text:
User talk:IKadlecik

 Hello IKadlecik, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
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 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 15:12, 27 December 2010 (UTC)

title: User talk:IKadlecik


text:
Egyptology/Hatshepsut Project

{|cellpadding="0" cellspacing="0" align=right style="margin: 5px; border:2px solid ;" !style="border-bottom:2px solid ; background:; padding:5px; text-align:center;"|Hatshepsut in hieroglyphs {|border="0" cellspacing="0" cellpadding="0" align="center" |  |    |
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|serekh or Horus name: Weseretkau
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|Nebty name: Wadjrenput
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|Nebty name: Wadjrenput
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|Golden Horus name: Netjeretkhau
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|Golden Horus name: Netjeretkhau
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|

| |align="center" valign="middle"| |} {|border="0" cellspacing="0" cellpadding="0" align="center" |  |  |
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|praenomen or throne name": Maatkare
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|praenomen or throne name": Maatkare
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|nomen or birth name: (Khnumet-Amun) Hatshepsut
 * style="text-align:center;text-valign:middle;padding:4px 10px 10px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:5px;background: white;"|nomen or birth name: (Khnumet-Amun) Hatshepsut
 * style="text-align:center;text-valign:middle;padding:4px 10px 10px 10px;background: white;"|
 * style="text-align:center;text-valign:middle;padding:4px 10px 10px 10px;background: white;"|

| |align="center" valign="middle"| |}
 * }
 * }

Hatshepsut was the daughter of Thutmose I and his wife Queen Ahmose. Hatshepsut married her half-brother Pharaoh Thutmose II. During the reign of her husband and brother Hatshepsut served as Great Royal Wife. When Thutmose II died, his son Thutmose III was very young and Hatshepsut assumed the role of regent for her young stepson.

At some point in or before year 7 Hatshepsut assumed the role of Pharaoh. She would rule for approximately 22 years. Hatshepsut is one of only a handful of royal women who ruled Ancient Egypt. Hatshepsut took on the five-fold royal titulary of a Pharaoh

Museum collections
Items from the reign of Hatshepsut can be found in many museums across the world. A large collection of important artifacts can be found in the Egyptian Museum in Cairo. The Metropolitan Museum of Art in New York similarly has a large number of statues and other artifacts.

A list of object in museums around the world related to Hatshepsut is given on the Hatshepsut: Museum Collections page.

People associated with Hatshepsut
As Queen and Pharaoh Hatsepsut would have been surrounded with court officials and other notable people. A short (and necessarily incomplete) list:

 * Princess Neferure, daughter of Hatshepsut. Served as God's Wife of Amun.
 * Sitre called In, nurse of Hatshepsut
 * Senenmut, tutor of princess Neferure and Architect
 * Hapuseneb, High Priest of Amun and possibly vizier
 * Ahmose Pen-nekhbet, Army leader and tutor of princess Neferure.
 * Amethu called Ahmose, Vizier
 * Useramen, Vizier and son of Amethu called Ahmose
 * Seni, Viceroy of Kush during the early years
 * Ineni, Architect
 * Neshi, Chancellor, leader of the expedition to Punt, Chief treasurer. Buried in Saqqara (tomb discovered by Zivie)

Temples

 * Deir el-Bahari Temple: Mortuary temple built for Hatshepsut in Thebes. The temple was called Djeser Djeseru in ancient times.
 * Chapelle Rouge, Karnak: Also called the Red Chapel, was originally constructed as a barque shrine.
 * Obelisk at Karnak Temple
 * Speos Artemidos: Temple at Beni Hasan (Batn el-Bakarah) dedicated to the lion goddess Pakhet The Artemidos had an outer pillared hall and an inner sanctuary cut into the rock. The temple was unfinished, but contained a dedication text denouncing the Hyksos rulers. The pillared hall shows Hatshepsut before various deities. An image of the goddess Pakhet was carved from the rock in a niche in the inner room. The temple was later usurped by Pharaoh Seti I
 * Horus temple at Buhen:The ancient settlement of Buhen on the West Bank of Luxor was submerged with water during the flooding of Lake Nassar during the building of the Aswan Dam. Before the flooding, Buhen was excavated by British Egyptologist Walter Bryan Emery. Saved from the flood waters, a temple to Horus which was built by Hatshepsut was dismantled and moved to the Sudan National Museum. Today, it resides there for all to see. (from: Hatshepsut Project blogspot)
 * Temple Building in Faras, Nubia

Inscriptions

 * Sinai: Year 11 stela showing Neferure and Senmut

Tombs

 * KV 20 was one of the first tomb to be constructed in the Valley of the Kings in Thebes. Originally constructed for Thutmose I, the tomb was later adapted for the burial of both Thutmose I and Hatshepsut.
 * KV 60 was the tomb of Hatshepsut's nurse Sitre-In. Hatshepsut's mummy was found in this tomb.
 * Rock Cut Tomb, in Thebes. This tomb was started for Hatshepsut when she was the great royal wife to Thutmose II. (Photographs)w

Links

 * Hatshepsut Project Blog by Stuart Tyler
 * Maat-ka-Ra Hatshepsut by Dr Karl Leser

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

title: Egyptology/Hatshepsut Project


text:
Hatshepsut: Museum Collections

Below is a table listing the objects as found in museums. When possible the museum number has been given. If the artifact is described in the book Hatshepsut: From Queen to Pharaoh edited by Catherine Roehrig then the catalog number will be included.

In some cases a link to a photograph from the Metropolitan Museum is given. Due to copyright issues the image is not displayed here. The MET has more than 250 objects related to Hatshepsut in their database, the artifacts listed below are a sample of those.

Remark: This is a work in progress!

