg3.djvu|Page 3-2]]
[[media:Pea1.f11.mtg2.djvu|Eqn(2) p.2-4:]]

With the acceleration a being one of the following quantities $$\ddot u^1$$, $$\ddot u^2$$, $$\ddot Y^1$$ R*1.2: Derive (3) p.2-4 and (1) p.2-5, and show the similarity with the derivation of the Coriolis force. Note: Number (1) P. 2-5 is similar to the acceleration of a particle in a rotating frame that gives rise to the Coriolis force (undergraduate dynamics). Number (3) p.2-4 is related to the material time derivative and the Reynolds Transport Theorem in Continuum Mechanics (which covers Heat, Solids, Fluids, Electromagnetics) Intelligence consists of this; that we recognize the similarity between different things and the difference between similar things. Montesquieu (1689-1755)

=Equations of Motion (EOM) of Wheel/Magnet =

[[media:Pea1.f11.mtg3.djvu|Page 3-3]]
Note: Terms with $$\dot Y^1$$ are not due to damping, but to convection. Ref: VQ&O 1989 CMAME Eqs.(2.1bc). There is a misprint in (2.1b).

[[media:Pea1.f11.mtg3.djvu|Page 3-4]]
M = mass of wheel/magnet. From (2) p.3-3:   is horizontal force acting on wheel/magnet

Dimension of $$F^1$$		force [1] = 1

Second derivative $$\frac{\partial^2 u^2}{\partial S^2}$$ superscript Curvature	power $$(\partial S)^2$$ $$\displaystyle [u^2] = L \Rightarrow \underbrace{[u^2_{,SS}]}_{\color{blue}{\displaystyle \frac{\partial^2 u^2}{\partial S^2}}}$$

[[media:Pea1.f11.mtg3.djvu|Page 3-5]]
R*1.3: Analyze the dimension of all terms in (2) p.3-3, and provide the physical meaning. Numer (1) P.3-3: “Ordinary” Differential Equation (ODE) Order = Highest order of derivative Nonlinearity = What is linearity? Use intuition for now, formal definition coming soon. EOM (1) p.3-3 = Nonlinear 2nd-order ODE(N2-ODE) System has 3 unknown functions: $$Y^1(t), \ u^1(S,t), \ u^2S,t)$$ 1 nonlinear ODE for $$Y^1(t)$$ 2 PDEs for $$u^1(S,t)$$ and $$u^2(S,t)$$ . PDE = Partial Differential Equations 3 equations are coupled: use numerical methods; see Vu-Quoc & Olsson 1989 (2.7ab)

title: User:Egm6322.s12.team2.steele.m2/Mtg3


text:
User:Egm6322.s12.team2.steele.m2/Mtg4

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg4.djvu|Mtg 4]] Tue, 30 Aug 11
=Simplify for Analytical Solution=

[[media:Pea1.f11.mtg4.djvu|Page 4-1]]
=Simplify for Analytical Solution= Complex 		 Simplified 2nd order		=	2nd order Nonlinear		-> 	linear Unknown varying	->	known varying coefficients Coefficients Note: The formulas for the coefficients $$c_i (Y^1,t)$$ for i=0, 1, 2, 3 are known, but not their values until $$u^1$$ and $$u^2$$ are known (i.e., solved for). General structure of Linear 2nd-order ODEs with varying coefficients (L2-ODE-VC)

X = independent variables Y(x) = dependent variable (unknown function to be solved for)

[[media:Pea1.f11.mtg4.djvu|Page 4-2]]
Ref: King 2003, Differential equations, p.3. Caveat: “… “exact” solutions of a differential equation … Winsberg appears to believe that numerically evaluating the closed-form solutions to the equation provides the best results. However, it’s well known that for some functions, including the Airy functions and the Hankel functions, approximating the differential equation is more numerically precise than evaluating the closed-form function directly. In general, closed-form solutions won’t give more insight than will the differential equation itself.” F. Sullivan, Reviewer of Science in the Age of Computer Simulation by E. Winsberg 2010, Physics Today, Aug. 2011, p. 50. See, e.g., SPECFUN – Special Function Evaluation http://people.sc.fsu.edu/~jburkardt/f_src/specfun/specfun.html

[[media:Pea1.f11.mtg4.djvu|Page 4-3]]
Motivation: Many linear PDEs in engineering applications can be solved by separation of variables. Some examples include, but are not limited to: Heat, solids, fluids, acoustics and electromagnetic. Examples of these PDEs are:

Ref: PEA1 F09 Mtgs 28, 29, 30 A point x in 3-D has Cartesian coordinates $$(x_1, x_2, x_3)$$ and is written as

Consider the separation of variables $$X(x) = X_1(x_1) X_2(x_2) X_3(x_3)$$ $$x = (x_1, x_2, x_3)$$

[[media:Pea1.f11.mtg4.djvu|Page 4-4]]
See Cartesian curve for $$x = (x_1, x_2, x_3)$$. See Curvilinear curve for $$\xi = (\xi_1, \xi_2, \xi_3)$$. A point $$\xi$$ in 3-D has curvilinear coordinates $$(\xi_1, \xi_2, \xi_3)$$ and is written as

Consider the separation of variables:

Separated equations for i=1,2,3

[[media:Pea1.f11.mtg4.djvu|Page 4-5]]
R1.4: Draw the polar coordinate lines, in a 2-D plane emanating from a point, not at the origin. Change symbols to simplify: $$\xi_i \rightarrow x$$ $$X_i(\xi_i) \rightarrow y(x)$$

$$f_i(\xi_i) \rightarrow a_0(x)$$ R1.5: Show that (3) p.4-4 becomes

Note that $$a_1(x) = \frac{g^\prime (x)}{g(x)}$$ Number (2) is a particular case of (1) p.4-1, which in turn can be written as

[[media:Pea1.f11.mtg4.djvu|Page 4-6]]
=Definition: Linearity= Consider an operator $$F(\cdot)$$ is linear if and only if (iff)

Note: $$\forall$$ = forall $$\in$$ = in, belonging to 	$$\exists$$ = there exists $$\mathbb R$$ = set of all real numbers (bb = blackboard for \mathbb)

title: User:Egm6322.s12.team2.steele.m2/Mtg4


text:
User:Egm6322.s12.team2.steele.m2/Mtg5

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg5.djvu|Page 5-1]]
Example: Derivative operator

Let $$u (x)$$ and $$v(x)$$ be any 2 functions of x. Let $$\alpha$$ and $$\beta$$ be any 2 real numbers. We write $$\forall u,v: \mathbb R \rightarrow \mathbb R$$ Domain		range

$$x \mapsto u(x)$$ $$x \in \mathbb{R}$$			$$u(x) \in \mathbb{R}$$ Note: Vertical bar	value (image) of u at x

$$x \mapsto v(x)$$ $$\forall \alpha, \beta \in \mathbb{R}$$

[[media:Pea1.f11.mtg5.djvu|Page 5-2]]
Note: $$\rightarrow$$	set-to-set mapping $$\mapsto$$	point-to-point mapping (end Note)	/// (end Example) /// Example: Matrix algebra Consider the following operator:

[[media:Pea1.f11.mtg5.djvu|Page 5-3]]
$$\forall \mathbf{u}, \mathbf{v} \in \mathbb{R}^{m \times 1}$$ And any two real numbers, i.e., $$\forall \alpha, \beta \in \mathbb{R}$$ We have the matrix A as a linear operator since:

[[media:Pea1.f11.mtg5.djvu|Page 5-4]]
Nonlinear = not linear, i.e., not satisfying [[media:Pea1.f11.mtg4.djvu|Eqn(3) p.4-6:]]

Example: In the EOM (1) p. 3-3, the first term $$c_3(Y^1, t)\ddot{Y^1}$$ is nonlinear 2nd-order, thus making [[media:Pea1.f11.mtg3.djvu|Eqn(1) p.3-3]]a Nonlinear 2nd-order ODE (N2-ODE).