Return to Hatshepsut Project

title: Hatshepsut: Museum Collections


text:
User talk:Wendishka

Hello, I joined Wikiversity to learn about Technical Writing as a potential career. Thanks to the contributors, and I hope to have something to add at some point. Wendishka 16:29, 27 December 2010 (UTC)

Welcome
 Hello Wendishka, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 17:32, 27 December 2010 (UTC)

title: User talk:Wendishka


text:
Introduction to Hebrew

Common Phrases
Shalom - שלום - Hello.

Ma shlomkha? - ?מה שלומך - How are you? (Formal).

Ma kore? - ?מה קורה - How are you? (Informal).

Koreem li... - ...קוראים לי - My name is... (Literal: I'm called...).

Ekh koreem lekha? - ?איך קוראים לך - What's your name? (Literal: How are you called?).

Ani lo medaber ivrit - אני לא מדבר עברית - I don't speak Hebrew.

Ani medaber ktsat ivrit - אני מדבר קצת עברית - I speak a little Hebrew.

Pronouns
I - Ani - אני.

You (singular) - Ata - אתה.

He - Hoo - הוא.

She - Hee - היא.

We - Anakhnu - אנחנו.

You (plural) - Atem - אתם.

They (Masculine) - Hem - הם.

They (Feminine) - Hen - הן.

Prepositions
With - Eem - עם.

In (Used to place and time) - Be... (no space before next word) - ...ב.

On (Used only as "the book is on the table") - Al - על.

Of/'s - Shel - של.

From - Mi... (No space) - ...מ.

For - Bishvil/ Le...(No space) - ל/בשביל.

Conjunctions
And - Ve... (No space, can be pronounced Va, Vi and Oo too.) - ....ו.

Or - O - או.

But - Aval - אבל.

Note
For Hebrew Alphabet go to Reading and Writing Hebrew.

title: Introduction to Hebrew


text:
User talk:Nitayyarn

 Hello Nitayyarn, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

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title: User talk:Nitayyarn


text:
User:Doctorbee~enwikiversity

Brief Information
Hello, my name is Neil Bhattarai. I am a B.S. (Biology) graduate from Creighton University. I am currently studying Surgical Technology at Mercy College of Health Sciences in Des Moines, IA. I have a strong interest in Anatomy and Physiology. My ultimate goal is to become a physician and I am currently in the process of applying to medical schools.

title: User:Doctorbee~enwikiversity


text:
User talk:Doctorbee~enwikiversity

 Hello Doctorbee, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Geoff 01:05, 28 December 2010 (UTC) Geoff 01:05, 28 December 2010 (UTC)

Hello!
Hi, I have been following Wikiversity for a while, and I figured it is finally time to contribute. Let's collaborate!

--Doctorbee 18:37, 28 December 2010 (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 Doctorbee. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Doctorbee~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 23:31, 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) 06:06, 19 April 2015 (UTC)

title: User talk:Doctorbee~enwikiversity


text:
Overview of Musculoskeletal Anatomy

Welcome to Overview of Musculoskeletal Anatomy

In this course, we will explore the musculoskeletal anatomy. This course has been divided into seven lessons, each taking up to one hour each. The sidebar in each of the lessons will help you understand the prerequisites for the lesson, estimate time for completion of the lesson, and level of the course. There will be a quiz after the end of each of the lessons. This course is designed for students with prior learning in regional anatomy. However, students that do not have prior knowledge can also find the lessons as each of the lessons have summarizing points.

Lesson List
Lesson 1
 * An introduction to the bones and muscles of the head and neck

Lesson 2
 * An introduction to the bones and muscles of the thorax

Lesson 3
 * An introduction to the bones and muscles of the back

Lesson 4
 * An introduction to the muscles of the abdomen

Lesson 5
 * An introduction to the bones and muscles of the pelvis and perineum

Lesson 6
 * An introduction to the bones and muscles of the upper limb

Lesson 7
 * An introduction to the bones and muscles of the lower limb

Parent Department

title: Overview of Musculoskeletal Anatomy


text:
Overview of Musculoskeletal Anatomy/Lesson 1

Overview of Musculoskeletal Anatomy - Lesson 1 Bone and muscles of the head and neck

Skull
The skull, also known as the cranium, contains 22 bones. There are two sets of bones in the skull: eight cranial and fourteen facial bones.

Cranial Bones
There are eight cranial bones. They form the cranial cavity, which functions to hold and protect the brain.
 * 1) Frontal (1)
 * 2) Parietal (2)
 * 3) Temporal (2)
 * 4) Occipetal (2)
 * 5) Sphenoid (1)
 * 6) Ethmoid (1)
 * 7) Inferior nasal conchae (2)
 * 8) Vomer (1)

Frontal Bone

 * Vertical portion is also known as squama frontalis. It corresponds with the forehead region. There are two surfaces to the squama frontalis as well: external surface and internal surface.
 * External surface of the squama frontalis


text: Template:Hiero

title: Template:Hiero


text:
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palegoldenrod

title: Template:Hiero/nk/bgcolour


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crimson

title: Template:Hiero/nk/bordercolour


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User talk:Threetwoone~enwikiversity

 Hello Threetwoone, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 13:16, 28 December 2010 (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 Threetwoone. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Threetwoone~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 23:45, 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) 06:24, 19 April 2015 (UTC)

title: User talk:Threetwoone~enwikiversity


text:
Extraterrestrial life/Astrobiology

Department of Astrobiology

Astrobiology (or exobiology) is the study of the origin, evolution, distribution, and future of life in the universe. Earth is the only place in the universe known to harbour life. However, advancements in the fields of astrobiology, observational astronomy and discovery of large varieties of extremophiles with extraordinary capability to thrive in the harshest environments on Earth, have led to speculation that life may possibly be thriving on many of the extraterrestrial bodies in the universe. This interdisciplinary field encompasses the search for habitable environments in our Solar System and habitable planets outside our Solar System, the search for evidence of prebiotic chemistry, laboratory and field research into the origins and early evolution of life on Earth, and studies of the potential for life to adapt to challenges on Earth and in outer space.[2]

This discipline could include •  Astrophysics •  Astropaleontology •  Bioastronomy •  Cosmochemistry •  Ecogenomics •  Exobiology •  Extremophiles •  Geomicrobiology •  Gravitational biology •  Life detection technology •  Meteoritics •  Origins of life •  Planetary geoscience •  Planetary protection •  Prebiotic chemistry •  Space exploration technology •  Terraforming

Journals
 * Astrobiology
 * International Journal of Astrobiology

Theoretical Related Approaches
 * Astrosociobiology

title: Extraterrestrial life/Astrobiology


text:
User talk:Xideral

 Hello Xideral, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --mikeu talk 16:47, 28 December 2010 (UTC)

title: User talk:Xideral


text:
User talk:Pattyb42

Hi everyone. My name is Pat. I joined here because I heard that students may be able to get free stuff for school. Is that really true? I'm a student at Mendocino College in Ukiah, Ca (Mendocino County) & I'm interested in stuff that will help me further my education. So, please let me know what you know. Thanks so much & take care. Dec. 28, 2010, 9:34 a.m. Pattyb42 17:35, 28 December 2010 (UTC)

title: User talk:Pattyb42


text:
User talk:Marcus Qwertyus

 Hello Marcus Qwertyus, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Marcus Qwertyus


text:
Master student

Link to any subpages this page might have

ACA - 118

College Course ACA - 118 Central Piedmont Community College (CPCC) Welcome, ACA - 118 is a course in the college, it's for developing master student skills. It's divided into sections; 12, 04... Sections depending on instructors. This course is how to be a master student.

Book Title: Becoming a Master Student 13th edition. Author: Dave Ellis. There's a textbox edition and a kindle edition. If you are a student at CPCC, you can get to access the e-book.

Master Student

Master students are people that do what they need to do. They're on time for class, they do their work and do a good job. They make good grades. They learn from instructor's lectures. They access blackboard and print off the notebook handouts. They go to the campus library and information on the topic containing the subject. You should start at the Central High building if your at central campus. You should ask for information packages about the class and preview information. Before class starts, you should take a tour of the campus. You should also get a map of the campus. The campuses all have different layouts. The are spread out across Mecklenburg County, NC.

Instructor's Lectures

An Instructor will lecture a lot and sometimes it will be hard learn if you are a visual learner. Sometimes instructors will discuss information that is not in the book. They'll sometimes let you get a tape recorder and record a lecture. You have to get permission for that. Some instructors might have emotional states and they don't want to be recorded.

Memory Memory is a record of thoughts stored. Memory is in your cerebral cortex. Brain hemispheres; frontal lobes, parietal lobes, occipital lobes and temporal lobes. Those parts of the brain are a part of memory.

Time Management

Time management is where you manage you time correctly. You plan out your day hour after hour. The instructor may assign an activity on the topic.

Learning Styles (VAK) Visual - seeing colors and visual memory. Auditory - listening to lectures. Kintestic / Tactile - grasping and learning with you hands.

Health Emotional states depression suicide is not a positive solution to your problems. Laughing is a good helping device for emotions. It's good to have good physical health.

Communication Talking is a kind of communication (verbal). Non-verbal is when you use body language to communicate. Instructors' may come up with games to play teaching the topic.

title: Master student


text:
User talk:Camsipe

 Hello Camsipe, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Camsipe


text:
User talk:Livefreelovehard

 Hello Livefreelovehard, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Livefreelovehard


text:
Power Systems/Synchronous Generator

synchronous generator are efficient form for generating electricity than conventional DC generator. Many industries started generating through synchronous generator for constant voltage at constant frequency. it is better than alternator in aspects like large power rating and efficient control.

title: Power Systems/Synchronous Generator


text:
User talk:Pretty~enwikiversity

 Hello Pretty, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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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 Pretty. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Pretty~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.

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Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:47, 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) 06:18, 19 April 2015 (UTC)

title: User talk:Pretty~enwikiversity


text:
User talk:Wutsje

 Hello Wutsje, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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 * Thanks. Wutsje 01:08, 30 December 2010 (UTC)

Aqib Aze storie test
Wutsje. I know you are in the existing history of a page of mine, Mabuabsdd--Mabuabsdd 15:22, 8 February 2011 (UTC)

 * Mabuabsdd is apparently an elementary school student, creating many WV pages in inappropriate places. I've been trying to help him learn to follow Wikiversity policies and traditions, with some small success. Obviously not complete success, yet, but that's the thing about kids. While they may learn very quickly, sometimes they don't. My crystal ball is broken. In any case, he's learning to write wikitext and various things about how to use the MediaWiki interface, if nothing else! He's obviously learned how to add a signature.... He doesn't know, yet, that he doesn't need to manually add the user name, that's common for him to do both. --Abd 15:11, 9 February 2011 (UTC)
 * Ah well, he doesn't bother me. Thanks for the explanation. Regards, Wutsje 15:23, 10 February 2011 (UTC)

title: User talk:Wutsje


text:
Heat equation

The heat equation is a very important equation in physics and engineering. The following pages will allow for a deeper understanding of the mathematics behind solving the heat equation.