Curvature	$$\chi = \frac{1}{R}$$

[[media:Pea1.f11.mtg5.djvu|Page 5-5]]
R*1.6: Show $$c_e(Y^1,t)\ddot{Y^1}$$ is nonlinear with respect to. Back to the general L2-ODE-VC [[media:Pea1.f11.mtg4.djvu|Eqn(1) p.4-1.]]

title: User:Egm6322.s12.team2.steele.m2/Mtg5


text:
User:Egm6322.s12.team2.steele.m2/Mtg6

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg6.djvu|Page 6-1]]
L2-ODE-VC: 2nd order Need 2 conditions to solve for 2 constants 2 types of problems: BVP and IVP 1.	Boundary Value Problem (BVP) Solution domain: Closed interval [a,b] Closed = End points a and b are included in domain. Boundary conditions: Prescribe known values

2.	Initial Value Problem (IVP) Solution domain: (Semi) Open interval $$[a, +\infty) \equiv [a,+\infty[$$ .	Open = End point $$+\infty$$ included in domain. Initial conditions: Prescribe known values $$y(a) \alpha, \ y\prime (a) = \beta$$

[[media:Pea1.f11.mtg6.djvu|Page 6-2]]
For BVPs and IVPs: Two facts A.	Existence and uniqueness of solution Solution exists and is unique; no proof here; accept this result. B.	Superposition of solutions L2-ODE-VC [[media:Pea1.f11.mtg5.djvu|Eqn(1) p.5-5:]] (2)

[[media:Pea1.f11.mtg6.djvu|Page 6-3]]
$$L_2(\cdot)$$ is linear, thus superposition is applicable. R1.7: Show that $$L_2(\cdot)$$ is linear. 2 types of solutions: Homogeneous and particular

B1. Homogeneous solution

B2. Particular solution $$y_P(x)$$

Solution:

$$L_2(\cdot)$$ is linear; hence

[[media:Pea1.f11.mtg6.djvu|Page 6-4]]
Since the solution for (1) p.6-2 exists and is unique, then, due to (4) – (5) p.6-3, the solution for (1) p.6-2 should be Question: $$L_2(\cdot)$$ is 2nd-order differential operator; there are 2 integration constants, thus we need 2 boundary conditions or 2 initial conditions to solve for these 2 constants; where are those 2 constants? Answer: There are actually 2 linearly independent homogeneous solutions: $$y^1_H(x)$$ and $$y^2_H(x)$$. The homogeneous solution $$y_H(x)$$ is a linear combination of $$y^1_H(x)$$ and $$y^2_H(x)$$: 2 integration constants

[[media:Pea1.f11.mtg6.djvu|Page 6-5]]
The 2 integration constants are determined by requiring the solution y(x) to satisfy the boundary conditions (1) p.6-1 or the initial conditions (2) p.6-1: Ref: cf. King 2003 p.4 R1.8: Find the integration constants in terms of The boundary conditions (1) p.6-1	-> F10 The initial conditions (2) p.6-1 		-> F11 Example: Legendre differential operator/equation King 2003 p. 31

title: User:Egm6322.s12.team2.steele.m2/Mtg6


text:
User:Egm6322.s12.team2.steele.m2/Mtg7

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg7.djvu|Page 7-1]]
Example: Legendre differential operator/equation King 2003 p.31 Two homogeneous solutions:

R*2.1: Verify that $$L_2(y^1_H) = L_2(y^2_H) = 0$$ Note: cf. King 2003 p. 33 $$\{P_n(x), n=0, 1, 2, \ldots\}$$ Legendre polynomials $$\{Q_n(x), n=0,1,2, \ldots\}$$ Legendre functions $$\{[P_n(x), Q_n(x)], n=0,1,2, \ldots\}$$ Legendre functions =Methods of reduction of order=

[[media:Pea1.f11.mtg7.djvu|Page 7-2]]
It may be easier to solve an ODE with lower order than the original ODE with higher order. =Reduction-Of-Order Method 0 (Zero)= Missing dependent variable y(x) Independent variable General nonlinear ODE of order n (Nn-ODE) (without missing y(x)) Nonlinear function Example:

Case of missing dependent variable y(x)

Missing y(x)

[[media:Pea1.f11.mtg7.djvu|Page 7-3]]
Example: See Example p.7-2; delete log y in (2) p.7-2 (1) 						Y(x) missing Reduce order by defining Order (n-1), reduced from order n If (3) can be solved for p(x), then

Integration constant Primitive or antiderivative or indefinite integral

[[media:Pea1.f11.mtg7.djvu|Page 7-4]]
Also		dummy integration variable (2) p.7-3:

Primitive cf. King 2003 p.5 Example: L2-ODE-CC (Constant Coefficients) Y(x) missing

[[media:Pea1.f11.mtg7.djvu|Page 7-5]]
Integration constant is k Note: Method: Euler integration factor, coming soon. For the impatient: See F10 Mtg 7. R*2.2: Verify that (1) is indeed the solution for (6) p.7-4. Solution for (5) p.7-4: =Reduction-Of-Order Method 1: Exact N1-ODEs=

[[media:Pea1.f11.mtg7.djvu|Page 7-6]]
Euler Integrating factor method (IFM) Particular class of N1-ODEs: Linear in $$y\prime$$

$$\frac{dy}{dx}$$

R*2.3: Show that (2) is linear in y’, and that (2) is in general an N1-ODE. But (2) is not the most general N1-ODE as represented by (1). Give an example of a ore general N1-ODE.

title: User:Egm6322.s12.team2.steele.m2/Mtg7


text:
User:Egm6322.s12.team2.steele.m2/Mtg8

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg8.djvu|Page 8-1]]
Euler integrating factor method (cont’d p.7-6) Example: for [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6,]] a particular case of N1-ODE R*2.4: Verify that (3) is a N1-ODE.

Note: While (2) p.7-6 is in general an N1-ODE, it is still a particular N1-ODE, not the most general N1-ODE, which has the form [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-6.]] Satisfied (1) p.7-6 but can be converted to (2) p.7-6 with the definition:

[[media:Pea1.f11.mtg8.djvu|Page 8-2]]
Satisifies (1) p.7-6 but can be converted to (2) p.7-6 with Condition for (4) to hold If (5) is satisfied, then (4), which satisfies (1) p.7-6, can be converted to (2) p.7-6 with

[[media:Pea1.f11.mtg8.djvu|Page 8-3]]
Note: Functions like M(x,y) and N(x,y) include constants (think of $$x^0 = y^0 = 1$$ ). Power zero Q: So when do we have a general N1-ODE of the form (1) p.7-6 that cannot be converted into the particular form (2) p.7-6? A.	Eqs. (3) – (4) p.8-1, and (2), (4) p.8-2 can all be expressed more generally as Where F(y’) is any function of y’ (nonlinear in general). If the function $$F(\cdot)$$ has no explicit inverse $$F^{-1}(\cdot)$$, then (2) cannot be put in the particular form (2) p.7-6.

[[media:Pea1.f11.mtg8.djvu|Page 8-4]]
Definition: Exact N1-ODE A general N1-ODE of the form (1) p.7-6 is exact if there exists a function $$\phi(x,y)$$ such that

For any constant k, the level set

Leads to the N1-ODE (1) p.7-6: i.e. (4) is a first integral of (5).

[[media:Pea1.f11.mtg8.djvu|Page 8-5]]
Note: Given a general N1-ODE (1) p.7-6, if a function  can be found such that (3)-(4) p.8-4 are satisfied, then we have effectively reduced the order of the N1-ODE (1) p.7-6 from 1 to 0, as represented by (4) p.8-4. The solution y(x) is then obtained if we can solve (4) p.8-4 for y in terms of x, with k being the integration constant.

Let’s carry out the total derivative in (4) p. 8-4 (remember the first total time derivative [[media:Pea1.f11.mtg2.djvu|Eqn(3) p.2-4,]] in high-speed train modeling and Montesquieu p.3-2) Which has the particular form (2) p.7-6 M(x,y) + N(x,y)y’ = 0 With

[[media:Pea1.f11.mtg8.djvu|Page 8-6]]
=First Exactness Condition for N1-ODEs:= Thus the necessary condition for an N1-ODE to be exact is that it has the particular form (2) p.7-6. In other words, if an N1-ODE cannot be put in the particular form (2) p.7-6, then it cannot be exact. Example: Reverse engineering:  Given $$\phi (x,y)$$, find the N1-ODE with particular form (2) p.7-6.

M(x,y)		N(x,y)

title: User:Egm6322.s12.team2.steele.m2/Mtg8


text:
User:Egm6322.s12.team2.steele.m2/Mtg9

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg9.djvu|Page 9-1]]
Note: Linearly-independent functions, p.6-4 1)	Case of vectors (restricted to $$\mathbb R^2$$ for simplicity) Linearly dependent		Linearly independent 2)	Case of functions R*2.6: The homogeneous solutions $$y^1_H(x)$$ in [[media:Pea1.f11.mtg7.djvu|Eqn(3) p.7-1]] and $$y^2_H(x)$$ in [[media:Pea1.f11.mtg7.djvu|Eqn(4) p.7-1]]are linearly independent, i.e., show that i.e. for any given $$\alpha$$, show that

[[media:Pea1.f11.mtg9.djvu|Page 9-2]]
Plot $$y^1_H(x)$$ and $$y^2_H(x)$$ HW: Read King 2003, Appendix 5, ODEs (particular cases of lectures), p.511-516. For example, compare $$g(y) \frac{dy}{dt} = f(t)$$		King 2003 p.511(A5.1) Or translated into our notation as To the particular form [[media:Pea1.f11.mtg6.djvu|Eqn(2) p.6-6]] $$M(x,y) + N(x,y) y\prime = 0$$ R*2.7: Consider the following function

[[media:Pea1.f11.mtg9.djvu|Page 9-3]]
And show that (3) p.9-2 is an N1-ODE. R*2.8 Does the following N1-ODE satisfy the first exactness condition? Hint: See [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-3]] and [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-4]].