Solutions to the heat equation

 * Rectangular Coordinates
 * The 1-D heat equation
 * The 2-D heat equation

 * Cylindrical Coordinates
 * The 2-D heat equation
 * The 3-D heat equation

title: Heat equation


text:
User:MD Wagner

This site is under construction. de:Benutzer:MD_Wagner

title: User:MD Wagner


text:
User talk:Ryaminal

 Hello Ryaminal, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Ryaminal


text:
User talk:MD Wagner

 Hello MD Wagner, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:MD Wagner


text:
User talk:Ninth~enwikiversity

 Hello Ninth, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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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 Ninth. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Ninth~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 23:45, 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) 06:17, 19 April 2015 (UTC)

title: User talk:Ninth~enwikiversity


text:
Renewable energy/Solar thermal

About solar collector
A solar thermal collector is a solar collector designed to collect heat by absorbing sunlight. The term is applied to solar hot water panels, but may also be used to denote more complex installations such as solar parabolic, solar trough and solar towers or simpler installations such as solar air heat. The more complex collectors are generally used in solar power plants where solar heat is used to generate electricity by heating water to produce steam which drives a turbine connected to an electrical generator. The simpler collectors are typically used for supplemental space heating in residential and commercial buildings. A collector is a device for converting the energy in solar radiation into a more usable or storable form. The energy in sunlight is in the form of electromagnetic radiation from the infrared (long) to the ultraviolet (short) wavelengths. The solar energy striking the Earth's surface depends on weather conditions, as well as location and orientation of the surface, but overall, it averages about 1,000 watts per square meter under clear skies with the surface directly perpendicular to the sun's rays.

Flat plate collectors
Flat plate collectors, developed by Hottel and Whillier in the 1950s, are the most common type. They consist of (1) a dark flat-plate absorber of solar energy, (2) a transparent cover that allows solar energy to pass through but reduces heat losses, (3) a heat-transport fluid (air, antifreeze or water) to remove heat from the absorber, and (4) a heat insulating backing. The absorber consists of a thin absorber sheet (of thermally stable polymers, aluminum, steel or copper, to which a matt black or selective coating is applied) often backed by a grid or coil of fluid tubing placed in an insulated casing with a glass or polycarbonate cover. In water heat panels, fluid is usually circulated through tubing to transfer heat from the absorber to an insulated water tank. This may be achieved directly or through a heat exchanger. Most air heat fabricates and some water heat manufacturers have a completely flooded absorber consisting of two sheets of metal which the fluid passes between. Because the heat exchange area is greater they may be marginally more efficient than traditional absorbers.[1]

There are a number of absorber piping configurations:

* harp - traditional design with bottom pipe risers and top collection pipe, used in low pressure thermosyphon and pumped systems * serpentine - one continuous S that maximizes temperature but not total energy yield in variable flow systems, used in compact solar domestic hot water only systems (no space heating role) * completely flooded absorber consisting of two sheets of metal stamped to produce a circulation zone. Because the heat exchange area is greater they may be marginally more efficient than traditional absorbers.

As an alternative to metal collectors, new polymer flat plate collectors are now being produced in Europe. These may be wholly polymer, or they may include metal plates in front of freeze-tolerant water channels made of silicone rubber. Polymers, being flexible and therefore freeze-tolerant, are able to contain plain water instead of antifreeze, so that they may be plumbed directly into existing water tanks instead of needing to use heat exchangers which lower efficiency. By dispensing with a heat exchanger in these flat plate panels, temperatures need not be quite so high for the circulation system to be switched on, so such direct circulation panels, whether polymer or otherwise, can be more efficient, particularly at low light levels.

Some early selectively coated polymer collectors suffered from overheating when insulated, as stagnation temperatures can exceed the melting point of the polymer.[1] [2] For example, the melting point of polypropylene is 160°C, while the stagnation temperature of insulated thermal collectors can exceed 180°C if control strategies are not used. For this reason polypropylene is not often used in glazed selectively coated solar collectors. Increasingly polymers such as high temperate silicones (which melt at over 250C) are being used. Some non polypropylene polymer based glazed solar collectors are matt black coated rather than selectively coated to reduce the stagnation temperature to 150C or less.

In areas where freezing is a possibility, freeze-tolerance (the capability to freeze repeatedly without cracking) can be delivered by the use of flexible polymers. Silicone rubber pipes have been used for this purpose in UK since 1999. Conventional metal collectors are vulnerable to damage from freezing, so if they are water filled they must be carefully plumbed so they completely drain down using gravity before freezing is expected, so that they do not crack. Many metal collectors are installed as part of a sealed heat exchanger system. Rather than having the potable water flow directly through the collectors, a mixture of water and antifreeze such as propylene glycol (which is used in the food industry) is used as a heat exchange fluid to protect against freeze damage down to a locally determined risk temperature that depends on the proportion of propylene glycol in the mixture. The use of glycol lowers the water's heat carrying capacity marginally, while the addition of an extra heat exchanger may lower system performance at low light levels.

A pool or unglazed collector is a simple form of flat-plate collector without a transparent cover. Typically polypropylene or EPDM rubber or silicone rubber is used as an absorber. Used for pool heating it can work quite well when the desired output temperature is near the ambient temperature (that is, when it is warm outside). As the ambient temperature gets cooler, these collectors become less effective.

Most flat plate collectors have a life expectancy of over 25 years.

Evacuated tube collectors
Most (if not all) vacuum tube collectors use heat pipes for their core instead of passing liquid directly through them. Evacuated heat pipe tubes (EHPT's) are composed of multiple evacuated glass tubes each containing an absorber plate fused to a heat pipe.[2] The heat from the hot end of the heat pipes is transferred to the transfer fluid (water or an antifreeze mix—typically propylene glycol) of a domestic hot water or hydronic space heating system in a heat exchanger called a "manifold". The manifold is wrapped in insulation and covered by a sheet metal or plastic case to protect it from the elements.

The vacuum that surrounds the outside of the tube greatly reduces convection and conduction heat loss to the outside, therefore achieving greater efficiency than flat-plate collectors, especially in colder conditions. This advantage is largely lost in warmer climates, except in those cases where very hot water is desirable, for example commercial process water. The high temperatures that can occur may require special system design to avoid or mitigate overheating conditions. Glass-glass evacuated tube

Some evacuated tubes (glass-metal) are made with one layer of glass that fuses to the heat pipe at the upper end and encloses the heat pipe and absorber in the vacuum. Others (glass-glass) are made with a double layer of glass fused together at one or both ends with a vacuum between the layers (like a vacuum bottle or flask) with the absorber and heat pipe contained at normal atmospheric pressure. Glass-glass tubes have a highly reliable vacuum seal but the two layers of glass reduce the light that reaches the absorber and there is some possibility that moisture will enter the non-evacuated area of the tube and cause absorber corrosion. Glass-metal tubes allow more light to reach the absorber and protect the absorber and heat pipe (contained in the vacuum) from corrosion even if they are made from dissimilar materials (see galvanic corrosion).

The gaps between the tubes may allow for snow to fall through the collector, minimizing the loss of production in some snowy conditions, though the lack of radiated heat from the tubes can also prevent effective shedding of accumulated snow.

title: Renewable energy/Solar thermal


text:
User talk:Tom0571

 Hello Tom0571, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Tom0571


text:
User talk:Jones~enwikiversity

 Hello Jones, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 06:08, 30 December 2010 (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 Jones. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Jones~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 23:35, 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) 06:12, 19 April 2015 (UTC)

title: User talk:Jones~enwikiversity


text:
Molecular Biology/Milestones

Milestones
Friedrich Miescher, 1869 identified Nuclein from pus cells.

Feulgan, 1927-28 demonstrated two types of Nucleic acids, i.e. DNA and RNA.

Frederick Grifith, 1928 demonstrated genetic transformation principle.

Erwin Chargaff, 1940, showed quantitative equivalence of total purine to total pyrimidines, i.e. A=T, G=C and A+T = G+C,

Oswald Avery, Colin McLeod and MacLean McCarty, 1944, demonstrated experimentally that DNA as the genetic material, they used Pneumococcus virulence and nonvirulent strains.

Alfred Hershey and Martha Chase, 1952, ultimate demonstration of DNA as the genetic material, using radioisotope labeled T2 phages.

James Watson and Francis Crick, 1952-53, a crowning achievement in proposing a 3-D model of DNA, a double helix. Rosalind Franklin and Maurice Wilkins provided x-ray diffraction data. Jerry Donahue an office mate of J.D.Watson provided the clue about the correct base tautomeric states.

George Beadle, Joshua Lederberg, Edward L.Tatum, 1958, for work on gene expression, in Neurospora, in medicine and Physiology.

Frederick Sanger, 1958, in chemistry for his work on primary structure of proteins; amino acid sequence of Insulin.

Arthur Korenberg, 1959, did in vitro synthesis of DNA using DNA polymerase, which is now called DNA polymerase-I. Severo Ochoa did an excellent job on RNA polymerases; also he worked on in initiation of translation in eukaryotes. Nobel Prize in medicine and physiology.

F. MacFarlane and Peter B. Medawar, 1960, for their excellent work on clonal selection theory of Antibody formation; Nobel Prize in Medicine and physiology.

Maramur Dotty, 1961 conducted historical DNA-DNA and DNA-RNA hybridization experiments.

Melvin Calvin, 1961, Nobel Prize in chemistry for his work on Biochemistry of carbon fixation,

Werner Arber, 1962, showed evidence for restriction enzymes; Nathan and H.Smith showed that restriction enzymes cut DNA in sequence specific manner; Nobel Prize in chemistry.

'''F.C. Crick. J.D.Watson and M.H.F. Wilkins''', 1962 awarded Nobel Prize in medicine and Physiology, for their work on 3-D structure of DNA.

John Kendrew, Max F. Perutz, 1962, for their great work on 3-D structure of globular proteins, awarded Nobel Prize in chemistry.

John C. Eccles, Alan L.Hodkin and Andrew F. Huxley, 1963, awarded Nobel Prize in medicine and Physiology for their work on ionic basis of nerve membrane potentials.

Dorothy L.Hodkin, 1964, awarded Nobel Prize in Chemistry for their work in structure of complex organic molecules.

Francois Jacob, Andre M. Lwoff, and Jacques Monod, 1965, awarded Nobel Prize in medicine and physiology for their excellent work on bacterial operons, Lac-operon, and mRNAs.

Peyton Rous, 1966, for the discovery of Rous sarcoma virus, awarded Nobel Prize for this pioneering discovery of retroviruses.

Marshal Nierenberg, Severo Ochoa and Har Gobind Khorana, 1966, elucidated genetic code; H.G Khorana synthesized alanine tRNA, he also deciphered and decoded genetic code dictionary by filter based assays.

Esther Gillert, 1967, discovered DNA ligase.

H. Gobind Khorona, Marshal W. Nierenberg and Robert W. Holley, 1968, awarded Nobel Prize in Medicine and Physiology, for their path breaking work in elucidating Genetic code and tRNA structure. H.G Khorona elucidated the genetic code dictionary, by filter paper assays and also responsible for the synthesis of first gene in test tube. M.W Nierenberg was the first to elucidate the meaning of the first codon UUU means Phenylalanine. Robert W. Holley elucidated the structure of ala tRNA and proposed clover leaf model.

Max Delbruck, Alfred D.Hershey, Salvador E.Luria, 1969, awarded Nobel Prize in Medicine and Physiology for their work on genetics and structure of viruses.

Bernard Katz, Ulf S Voneuler, 1970, awarded Nobel Prize in Medicine and Physiology for their extraordinary work on Biochemistry of Nerve impulse propagation and transmission.

Luis F. Leloir, 1970, awarded Nobel Prize in Chemistry, for his work on the role of nucleotides in carbohydrate synthesis.

Earl Sutherland, 1971, Nobel Prize in Medicine and physiology for their work on the mechanism of hormone action and cyclic AMP.

Gerald Edelman, Rodney R. Porter, 1972, awarded Nobel Prize in medicine for their work on the structure of Immunoglobulins.

Christian B. Anfinsen, 1972, awarded Nobel Prize for his work on relationship between primary and tertiary structures of proteins.

Albert Claude, Christian de Duve, and George E. Palade, 1974 awarded Nobel Prize in medicine and physiology, for their work on internal structure of cells.