Q: If $$\phi (x,y)$$ exists, what is then the relationship between$$M(x,y) = \phi_x (x,y) \ and \ N(x,y) = \phi_x (x,y)$$ ? Can M(x,y) and N(x,y) be chosen arbitrarily? NO. Assuming that  is smooth, we have: [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-5]] and [[media:Pea1.f11.mtg8.djvu|Eqn(1) p.8-6]] ->    $$M_y(x,y)$$  $$N_x(x,y)$$

=Second Exactness Condition:=

[[media:Pea1.f11.mtg9.djvu|Page 9-4]]
R*2.9 (PEA2 S09) Review calculus, and find the minimum degree of differentiability of the function $$\phi (x,y)$$ such that (2) p.9-3 is satisfied. State the full theorem and provide a proof.

title: User:Egm6322.s12.team2.steele.m2/Mtg9


text:
User:Egm6322.s12.team2.steele.m2/Mtg10

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg10.djvu|Page 10-1]]
Note: Smooth functions are functions that are infinitely differentiable, e.g., cos x, ex, log x, x, x2, etc. Examples of non-smooth functions: 1.	Heaviside (step) function The derivative is a generalized function: (1)

[[media:Pea1.f11.mtg10.djvu|Page 10-2]]
2.	Hat (or roof) function (application in FEM): The derivative is a generalized function: The Hat function is not differentiable in the classical sense at $$(x_1, x_2, x_3)$$ (the “kinks”), i.e.,

[[media:Pea1.f11.mtg10.djvu|Page 10-3]]
Limit on left of :  	$$\lim_{x \to x_i^-} \frac{df(x)}{dx}$$ Think: $$x_i^- = x_i - \epsilon$$, where $$\epsilon$$ is very small. Limit on right of $$x_i$$: $$\lim_{x \to x_i^+} \frac{df(x)}{dx}$$ Think: $$x_i^+ = x_i + \epsilon$$, where $$\epsilon$$ is very small. So, smooth -> no kinks (in any derivative order) (end NOTE) Summary: Two exactness conditions for N1-ODEs 1.	1st condition: Particular form [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6]] (see also p.8-6) 2.	2nd condition: [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]]

[[media:Pea1.f11.mtg10.djvu|Page 10-4]]
Example: See Example on p.8-6 Verify 2 exactness conditions for [[media:Pea1.f11.mtg8.djvu|Eqn(2) p.8-6]]. 1.	1st exactness condition (2) p.7-6: (2) p.8-6 satisfies $$M(x,y) + N(x,y) y\prime = 0$$		(2) p.7-6 2.	2nd exactness condition [[media:Pea1.f11.mtg8.djvu|Eqn(3) p.8-3]]: $$M_y(x,y) = 0$$		Thus (2) p.8-6 is exact $$N_x(x,y) = 0$$ Pretend that $$\phi(x,y)$$ was not known, and could be found from the above data (by some method) to be as given in (2) p.8-6. Then the solution y(x) is obtained by solving (2) p.7-6 for y in terms of x

Integration constant

[[media:Pea1.f11.mtg10.djvu|Page 10-5]]
R*2.10: Verify that (1) p.10-4 is indeed the solution for the N1-ODE (2) p.8-6. Q: Suppose that an N1-ODE satisfies the 1st exactness condition (2) p.7-6, but not the 2nd exactness condition [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]]. Is it possible to transform the original N1-ODE into an exact N1-ODE? A: yes, use the following … Euler Integrating factor method (IFM) Consider an N1-ODE with the particular form $$M(x,y) + N(x,y) y\prime = 0$$			[[media:Pea1.f11.mtg6.djvu|Eqn(2) p.6-6]]

i.e., satisfying the 1st exactness condition, but such that i.e., not satisfying the 2nd exactness condition (3) p. 8-3.

=Integrating Factor=

[[media:Pea1.f11.mtg10.djvu|Page 10-6]]
Euler’s idea: Find an integrating factor h(x,y) such that the following N1-ODE is exact:

title: User:Egm6322.s12.team2.steele.m2/Mtg10


text:
User:Egm6322.s12.team2.steele.m2/Mtg11

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg11.djvu|Page 11-1]]
Euler Integrating factor method (IFM) p.10-6 cont’d [[media:Pea1.f11.mtg10.djvu|Eqn(1) p.10-6]]can be written as

Apply 2nd exactness condition [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]] to find h:

Thus for (1) to be exact: Solving (4) for the integrating factor h(x,y) is usually not easy. R2.11: Explain why.

[[media:Pea1.f11.mtg11.djvu|Page 11-2]]
Consider 2 cases: Case 1: Suppose $$h_y(x,y) = 0$$, thus h is a function of x only; then (4) p.11-1 becomes: See (3) p.7-4

[[media:Pea1.f11.mtg11.djvu|Page 11-3]]
Case 2: Suppose $$h_x(x,y) = 0$$, thus h is a function of y only; then (4) p. 11-1 becomes: R*2.12: Find h using (1). Application: General non-homogeneous L1-ODE-VC If $$P(x) \neq 0 \ {\rm then} \ \forall x$$

[[media:Pea1.f11.mtg11.djvu|Page 11-4]]
Check condition (2) p.11-2:

[[media:Pea1.f11.mtg11.djvu|Page 11-5]]
=Application: Specific Non-Homogeneous L1-ODE-VC= It can be shown that R*2.13: Show (3). R*2.14: Solve the general L1-ODE-VC: To be continued…

title: User:Egm6322.s12.team2.steele.m2/Mtg11


text:
User:Egm6322.s12.team2.steele.m2/Mtg12

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg12.djvu|Page 12-1]]
R*2.14: p.11-5 Cont’d 2.	Find y(x) in terms of $$a_x, \ a_0(x), \ b(x).$$ 3.	3. $$a_1(x) = x^2 + 1$$ $$a_0(x) = x$$ $$b(x) = 2x$$ Note: 1.	Cf. King 2003, Appendix 5, p.512, where the particular L1-ODE-VC in [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-3]] was considered: $$y\prime + P(x) y = Q(x)$$ “If we multiply through by$$\exp{\int^x P(t)dt}$$, where …” 2.	King 2003 did not derive the integrating factor [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-4]]; it was like pulling a rabbit out of a hat.

[[media:Pea1.f11.mtg12.djvu|Page 12-2]]
R*2.15: Since (2)-(3) p.11-3 is an L1-ODE-VC, there should be only one integration constant, not two. Show that the integration constant $$k_1$$ in (3) p.11-4 is not necessary, i.e., only $$k_2$$ in (1) p.11-5 is necessary. R*2.16: Show that the solution of (3) p.11-3 in [[media:Pea1.f11.mtg11.djvu|Eqn(1) p.11-5]] agrees with the result presented in King 2003 p.512, i.e., Use (3) p.11-4 and (1) p.11-5 to identify A, $$y_H(x)$$ and $$y_P(x)$$. Compare your results with those in King 2003 p.512. R*2.17: Instead of identifying $$y_H(x)$$ from (3) p.11-4 and (1) p.11-5, solve the homogeneous counterpart of (3) p.11-3, i.e.,

How about ? Variation of parameters (soon).

=A Class of Exact N1-ODEs=

[[media:Pea1.f11.mtg12.djvu|Page 12-3]]
Recall the condition [[media:Pea1.f11.mtg11.djvu|Eqn(2) p.11-2]] for Case 1 $$(h_y(x,y) = 0)$$ ( To satisfy the above condition, consider (1): N(x,y) = N(x) $$\Rightarrow k_1(y) = k_1, \bar b(x,y) = \bar b(x)$$

[[media:Pea1.f11.mtg12.djvu|Page 12-4]]
Which is a particular L1-ODE-VC (not necessarily exact, but can be made exact by using the IFM). Application: Consider

i.e., select $$k_1 = 0$$ Also select $$k_2(x) = 10$$, so (1) becomes Which is a L1-ODE-VC that is either exact, or can be made exact by using the IFM.