Richard Dickerson, 1974 did x-ray diffraction of synthesized oligos of CGCGAATTTCGCG for –B- DNA; GGCCGGCC, GGTATACC for A-DNA and CGCGCG sequences are used for Z-DNA.

David Baltimore, Renato Dulbecco and Howard M. Temin, 1975, awarded Nobel Prize for their work on Reverse transcriptase and tumor causing viral activities.

Ed Southern, 1975, Southern developed Southern blotting technique.

Sanger & Barrel, and Maxam & Walter Gilbert, 1975-75, devised techniques for DNA sequencing.

D. Carleton Gajdusek, 1976, awarded Nobel Prize in medicine and physiology for his work on Prion based diseases.

Herb Boyer, Stanley Cohen and Paul Berg, 1977-78, conducted DNA cloning experiments.

Werner Arber, Daniel Nathan and Hamilton Smith, 1978, awarded Nobel Prize for their work on restriction endonuclease technology.

Peter Mitchell, 1978, awarded Nobel Prize for his work on chemiosmotic mechanism in oxidative or photophosphorylation.

Paul Berg, Walter Gilbert, Frederick Sanger, 1980, awarded Nobel Prize in chemistry for their work on Recombinant DNA technology, DNA sequencing.

Baruj Bennacerraf, Jean Dausset, and George D.Snell, 1980, Nobel Prize in Medicine and Physiology, for their work on Major Histocompatibility complex.

Palmitter and Ralph Brienster, 1981-82, produced transgenic mice with human growth hormone gene.

Aaron Klug, 1982, Nobel Prize in chemistry, for his work on structure of Nucleic acid and Protein complex.

Barbara McClintock, 1983, Nobel Prize in medicine and Physiology for her work on mobile elements in genome especially in corn plants, her work at that time was ahead of her times, a Botanist par excellence.

George Kohler, Cesar Milstein, and Neils K. Jerne, 1984, Nobel Prize in medicine and physiology for their work on monoclonal antibodies.

Michael S. Brown and Joseph L.Goldstein, 1985, Nobel Prize in Medicine and Physiology for their work on regulation of cholesterol metabolism and endocytosis.

'''Kary Mullis et. al.,''', 1985, conducted the first PCR experiments.

Rita Levi-Montalcini and Stanley Cohen, 1986, Nobel Prize in Medicine and Physiology for their work on neuronal growth factors.

Susumu Tonegawa, 1987, Nobel Prize in medicine and physiology for the work on DNA rearrangement responsible for generating diverse antibodies.

Joahan Deisenhofer, Robert Huber and Hartmut Michel, 1988, Nobel Prize in chemistry, for their work on bacterial photosynthetic reaction center.

J. Michael Bishop, Harold Varmus, 1989, Nobel Prize chemistry, for their work on Oncogenes, that is cellular genes causing cancer or causing malignant transformation.

Thomas R. Cech, Sidney Altman, 1989, Nobel Prize in chemistry for their work on RNA as enzymes or work on Ribozymes.

Erwin Neher and Bert Sakmann, 1991, Nobel Prize in Medicine and Physiology for their work on measurement of ion flux by patch and clamp recording.

Edmond Fischer and Edwin Krebs, 1992, Nobel Prize in medicine and physiology for their work on alteration of enzyme activity by phosphorylation and dephosphorylation.

Kary Mullis and Michael Smith, 1993, Nobel Prize in chemistry for their pioneering work on polymerase chain reaction (PCR) experiments and site directed mutagenesis (SDM).

Rich Roberts and Phillip Sharp, 1993, Nobel Prize in Medicine and Physiology for their work on split genes and RNA processing or RNA splicing.

Alfred Gilman and Martin Rodbell, 1994, Nobel Prize in Medicine and Physiology for their work on structure and function of GTP-binding (G) proteins.

Edward B. Lewis, Christiane Nusslein-Volhard and Eric Wieschaus, 1995, Nobel Prize in Medicine and Physiology for their work on genetic control of embryonic development.

Jens C.Skou, Paul Boyer and John Walker, 1996, Nobel Prize in Chemistry for their work Na+/K+- ATPase and the mechanism of ATP synthesis.

Rolf M. Zinkernagel and Peter C. Doherty, 1997, Nobel Prize for their work on recognition of viral infected cell by immune system.

title: Molecular Biology/Milestones


text:
User talk:FinalCoyote

 Hello FinalCoyote, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:FinalCoyote


text:
User talk:Aeliana

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title: User talk:Aeliana


text:
Partial differential equations/Laplace Equation

Laplace Equation
The Laplace equation is a basic PDE that arises in the heat and diffusion equations. The Laplace equation is defined as: $$\nabla^2 u = 0 \Rightarrow \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0 ~.$$

Solution to Case with 1 Non-homogeneous Boundary Condition
In a condensed notation in $$(x,y,z)$$ rectangular coordinates, the Laplace equation in two dimensions reduces to: $$u_{xx}+u_{yy} = 0, u(x,y) ~.$$ The solution to the case with 1 non-homogeneous boundary condition is the most basic solution type. For the purposes of this example, we consider that the following boundary conditions hold true for this equation:
 * 1) $$x=0: u(0,y)=f(y)$$
 * 2) $$x=L: u(L,y)=0$$
 * 3) $$y=0: u(x,0)=0$$
 * 4) $$y=M: u(x,M)=0$$

Step 1: Separate Variables
To solve this equation, we assume that the function $$u(x,y)$$ is comprised of two functions $$\textstyle X(x)$$ and $$\textstyle Y(y)$$ such that $$\textstyle u(x,y)=X(x)Y(y)$$. Hence, $$u_{xx}=X(x)Y(y)$$ and $$u_{yy}=X(x)Y(y) ~.$$ Making the substitutions into the Laplace equation, we get: $$XY+XY=0$$ $$\Rightarrow \frac{X}{X} + \frac{Y}{Y} = 0$$ $$\Rightarrow \frac{X}{X} = -\frac{Y}{Y} = \mu$$ The $$\mu$$ is called a separation constant because the solution to the equation must yield a constant. Because of the separation constant, it yields two linear ODEs:
 * 1) $$X''-\mu X=0$$
 * 2) $$Y''+\mu Y=0$$

Step 2: Translate Boundary Conditions
Translating the boundary conditions allows us to divide the boundary conditions among the variables. This division yields:
 * $$u(0,y)=X(0)Y(y)=f(y) \quad \forall y \in \lbrack 0,M \rbrack$$
 * $$u(L,y)=X(L)Y(y)=0 \quad \forall y \in \lbrack 0,M \rbrack \Rightarrow X(L)=0$$
 * $$u(x,0)=X(x)Y(0)=0 \quad \forall x \in \lbrack 0,L \rbrack \Rightarrow Y(0)=0$$
 * $$u(x,M)=X(x)Y(M)=0 \quad \forall x \in \lbrack 0,L \rbrack \Rightarrow Y(M)=0$$

Step 3: Solve the Sturm-Liouville Problem
The last two boundary conditions are homogeneous boundary conditions for the function $$Y(y)$$. Using the solution to the Sturm-Liouville problems (SLP), we can easily get a function for $$Y(y)$$. The following is a fairly simple SLP: $$ \begin{cases} Y''+\mu Y = 0 \\ Y(0)=0 \\ Y(M)=0 \end{cases} $$ The solution to the SLP yields the following information: The solution we obtained is a family of solutions dependent on the value for n.
 * $$\mu = \frac{(n+1)^2\pi^2}{M^2},n=0,1,2,\cdots$$
 * $$Y_n(y)=\sin \frac{(n+1)\pi y}{M},n=0,1,2,\cdots$$

Step 4: Solve Remaining ODE
The remaining ODE that we have doesn't have a SLP solution to it because we only know one boundary condition for it. We have to use what we obtained from the SLP solution to help us solve this ODE. We obtained the following information from steps 1 and 2: $$ \begin{cases} X''-\mu X = 0 \\ X(L)=0 \end{cases} $$ From analyzing the second order ODE, we obtain the characteristic equation $$\textstyle m^2-\mu=0 \Rightarrow m^2=\mu \Rightarrow m = \pm \frac{(n+1)\pi}{M},n=0,1,2,\cdots ~.$$ Out of the solution set that results from the exponentials, the only viable solution that arises is: $$X_n(x)=C_1 \cosh \frac{(n+1)\pi(x-L)}{M} + C_2 \sinh \frac{(n+1)\pi(x-L)}{M} ~.$$ The substitution of (x-L) instead of x in the equation results only in a shift in the eigenspace, so it is valid and it helps us apply the boundary condition. Since $$X(L)=0, C_1=0$$. For convenience, we choose $$C_2=1~.$$ The resulting equation is: $$X_n(x)=\sinh \frac{(n+1)\pi(x-L)}{M}, n=0,1,2,\cdots ~.$$

Step 5: Combine Solutions
We obtained equations for our separate variable functions and now we can substitute into our assumption in step 1. The substitution yields: $$u_n(x,y)=\sinh \frac{(n+1)\pi(x-L)}{M} \sin \frac{(n+1)\pi y}{M}$$ This function only satisfies the 3 homogeneous boundary conditions, however. To solve for the solution to the non-homogeneous boundary condition, we must consider that the complete solution consists of the following infinite series of terms: $$ \begin{align} u(x,y) & :=\sum_{n=0}^\infty a_n u_n(x,y) \\ & = \sum_{n=0}^\infty a_n \sinh \frac{(n+1)\pi(x-L)}{M} \sin \frac{(n+1)\pi y}{M} \end{align} $$ At the non-homogeneous boundary condition: $$ f(y)=u(0,y)=\sum_{n=0}^\infty \underbrace {a_n \sinh \frac{(n+1)\pi(0-L)}{M}}_{A_n} \sin \frac{(n+1)\pi y}{M} $$ This is an orthogonal expansion of $$f$$ relative to the orthogonal basis of the sine function. The term $$A_n$$ is a Fourier coefficient which is defined as the inner product: $$A_n=\frac{\int\limits_{0}^{M} f(y)\sin(\frac{(n+1)\pi y}{M}) dy}{\int\limits_{0}^{M} \sin^2 \frac{(n+1)\pi y}{M}}=\frac{2}{M} \int\limits_{0}^{M} f(y)\sin(\frac{(n+1)\pi y}{M}) dy, n=0,1,2,\cdots$$ Thus, the coefficient of the infinite series solution $$a_n$$ is: $$a_n=-\frac{2}{M\sinh\frac{(n+1)\pi L}{M}}\int\limits_{0}^{M} f(y)\sin\left(\frac{(n+1)\pi y}{M}\right) dy, n=0,1,2,\cdots~.$$ So, the entire general solution to the Laplace equation is: $$u(x,y)=\sum_{n=0}^\infty \left [ -\frac{2}{M\sinh\frac{(n+1)\pi L}{M}}\int\limits_{0}^{M} f(y)\sin\left(\frac{(n+1)\pi y}{M}\right) dy \right ] \sinh \frac{(n+1)\pi(x-L)}{M} \sin \frac{(n+1)\pi y}{M}~.$$ This is the general solution for the specific set of boundary conditions we assumed at the beginning. Other boundary conditions will yield a different solution. You can see the solution graphically by entering in a partial sum (e.g. n starts at 0 and ends at 10) into a numerical solver like Mathematica or Maple.