[[media:Pea1.f11.mtg12.djvu|Page 12-5]]
R*2.18: If (4) p.12-4 is not exact, find the integrating factor h to make it exact. But (1) p.12-4 is a class of general L1-ODE-VC that are either exact, or can be made exact by the IFM. Q: Is it possible to exhibit a class of N1-ODEs that are either exact, or can be made exact by the IFM?

title: User:Egm6322.s12.team2.steele.m2/Mtg12


text:
User:Egm6322.s12.team2.steele.m2/Mtg13

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg13.djvu|Page 13-1]]
Note: Number [[media:Pea1.f11.mtg12.djvu|Eqn(1) p.12-4:]]is a particular L1-ODE-VC, more general than [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-3:]], which is similar to the L1-ODE-VC in King 2003 p. 512, since to obtain (3) p.11-3, we need to require that $$\bar b(x) \ne 0, \ \forall x$$ Q: Is it possible to exhibit a more general class of N1-ODEs that are either exact, or can be made exact by the IFM? (See p.12-5) IFM: Due to Euler, who made important contributions to many areas of mechanics (and applied mathematics), in particular celestial mechanics. “When a first order equation is not exact it is often possible to multiply the equation by a quantity, called an integrating factor, that makes it exact. Though integrating factors had been used in special problems of first order ordinary differential equations, it was Euler who realized (in the 1734/35 paper) that this concept furnishes a method … Clairaut independently introduced the idea of an integrating factor in his 1739 paper … All the elementary methods of solving first order equations were known by 1740.” Kline 1972, Mathematical thoughts …, v.2, p.476

[[media:Pea1.f11.mtg13.djvu|Page 13-2]]
Of Euler, Clairaut, and the Moon… “In a dispute with more than just scientific import, Alexis Clairaut, Leonhard Euler, and Jean le Rond d’ Alembert each employed their own strategies to establish that they were the first to understand a puzzling feature of the Moon’s orbit.” Bodenmann, The 18th-century battle over lunar motion, Physics Today, Jan 2010. Back to creating a more general class of N1-ODEs that are either exact, or can be made exact by the IFM. Consider a class of N1-ODEs of the form:

[[media:Pea1.f11.mtg13.djvu|Page 13-3]]
Where a(x), b(x), c(y) are arbitrary functions. R3.1 Show that the N1-ODE [[media:Pea1.f11.mtg13.djvu|Eqn(1) p.13-2]] satisfies the condition [[media:Pea1.f11.mtg11.djvu|Eqn(2) p.11-2:]] that an integrating factor h(x) can be found to render it exact, only if $$k_1(y) = d_1$$ (constant). Show that (1) p.13-2 includes (1) p.12-4 as particular case.

Integration constants

[[media:Pea1.f11.mtg13.djvu|Page 13-4]]
R*3.2: Show that (1) either is exact, or can be made exact by the IFM. Find the integrating factor h. R*3.3:

[[media:Pea1.f11.mtg13.djvu|Page 13-5]]
1.	Find an N1-ODE of the form (1) p.13-2 that is either exact or can be made exact by IFM. (See R3.1) 2.	Find the first integral $$\phi(x,y) = k$$ R3.4: Construct a class of N1-ODEs, which is the counterpart of (1) p.13-2, and satisfies the condition (1) p.11-3 that an integrating factor h(y) can be found to render it exact.

=Derivation of a more general class of N1-ODEs that are either exact, or can be made exact by the IFM= Based on the condition [[media:Pea1.f11.mtg11.djvu|Eqn(2) p.11-2]], consider:

[[media:Pea1.f11.mtg13.djvu|Page 13-6]]
If $$k_1(y) = d_1$$ (constant)  (see R3.1), then (2) p.11-2 is satisfied:

[[media:Pea1.f11.mtg13.djvu|Page 13-7]]
Hence a more general class of N1-ODEs that are either exact, or can be made exact by the IFM

Where  are arbitrary functions.

[[media:Pea1.f11.mtg13.djvu|Page 13-8]]
=L2-ODE-VC with missing dependent variable= (1) (2)	P.7-3: p(x) := y’(x)

title: User:Egm6322.s12.team2.steele.m2/Mtg13


text:
User:Egm6322.s12.team2.steele.m2/Mtg14

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg14.djvu|Page 14-0]]
Plan: Application (engineering): Motion of a particle in the air (e.g., soccer ball, projectile) Application: Control engineering – Linear systems Example: Coupled pendulums Note: Problem numbering in reports: Use the same numbering as in lecture notes, i.e., R*m.n or Rm.n (put the star when there is a star to distinguish starred problems from non-starred problems). Link to the corresponding lecture notes in all problem statements All figures: Category, license End of reports: Assignment table, signatures, tasks done. See F11 Team 4R1. \, in latex code

[[media:Pea1.f11.mtg14.djvu|Page 14-1]]
=Application (Engineering): Motion of a Particle in the Air (e.g., soccer ball, projectile)= See graph on p. 14-1. $$v := \parallel \boldsymbol v \parallel$$ Vector	norm = magnitude $$k, n \in \mathbb{R}$$	constants M	mass of particle=m G	acceleration of gravity=g

[[media:Pea1.f11.mtg14.djvu|Page 14-2]]
R*3.5: 1.	Derive the equations of motion

(1)-(2) form a System of Coupled N1-ODEs = SC-N1-ODEs (need numerical methods) 2. Particular case k=0 : Verify that y(x) is parabola. 3. Consider the case $$k \ne 0$$ and $$v_{x0} = 0$$

[[media:Pea1.f11.mtg14.djvu|Page 14-3]]
3.1 Is (1) either exact or can be made exact using IFM? Find $$v_y(t)$$ and $$y(t)$$ for m constant. (Note graph on p. 14-3). 3.2 Find $$v_y(t)$$ and  for “John Bernoulli posed other trajectory problems for the English, his particular  noire being Newton. Since the English and the Continentals were already at odds, the challenges were marked by bitterness and hostility. John Bernoulli then solved the problem of finding the motion of a projectile in a medium whose resistance is proportional to any power of the velocity.” Kline 1972, Mathematical thoughts…, v.2, p.475

[[media:Pea1.f11.mtg14.djvu|Page 14-4]]
=Application: Control Engineering – Linear Systems= Linear time-variant systems: $$\mathbf{x} : [t_0, +\infty) \rightarrow \mathbb R^{n \times 1}$$ 	state of system $$t \ \mapsto \ \mathbf x(t)$$ $$\mathbf{A}:[t_0, +\infty) \rightarrow \mathbb R^{n \times n}$$ $$\mathbf{B} : [t_0, +\infty) \rightarrow \mathbb R^{n \times m}$$ $$\mathbf{u} : [t_0 , +\infty) \rightarrow \mathbb R^{m \times 1}$$ Linear time-invariant systems: $$\mathbf{A}, \mathbf{B}$$ constant matrices

[[media:Pea1.f11.mtg14.djvu|Page 14-5]]
Example: Coupled pendulums See image with spring and pendulums.

c.f Bryson & Ho 1975, Applied optimal control, p.164.

[[media:Pea1.f11.mtg14.djvu|Page 14-6]]
R*3.6: 1.	Derive the equations of motion (1)-(2) p.14-5. 2.	Write (1)-(2) p.14-5 in matrix form (1) p.14-4 with

Col matrix		T=transpose 		row matrix

Col matrix Find $$\mathbf{A} \in \mathbb{R}^{4 \times 4}, \mathbf{B} \in \mathbb{R}^{4 \times 2}$$

title: User:Egm6322.s12.team2.steele.m2/Mtg14


text:
User:Egm6322.s12.team2.steele.m2/Mtg15

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg15.djvu|Page 15-0]]
Plan: SC-L1-ODE-VC : System of Coupled L1-ODE-VC SC-L1-ODE-CC: SC-L1-ODE with Constant Coefficients Particular case: when n=1 (L1-ODE-VC) Solution by IFM Generalization: SC-L1-ODE-CC Exponentiation of a matrix Generalization: SC-L1-ODE-VC State transition matrix Coupled pendulums: Integrate SC-L1-ODE-CC Note: p.14-0 cont’d

[[media:Pea1.f11.mtg15.djvu|Page 15-1]]
=SC-L1-ODE-VC: System of Coupled L1-ODE-VC= (linear time variant) SC-L1-ODE-CC: SC-L1-ODE with Constant Coefficients (linear time invariant) Particular case: When n=1 (L1-ODE-VC) L1-ODE-CC: R*3.7: Use IFM to show that the solution of (2) is

[[media:Pea1.f11.mtg15.djvu|Page 15-2]]
Identify the integrating factor, the homogeneous solution, and the particular solution. Show that the solution of the L1-ODE-VC (1) p.15-1 is Generalization: SC-L1-ODE-CC (constant coefficients)

[[media:Pea1.f11.mtg15.djvu|Page 15-3]]
Exponentiation of a matrix:

Nxn 	nxn		nxn		nxn		nxn Identity matrix R*3.8: Generalization (2) p.15-2 to the case of linear time-variant system [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4]]. See Eqn(1) p.15-2. Verify that your expression is indeed the solution for (1) p.14-4. Bryson & Ho 1975 (A4.9) p.450:

[[media:Pea1.f11.mtg15.djvu|Page 15-4]]
Compare (3) p.15-3 to (2) p.15-2 (for L1-ODE-CC):

The fundamental or state transition matrix $$\Phi$$ is related to the integrating factor. Properties of the state transition matrix $$\Phi$$ Nxn 		nxn

Nxn	nxn R*3.9: Verify that (1) satisfies (2)-(3).

[[media:Pea1.f11.mtg15.djvu|Page 15-5]]
R3.10: Free vibration of coupled pendulums p.14-5 Pendulums: $$m_1 g = 3, \ m_2 g = 6$$ No applied forces: $$u_1 = u_2 = 0$$ Initial conditions $$\theta_1(0) = 0, \ \dot \theta_1(0) = -2$$ $$\theta_2(0) = 0, \ \dot \theta_2(0) = +1$$ 1.	Use matlab’s ode45 command to integrate the system [[media:Pea1.f11.mtg14.djvu|Eqns(1)-(2) p.14-5]] in matrix form (1) p.14-4 for $$t \in [0,7]$$. Can also use equivalent Octave command. 2.	Use (2) p.15-2 to find the solution at the same time stations as in Q1. 3.	Plot $$\theta_1(t)$$ from Q1 and from Q2. 4.	Plot $$\theta_2(t)$$ from Q1 and from Q2.

title: User:Egm6322.s12.team2.steele.m2/Mtg15


text:
User:Egm6322.s12.team2.steele.m2/Mtg16

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg16.djvu|Page 16-0]]
Plan: SC-L1-ODE-CC: p.15-5 cont’d Example: Roll control of rocket Time constant 2nd-order ODEs (cont’d from p.13-8) 1st exactness condition for N2-ODEs 2nd exactness condition for N2-ODEs Application Solution for exact N2-ODE

[[media:Pea1.f11.mtg16.djvu|Page 16-1]]
SC-L1-ODE-CC: [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4:]] $$\mathbf{\dot X}(t) = \mathbf A(t) \mathbf x(t) + \mathbf B(t) \mathbf u(t)$$ R*3.11: use (2)-(3) p.15-4 [[media:Pea1.f11.mtg15.djvu|Eqns(2)-(3) p.15-4]] together with [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4:]] to show [[media:Pea1.f11.mtg15.djvu|Eqn(3) p.15-3]].

=Example: Roll Control of Rocket= Prevent rolling by activating the rocket ailerons $$\delta$$ = aileron deflection (angle) $$\phi$$ = roll angle $$\omega$$ = roll angular velocity Q = aileron effectiveness proportional to inverse of rocket roll intertia $$\tau$$ = roll time constant. See image of rocket in vertical position with air flow, force, and aileron.

[[media:Pea1.f11.mtg16.djvu|Page 16-2]]
Bryson & Ho 1975 p.169 Note: Time constant $$\tau$$

The smaller the time constant $$\tau$$ is, the larger the ratio $$\frac{1}{\tau}$$ is, and the faster $$\omega$$ goes down to zero.

[[media:Pea1.f11.mtg16.djvu|Page 16-3]]
R*3.12: Put (1)-(3) p.16-2 in the form of SC-L1-ODE-CC as in [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4:]] with A, B being constant matrices.

=2nd-order ODEs (cont’d from p.13-8)=

Number (1) is exact if there exists a function $$\phi(y\prime, y, x) = k_1$$

Use the same definition as in [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-3:]]

P(x) := y’(x)

[[media:Pea1.f11.mtg16.djvu|Page 16-4]]
Use (1) and (2) p.7-3 to rewrite (2) p.16-3 as

Number (2) is the required particular form of exact N2-ODEs, and is the equivalent counterpart of the particular form of exact N1-ODEs in [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6]].

[[media:Pea1.f11.mtg16.djvu|Page 16-5]]
1st exactness condition for N2-ODEs: Thus the necessary condition for an N2-ODE to be exact is that it has the particular form (2) p.16-4. In other words, if an N2-ODE cannot be put in the particular form (2) p.16-4, then it cannot be exact. Cf. p.8-6 2nd exactness condition for N2-ODEs

If both 1st and 2nd exactness conditions are satisfied, i.e., (2) p.16-4 and (1)-(2), then from (4) p.16-4, we have

[[media:Pea1.f11.mtg16.djvu|Page 16-6]]
Where h(x,y) is a function of integration (not necessarily a constant) to be determined using (3) p.16-4. Application: Consider the N2-ODE Comparing (1) to (2) p.16-4, it follows that

Thus (1) satisfies the 1st exactness condition. R*3.13: 2nd exactness condition 1.	Derive (2) p.16-5, i.e., the 2nd relation in the 2nd exactness condition, by differentiating the definition of g(x,y,y’) in (3) p.16-4 with respect to p := y’ defined in [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-3]]. (See F09 p.11-2)

[[media:Pea1.f11.mtg16.djvu|Page 16-7]]
2.	Derive (1) p.16-5, i.e, the 1st relation in the 2nd exactness condition. (See F09 p.11-3, Mtg 17) 3.	Verify that (1) p.16-6 satisfies the 2nd exactness condition

Solution for (1) p.16-6 (exact N2-ODE)

1st integration constant

Number (3) p.16-6:

(2) p.16-6

title: User:Egm6322.s12.team2.steele.m2/Mtg16


text:
User:Egm6322.s12.team2.steele.m2/Mtg17

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

Page 17-0
Plan: Application, exact N2-ODE (p.16-7 cont’d) Note: Comparing exactness conditions for N1-ODE vs those of N2-ODEs 2nd exactness condition of N2-ODEs: Derivation

[[media:Pea1.f11.mtg17.djvu|Page 17-1]]
=Application (p.16-7 cont’d): = [[media:Pea1.f11.mtg16.djvu|Eqns(4)-(5) p.16-7]]:

The first integral in [[media:Pea1.f11.mtg16.djvu|Eqn(3) p.16-7]] becomes

Step 2: Solve the reduced-order N1-ODE (4) (by separation of variables):

[[media:Pea1.f11.mtg17.djvu|Page 17-2]]
2 integration constants for the N2-ODE [[media:Pea1.f11.mtg16.djvu|Eqn(1) p.16-6.]] Note: Recall N1-ODE; 1st exactness condition [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6:]] $$G(x,y,y\prime) = 0$$ is exact if $$\exists \phi(x,y)$$ such that

=2nd Exactness Condition [[media:Pea1.f11.mtg9.djvu|Eqn(3) p.9-3]]: =

Cf. [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]]for N2-ODE Clairaut 1739-40

[[media:Pea1.f11.mtg17.djvu|Page 17-3]]
Then $$\phi(x,y) = k$$ is the 1st integral. 2nd exactness condition of N2-ODEs: Derivation 2nd relation (2) p.16-5: Easy, see R*3.13, Q1, p.16-6 1st relation (1) p.16-5: $$\phi(x,y,p)$$ has 3 arguments; hence 3 relations for the symmetry of mixed 2nd partial derivatives:

Circular permutation between x, y, and p Number [[media:Pea1.f11.mtg16.djvu|Eqn(3) p.16-4:]] $$g = \phi_x + \phi_y \, y\prime$$ Number [[media:Pea1.f11.mtg16.djvu|Eqn(4) p.16-4:]] $$f = \phi_p$$

[[media:Pea1.f11.mtg17.djvu|Page 17-4]]
Step 1: [[media:Pea1.f11.mtg17.djvu|Eqn(1c) p.17-3:]]

(1b) p.17-3 and (4) p.16-4:

Express $$\phi_y$$ in terms of (f,g,p):

[[media:Pea1.f11.mtg17.djvu|Page 17-5]]
Step 2: (1b) p.17-3 and (3)-(4) p.16-4:

With $$\phi_{xp} = \phi_{px} = f_x \, \phi_x$$ can be expressed in terms of $$(f,g,p)$$:

Step 3: Use (1a) p.17-3, i.e., $$\phi_{xy} = \phi_{yx}$$, and (5) p.17-4 and (3) to derive the 1st relation [[media:Pea1.f11.mtg16.djvu|Eqn(1) p.16-5]] of the 2nd exactness condition. (R*3.13, Question 2, p.16-6)

title: User:Egm6322.s12.team2.steele.m2/Mtg17


text:
User:Egm6322.s12.team2.steele.m2/Mtg18

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg18.djvu|Page 18-0]]
Plan: Application: Generate exact N2-ODEs Method: (reverse engineering) Examples

Notes: $$\Rightarrow$$Example: Roll control of rocket $$\mathcal{Q}$$ = aileron effectiveness, meaning, p.16-1 Time constant $$\tau$$

[[media:Pea1.f11.mtg18.djvu|Page 18-1]]
R*3.14: Find h(x,y). (1)	P.17-1: If we select $$h(x,y) = k_2.$$ (constant), Then $$h_x = h_y = 0$$, and (1) is satisfied. Without assuming a priority that h = constant, discuss the search for the solution of (1). =Application: Generate Exact N2-ODEs:  Method: (reverse engineering)= 1.	Select $$\phi(x,y,p)$$ and set: $$\phi(x,y,p) = k$$	[[media:Pea1.f11.mtg16.djvu|Eqn(3) p.16-7]] Differentiate (3) p.16-7 to obtain the corresponding exact N2-ODE: 2.	$$G(x,y,y\prime,y\prime\prime) = \frac{d}{dx} \phi(x,y,y\prime) = 0$$	[[media:Pea1.f11.mtg16.djvu|Eqn(2) p.16-3]]

[[media:Pea1.f11.mtg18.djvu|Page 18-2]]
Examples:

Exact N2-ODE: (2) p.16-3 -> [[media:Pea1.f11.mtg16.djvu|Eqns(2)-(4) p.16-4]]

[[media:Pea1.f11.mtg18.djvu|Page 18-3]]
(3)-(8) p.18-2:

Pretend that $$\phi$$ is not known, show that the N2-ODE (1) is exact. 1st exactness condition for N2-ODEs(2) p.16-4: (1)	Clearly satisfies (2) p.16-4 with:

(1)	And (3) p.16-4:

2nd exactness condition for N2-ODEs(1)-(2) p.16-5: R*3.15: Verify whether (1) satisfies the 2nd exactness condition [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]].

[[media:Pea1.f11.mtg18.djvu|Page 18-4]]
Find the first integral $$\phi$$
 * 1) 4 p.16-4:

h(x,y) can be found using the definition of g(x,y,p). For (2) and (3) p.18-3:

The problem of finding h(x,y) in (3) is even more general than that in (1) p.18-1. Note: (3) is a PDE for h with (x,y) treated as independent variables. Similar to the two cases to solve (4) p.11-1 for the integrating factor for N1-ODEs, there are two possible choices (assumptions) that can be made.

[[media:Pea1.f11.mtg18.djvu|Page 18-5]]
Note: Unlike the integrating factor h(x,y) in (4) p.11-1 for N1-ODEs, the function h(x,y) in (3) p.18-4 is not an integrating factor, but a function of the integration in (1) p.18-4. Yet, the solution procedure in both cases is similar. Remember Montesquieu p.3-2.

title: User:Egm6322.s12.team2.steele.m2/Mtg18


text:
User:Egm6322.s12.team2.steele.m2/Mtg19+20

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg19-20.djvu|Page 19-0]]
Notes, Sec 19 Plan: Example: Exact N2-ODE p.18-5 cont’d Linear systems and Reynolds Transport Theorem 1-D case 3-D case Material time derivative Transform integral on $$\mathcal{B}_t$$ to integral on $$\mathcal{B} \equiv \mathcal{B}_0$$ Velocity of material point $$X \equiv x_0$$ Material description Spatial description Summation convention on repeated indices Conservation of mass equation Reynolds Transport Theorem (RTT) Application					(Note p.3-2)

[[media:Pea1.f11.mtg19-20.djvu|Page 19-1]]
Example: Exact N2-ODE p.18-5 cont’d 2 Choices: 1)	From [[media:Pea1.f11.mtg18.djvu|Eqn(3) p.18-4]], assume that

[[media:Pea1.f11.mtg19-20.djvu|Page 19-2]]
2)	From (3) p.18-4, assume that

R*3.16: Finish the story.

[[media:Pea1.f11.mtg19-20.djvu|Page 19-3]]
=Linear Systems and Reynolds Transport Theorem= Recall [[media:Pea1.f11.mtg15.djvu|Eqn(1) p.15-2]] and R*3.7 p.15-1: Differentiate an integral with time-dependent integration limits.

See graphic on this page with the following parameters: $$\mathcal{B}_0, \mathcal{B}_{t1}, \mathcal{B}_{t2}, \mathcal{B}_{t3}, x_0, x_{t1} = \phi_{t1}(x_0), x_{t2}, x_{t3}$$

[[media:Pea1.f11.mtg19-20.djvu|Page 19-4]]
1-D case: More general than (1) p.15-2

$$\color{blue}{\alpha} = \frac{d}{dt} \left[F(t,b(t))-F(t,a(t))\right]$$ $$=\frac{\partial F(t,b(t))}{\partial s} \frac{db(t)}{dt}-\frac{\partial F(t,a(t))}{\partial s}\frac{da(t)}{dt}$$

[[media:Pea1.f11.mtg19-20.djvu|Page 19-5]]
$$=f(t,b(t)) \, \frac{db(t)}{dt}-f(t,a(t)) \, \frac{da(t)}{dt}$$ $$+\left[\frac{\partial}{\partial t} \int f(t,s) ds \right]_{s=a(t)}^{s=b(t)}$$ $$\left[ \int \frac{\partial f(t,s)}{\partial t} ds \right]_{s=a(t)}^{s=b(t)} = \int_{s=a(t)}^{s=b(t)} \frac{\partial f(t,s)}{\partial t} ds$$

[[media:Pea1.f11.mtg19-20.djvu|Page 19-6]]
2-D and 3-D cases: Reynolds transport theorem

Mass density

Conservation of mass: $$\frac{dM}{dt}|_{X \{\rm fixed} =: \frac{DM}{Dt} = 0$$	(4) $$\frac{d(\cdot)}{dt}|_{X \{\rm fixed} =: \frac{D(\cdot)}{Dt}$$ = Material time derivative, i.e., total time derivative keeping the material point $$X \equiv x_0$$ fixed.

[[media:Pea1.f11.mtg19-20.djvu|Page 19-7]]
Transform integral on $$\mathcal{B}_t$$ to integral on $$\mathcal{B} \equiv \mathcal{B}_0$$to take the total time derivative

See, e.g., Kolmogrov scales, Time rate of Jacobian http://en.wikiversity.org/wiki/User:Egm6936.f09/Kilmogorov_scales Velocity of material point$$X \equiv x_0$$:

U is material description

[[media:Pea1.f11.mtg19-20.djvu|Page 19-8]]
(2)-(4) p.19-5 and [[media:Pea1.f11.mtg2.djvu|Eqn(3) p.2-4]]:

Summation convention on repeated indices

Or evuivalently in component form:

[[media:Pea1.f11.mtg19-20.djvu|Page 19-9]]
$$\frac{DM}{Dt} = \int_{\mathcal B} \left[\frac{\partial \rho}{\partial t} + {\rm div} (\rho \mathbf u) \right] \, J \, d \mathcal B = 0$$
 * 1) (2) p. 19-7 and (1) p.19-8:

Hence the conservation of mass equation:

(3) Instead of the mass density $$\rho(x,t)$$ in (1) p.19-7 and (2), consider a general function f(x,t). Then the equivalent of (1) p.19-7 and (2) is the:

[[media:Pea1.f11.mtg19-20.djvu|Page 19-10]]
Reynolds Transport Theorem (RTT):

See, e.g., Gross & Reusken 2011, Numerical methods for incompressible two-phase flows, p.3, (1.3). Application: Consider

(1)	And (3) p.19-9:

See, e.g., Malvern 1969, Intro to the mechanics of a continuous medium, p.211, (5.2.18). R*3.17: Show(3).

[[media:Pea1.f11.mtg19-20.djvu|Page 19-11]]
How to relate the RTT to the 1-D case in (1) p.19-4? A: Apply the divergence theorem to the second term in (1) p.19-10: $$\frac{D}{Dt} \int_{\mathcal B_t}f(x,t) \, d \mathcal B_t = \int_{\mathcal B_t} \frac{\partial f}{\partial t} \, d \mathcal B_t + \int_{\partial \mathcal B_t} \mathbf n \cdot (f \mathbf u) \, d(\partial \mathcal B_t)$$	(1) See illustration of volume with the following parameters: a normal vector – an outward normal to boundary $$\partial \mathcal{B}_t, \partial{\mathcal{B}_t}, \mathcal{B}_t, x_t = \phi(X,t)$$

[[media:Pea1.f11.mtg19-20.djvu|Page 19-12]]
(1)	P.19-11 is another version of the RTT stated in (1) p.19-10; see a different and direct derivation of (1) p.19-11 in Malvern 1969 p.211 (5.2.17).

R*3.18: Obtain (1) p.19-4 from (1) p.19-11.

R4.1: Provide a different and direct derivation of (1) p.19-11; see, e.g., Malvern 1969 p.211 (5.2.17).


text:
User:Egm6322.s12.team2.steele.m2/Mtg21

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg21.djvu|Page 21-0]]
Plan: 2nd version of RTT: p.19-12 cont’d, R4.1 Non-exact L2-ODE-VC Example p.21-1 ODEs in power form: N1-ODE, L2-ODE-VC Integrating factor Non-exact L2-ODE-VC, integrating factor Example p.21-1 cont’d First integral for a class of exact L2-ODE-VC Example Application Exact Nn-ODE’s (Non-linear nth order) Case n=1: N1-ODE

[[media:Pea1.f11.mtg21.djvu|Page 21-1]]
=Non-Exact L2-ODE-VC= Example: Consider the following ODE R*4.2: Verify the exactness of (1). Method (recipe) ODEs in special power form: $$a, b, c, d \in \mathbb R$$ $$p, q, r, s \in \mathbb R$$ $$r, s, t \in \mathbb R$$ $$\alpha, \beta, \gamma \in \mathbb R$$

[[media:Pea1.f11.mtg21.djvu|Page 21-2]]
For ODEs of the form (2)-(3) p.21-1, use the integrating factor of the form

With $$m, n \in \mathbb R$$ to be determined.

[[media:Pea1.f11.mtg21.djvu|Page 21-3]]
Non-exact L2-ODE-VC, integrating factor Example P.21-1 cont’d L2-ODE-VC (1) p.21-1 and integrating factor (1) p.21-2. R*4.3: Find $$m, n \in \mathbb R$$ such that the following N2-ODE is exact: Show that the first integral is a L1-ODE-VC

With $$p(x) := y\prime(x)$$	[[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-3]] Solve (2) for y(x). (end R*4.3) /// (end Example) ///

[[media:Pea1.f11.mtg21.djvu|Page 21-4]]
First integral for a class of exact L2-ODE-VC $$\phi(x,y,p)$$	First integral Recall [[media:Pea1.f11.mtg16.djvu|Eqns(2)-(3) p.16-3:]] Consider the L2-ODE-VC:

[[media:Pea1.f11.mtg21.djvu|Page 21-5]]
The following function generates a class of exact L2-ODE-VC: See F09 p.13-1 for the problem statement on the derivation of (1). R*4.4: Show that (2)-(3) p.21-4 leads to (1). Example p.21-3

Application Select $$\phi$$ satisfying (1):

[[media:Pea1.f11.mtg21.djvu|Page 21-6]]
R*4.5: L2-ODE-VC

1.	Show (5) is exact. 2.	Find $$\phi$$ 3.	Solve for y(x) 1.	(end R*4.5)		/// 2.	(end Application)	/// 3.

[[media:Pea1.f11.mtg21.djvu|Page 21-7]]
=Exact Nn-ODE’s (Non-linear nth order)= $$G(x,y^{(0)}, y^{(1)}, y^{(2)}, \cdots, y^{(n)}) = 0$$ [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-2]] $$\displaystyle y^{(k)} (x) := \frac{d^ky(x)}{dx^k}$$	. [[media:Pea1.f11.mtg4.djvu|Eqn(2) p.4-6]] Exactness Condition 1:

Functions of $$(x, y^{(0)}, y^{(1)}, \cdots, y^{(n-1)})$$ Exactness Condition 2: Partial derivative

[[media:Pea1.f11.mtg21.djvu|Page 21-8]]
Total derivatives

Equivalently in more compact form: Case n=1: N1-ODE 2nd exactness condition (1)-(2): Which is equivalent to the 2nd exactness condition for N1-ODE in [[media:Pea1.f11.mtg9.djvu|Eqns(2)-(3) p.9-3:]] $$\phi_{xy} = \phi_{yx} \Leftrightarrow M_y = N_x$$		(2)-(3) p.93

title: User:Egm6322.s12.team2.steele.m2/Mtg21


text:
User:Egm6322.s12.team2.steele.m2/Mtg22+23

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg22.23.djvu|Page 22-0]]
Notes, Sec 22 Plan: Model assignment table/contribution table: Team 1 R3 	Exact Nn-ODE’s (Non-linear nth order) Case n=1: N1-ODE 2nd exactness condition p. 22-6 cont’d .		Method 1 Method 2 Case n=2: N2-ODE 2nd exactness condition Method 1 Method 2 Particle moving with air resistance: p.14-1 cont’d WolframAlpha: Symbolic computation, and more A beautiful game: Soccer (Magnus force, Bernoulli’s principle)

[[media:Pea1.f11.mtg22.23.djvu|Page 22-1]]
=Case n=1: N1-ODE p.21-8 cont’d= Equivalence of two forms of 2nd exactness condition, i.e., [[media:Pea1.f11.mtg21.djvu|Eqn(3) p.21-8]] and [[media:Pea1.f11.mtg16.djvu|Eqns(2)-(3) p.9-3]]. (2)	P.21-8 and [[media:Pea1.f11.mtg21.djvu|Eqn(3) p.21-7]]:

(3)	[[media:Pea1.f11.mtg21.djvu|Eqns(1)-(2) p.21-8,]] $$\phi_{xy} = \phi_{yx}$$	[[media:Pea1.f11.mtg9.djvu|Eqn(2) p.9-3]] (end Method 1) ///

[[media:Pea1.f11.mtg22.23.djvu|Page 22-2]]
Method 2: Work with coefficients in 1st exactness condition.

 * 1) (3) p.21-8 and (3) p.21-7:

[[media:Pea1.f11.mtg21.djvu|Eqn(3) p.21-8]], (2) and (4): $$M_y = N_x$$		(3) p.9-3 (end Method 2) ///

[[media:Pea1.f11.mtg22.23.djvu|Page 22-3]]
=Case n=2: N2-ODE= Equivalence of two forms of 2nd exactness condition: Form 2: (1)-(2) p.21-8:

Form 1: [[media:Pea1.f11.mtg17.djvu|Eqn(1) p.17-3]]: $$\phi_{xy} = \phi_{yx}, \phi_{yp} = \phi_{py}, \phi_{px} = \phi_{xp}$$		(1) p.17-3 Or equivalently [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]]: $$f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp}-g_y$$				(1) p.16-5 $$f_{xp} + pf_{yp} + 2f_y = g_{pp}$$					(2) p.16-5

[[media:Pea1.f11.mtg22.23.djvu|Page 22-4]]
Method 1: Equivalence to symmetry of mixed 2nd partial derivatives of first integral $$\phi$$, i.e., (1) p.17-3. R*14.6: Show this equivalence; see p.22-1. Hints: Don’t expand the derivatives too far toward the detail (bottom) level; stay at the more general (top) level as much as possible. Specifically, first keep in mind that the first integral $$\phi$$ is a function of (x,y,y’) which are considered as 3 independent variables. Second, show that $$g_0 := \frac{\partial}{\partial y}\left(\frac{d \phi}{dx} \right)$$		(3) p.21-7

(end Hints) /// (end Method 1) ///

[[media:Pea1.f11.mtg22.23.djvu|Page 22-5]]
=Method 2:= Work with coefficients in 1st exactness condition (1)	P.22-3 is equivalent to the 2 relations (1)-(2) p.16-5 in the 2nd exactness condition for N2-ODEs: See F09 Mtgs 13-14 for the details on the homework problem statement. Below is a detailed explanation. In addition to p(x) := y’(x)					(2) p.73
 * 1) (3) p.21-7:

[[media:Pea1.f11.mtg22.23.djvu|Page 22-6]]
R*5.1: 	(1) p.22-3:

Define: (2)

Since 1 and q are linearly independent (q = y’’ is in general not a constant), we must have $$\bar g = 0 \Leftrightarrow$$(1) p.16-5 $$\bar f = 0 \Leftrightarrow$$(2) p.16-5 (end Method 2)///

[[media:Pea1.f11.mtg22.23.djvu|Page 22-7]]
=Particle moving with air resistance: p.14-1 cont’d= A beautiful game: Soccer (Magnus force, Bernoulli’s principle) Soccer physics: Sport projectiles, must account for air resistance; see Mtg 14 (See illustration for $$F_D$$, D and U with vector diagram.)

Variable $$\rho$$ = air mass density			U = velocity magnitude A = cross section area		$$C_D$$ = drag coefficient D = ball diameter		$$\nu$$ = kinematic viscosity $$\mu$$ = dynamic / kinetic viscosity

[[media:Pea1.f11.mtg22.23.djvu|Page 22-8]]
Goff, Power and spin in the beautiful game, Physics Today, Jul 2010. See graph of drag crisis with $$\nu = \frac{\mu}{\rho}$$ and Re = $$\frac{UD}{\nu}$$ See graph of $$\omega x U$$ and $$F_M$$ = Magnus force

[[media:Pea1.f11.mtg22.23.djvu|Page 22-9]]
See graph of midfield free kick, soccer field with top view, and corner kick.

[[media:Pea1.f11.mtg22.23.djvu|Page 22-10]]
See graph of Bernoulli’s principle, increased velocity, and decreased pressure.

See graph of life, wing, presser exerted by faster moving air, and flight.


text:
User:Egm6322.s12.team2.steele.m2/Mtg24

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg24.djvu|Mtg24]] Tue, 11 Oct 11
'''Note: Images could not be found for these complicated flow patterns of using differential equations. So only text entry was done. (Manuel Steele - summer 2012). For images and decision flow, please see djvu file above in bold link.'''

[[media:Pea1.f11.mtg24.djvu|Page 24-0]]
Mtg 24: Tue, 11 Oct 11

Plan: Summary of ODE solution methods Flow Chart Exact Nn-ODE’s (Non-linear nth order) Case n=2: N2-ODE 2nd exactness condition Method 1 p.22-4 cont’d Method 2

[[media:Pea1.f11.mtg24.djvu|Page 24-1]]
=General ODE nth order 2nd order Nn-ODE LN-oDE-VC= 1st order N1-ODE N2-ODE p.24-3 L2-ODE-VC p.24-2 L1-ODE-VC Missing y(x) Exact or made exact by IFM  Non-exact 0th order eq. for p(x) 0th order eq. for y(x) Other methods analytical numerical P(x) y(x) $$y(x) = \int p(x)dx$$

[[media:Pea1.f11.mtg24.djvu|Page 24-2]]
2nd order p.24-1 N2-ODE p.24-3 L2-ODE-VC Non-exact Missing y(x)                                  Power form Exact or made exact by IFM

L1-ODE-VC for p(x)                      L1-ODE-VC for y(x)           N1-ODE p.241

P(x)                                                     y(x)                      Other methods:  analytical, numerical

$$y(x) = \int p(x)dx$$                                             Special L2-ODE-VC:  Legendre, Bessel, Laguerre, Hermit, etc.

[[media:Pea1.f11.mtg24.djvu|Page 24-3]]
N2-ODE p.24-1

Missing                                                          Exact                                                             Non-Exact

N1-ODE for p(x)                                       N1-ODE p.24-1                       Other methods:  analytical, numerical

p. 24-1

[[media:Pea1.f11.mtg24.djvu|Page 24-4]]
Nth order p.24-1 Ln-ODE-VC special L2-ODE-VC Legendre Bessel, Laguerre, Hermite, etc. L2-ODE-VC Non-exact Inverse method: Start w/ solution  Other methods analytical Transformation of variable Trial solution (undetermined coeff.) Single Term                              Polynomial Series:  Frobenius $$x^c e^{rx}$$                                             $$x^c \sum d_i x^i$$

title: User:Egm6322.s12.team2.steele.m2/Mtg24


text:
User:Egm6322.s12.team2.steele.m2/Mtg27+28+29

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-0]]
Notes, Sec 27 Plan: Are the Legendre and Bessel L2-ODEs-VC exact? Hermite L2-ODE-VC? Vibration: Chladni patterns Vibrating membrane: 2-D case, Cartesian coordinates 1-D case Separation of variables Separated equation: 1-D Helmholtz eq. Trial solutions: Sine waves More general solution Wave eq. in 1-D: Shape preserving Heat equation (1-D, 2-D, 3-D) Euler-Bernoulli beam: Equation of motion Free vibration

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-1]]
Are the Legendre and Bessel L2-ODEs-VC exact? Hermite L2-ODE-VC? *5.2: Consider the following governing L2-ODEs-VC for some classical special functions: Legendre: F09, F11 $$G=(1-x^2)y\prime \prime - 2xy\prime + n(n+1)y = 0$$ [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7]] Bessel: F10

Hermite: F11 1.	Verify the exactness of the designated L2-ODEs-VC. For the 2nd exactness condition, use 2 methods: 1a. [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]] 2b. [[media:Pea1.f11.mtg22.djvu|Eqn(1) p.22-3]]

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-2]]
2. If (2) p.27-1 is not exact, check whether it is in power form, and see whether if it can be made exact using IFM with $$h(x,y) = x^m y^n$$; see p.21-2. 3. The first few Hermite polynomials are

Verify that (1)-(3) are homogeneous solutions of the Hermite differential equation (2) p.27-1.

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-3]]
=Vibration: Chladni Patterns= See image with power and flexible plate showing sound wave. Sound wave is generated by acoustic speaker with tunable frequency

See sinusoidal wave in image with “Powder falling toward the ‘node’” and “Node.” Note the single vibrational mode of the plate. Vibrating membrane: $$\rho $$ = mass density (per unit membrane area) T = uniform tension in membrane

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-4]]
See image for (x,y) and u on Cartesian space of x, y, and z.

Note u(x,y,t) = transverse displacement of membrane at point (x,y) and at time t.

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-5]]
=Equation of Motion=

Coordinate independent See EML 5526 S11 p.38-4, p. 40-4, p.42-2 2-D case, Cartesian coordinates

1-D case: Two independent variables (x, t)

[[media:Pea1.f11.mtg27.28.29.djvu|Page 27-6]]
=Separation of variables: ansatz = trial solution = guess=

Separated equation for X(x): 1-D Helmholtz eq. See [[media:Pea1.f11.mtg4.djvu|Eqn(1) p.4-3]]; homogeneous L2-ODE-CC:

Trial solutions: $$\cos kx, \ \sin kx$$

Euler’s formula http://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysis

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i.e., a linear combination of $$\cos kx, \sin kx$$ with complex coefficients. Note: (1) p.24-6:

Same result is obtained if the ansatz had a phase shift, i.e.,

A is amplitude above.

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$$\omega$$ = (angular) frequency in time K = frequency in space (wave number)

King 2003 p.82 See diagram with parameters of F, speed c, and separate patterns for $$t = t_1$$ and $$t = t_2 > t_1$$ [[image: [[image: Pea1.f11.wave.propagation.svg|500px]]

Wave equation in 1-D: Shape preserving Speed c>0 + = outgoing wave (+ x direction) -	= incoming wave (- x direction)

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Wave eq. in 2-D: More complex Wave eq. in 3-D: Even more complex Wave eq. in curvilinear coordinates: Even more complex =Heat equation (1-D, 2-D, 3-D):=

$$\kappa$$ = conductivity tensor (matrix) Note u = temperature Note f = distributed heat source Note ρ = mass density Note c = specific heat (heat capacity per unit mass)

Note the plus sign for the diffusion term $${\color{red} +}{\rm div( \boldsymbol \kappa \cdot grad}) \, u$$ See EML 5526 S11 p.33-2, p.32-3

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Sum on i and sum on j

Kronecker delta: Note (1) p.27-9, and (1)-(3):

Sum on i

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Note: Finite-element discretization of (1) p.27-9

$$\mathbf M \mathbf{\dot d} + \mathbf K \mathbf d = \mathbf F$$ Plus sign becomes minus sign after integration by parts; see EML 5526 S11 p.34-2.

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=Euler-Bernoulli Beam:= See beam for f(x,t) and curves shown for undeformed and deformed states. See u(x,t). [[image: [[image: Pea1.f11.euler.bernoulli.beam.svg|500px]]

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Euler-Bernoulli beam: Equation of motion $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} + f = m \frac{\partial^2 u}{\partial t^2}$$	(1) Note the minus sign for the internal force term $$EI \frac{\partial^4 u}{\partial x^4}$$ E = Young’s Modulus I = moment inertia of cross section U = transverse displacement F = transverse distributed force M = mass per unit length Free vibration: f = 0 $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} = m \frac{\partial^2 u}{\partial t^2}$$	(2)

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=Note: Finite-element discretization of (1) p.27-12	= $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} + f = m \frac{\partial^2u}{\partial t^2}$$	(1) p.27-12 $$\mathbf M \mathbf{\ddot d} + \mathbf K \mathbf d = \mathbf F$$	(1) Minus sign stays minus after integration by parts twice; see EML 4500 F08 p.38-1.