Solution to Case with 4 Non-homogeneous Boundary Conditions
Because $$\nabla^2$$ is a linear operator, any solutions to the equation $$\nabla^2u=0$$ can be added together and the result will also be a solution to the equation. This is the superposition principle. Because of superposition, we can solve the case where all four boundary conditions are non-homogeneous. To illustrate how this will work, let's take the boundary conditions: $$ \begin{array}{lcl} u(x,0) & = & f_1(x) \\ u(x,M) & = & f_2(x) \\ u(0,y) & = & f_3(x) \\ u(L,y) & = & f_4(x) \end{array} $$ We divide this problem into 4 sub-problems, each one containing one of the non-homogeneous boundary conditions and each one subject to the Laplace equation condition, $$\frac{\partial^2}{\partial x^2}(u_n)+\frac{\partial^2}{\partial y^2}(u_n)=0~.$$ We get the following boundary conditions for the 4 sub-problems: u_1(x,0) & = & f_1(x) \\ u_1(x,M) & = & 0 \\ u_1(0,y) & = & 0 \\ u_1(L,y) & = & 0 \end{array} \right .$$ u_2(x,0) & = & 0 \\ u_2(x,M) & = & f_2(x) \\ u_2(0,y) & = & 0 \\ u_2(L,y) & = & 0 \end{array} \right .$$ u_3(x,0) & = & 0 \\ u_3(x,M) & = & 0 \\ u_3(0,y) & = & f_3(x) \\ u_3(L,y) & = & 0 \end{array} \right .$$ u_4(x,0) & = & 0 \\ u_4(x,M) & = & 0 \\ u_4(0,y) & = & 0 \\ u_4(L,y) & = & f_4(x) \end{array} \right .$$ In this case, we used all Dirichlet type boundary conditions, meaning that the boundary condition depends on the function value (e.g. $$u(x,y)=f(x)$$ ). If you are using any combination of Dirichlet, Neumann (e.g. $$u_x(x,y)=f(x) ~.$$ ), Robin (e.g. $$u(x,y)+u_x(x,y)=f(x)$$ ) types of boundary conditions, the types of the boundary conditions should be preserved in every sub-problem (e.g. if $$u_x(L,y)=f(x)$$, then the boundary condition is $$u_x(L,y)=0$$ when the boundary condition is made homogeneous and $$u_x(L,y)=f(x)$$ when it is the non-homogeneous boundary condition in a sub-problem). Using the superposition principle, the complete solution to the 4 non-homogeneous boundary condition case is constructed by adding up all the solutions from the 4 sub-problems. In equation form, $$u(x,y)=u_1(x,y)+u_2(x,y)+u_3(x,y)+u_4(x,y)~.$$ Each sub-problem can be solved using the method for the case with 1 non-homogeneous boundary condition as shown above.
 * 1) $$\left \{ \begin{array}{lcl}
 * 1) $$\left \{ \begin{array}{lcl}
 * 1) $$\left \{ \begin{array}{lcl}
 * 1) $$\left \{ \begin{array}{lcl}


text:
Partial differential equations/Poisson Equation

Definition
$$ \nabla^2 u = f \Rightarrow \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = f ~. $$

Description
Appears in almost every field of physics.

Solution to Case with 4 Homogeneous Boundary Conditions
Let's consider the following example, where $$u_{xx}+u_{yy}=F(x,y), (x,y) \in \lbrack 0,L \rbrack \times \lbrack 0,M \rbrack~.$$ and the Dirichlet boundary conditions are as follows: $$ \begin{align} u(0,y)&=&0 \\ u(L,y)&=&0 \\ u(x,0)&=&0 \\ u(x,M)&=&0 \\ \end{align} $$ In order to solve this equation, let's consider that the solution to the homogeneous equation will allow us to obtain a system of basis functions that satisfy the given boundary conditions. We start with the Laplace equation: $$u_{xx}+u_{yy}=0~.$$

Step 1: Separate Variables
Consider the solution to the Poisson equation as $$u(x,y)=X(x)Y(y)~.$$ Separating variables as in the solution to the Laplace equation yields: $$X''-\mu X=0$$ $$Y''+\mu Y=0$$

Step 2: Translate Boundary Conditions
As in the solution to the Laplace equation, translation of the boundary conditions yields: $$ \begin{alignat}{2} X(0) & = & 0\\ X(L) & = & 0\\ Y(0) & = & 0\\ Y(M) & = & 0 \end{alignat} $$

Step 3: Solve Both SLPs
Because all of the boundary conditions are homogeneous, we can solve both SLPs separately. $$ \left. \begin{alignat}{2} X''-\mu X & = & 0 \\ X(0) & = & 0 \\ X(L) & = & 0 \end{alignat} \right \} X_m(x) = \sinh\frac{m\pi x}{L},m=1,2,3,\cdots$$ $$ \left. \begin{alignat}{2} Y''+\mu Y & = & 0 \\ Y(0) & = & 0 \\ Y(M) & = & 0 \end{alignat} \right \} Y_n(y) = \sin \frac{n\pi y}{M},n=1,2,3,\cdots$$

Step 4: Solve Non-homogeneous Equation
Consider the solution to the non-homogeneous equation as follows: $$ \begin{align} u(x,y) & := \sum_{m,n=1}^\infty a_{mn}X_m(x)Y_n(y) \\ & = \sum_{m,n=1}^\infty a_{mn}\sinh \frac{m\pi x}{L}\sin \frac{n\pi y}{M} \end{align} $$ We substitute this into the Poisson equation and solve: $$ \begin{align} F(x,y)& = u_{xx}+u_{yy} \\

& = \sum_{m,n=1}^\infty \left \{ a_{mn}\left \lbrack \frac{m^2\pi^2}{L^2} \right \rbrack \sinh \frac{m\pi x}{L} \sin \frac{n\pi y}{M} \right \} + \left \{ a_{mn}\left \lbrack -\frac{n^2\pi^2}{M^2} \right \rbrack \sinh \frac{m\pi x}{L} \sin \frac{n\pi y}{M} \right \} \\

& = \sum_{m,n=1}^\infty \underbrace{\left [ a_{mn} \left ( \frac{m^2\pi^2}{L^2}-\frac{n^2\pi^2}{M^2} \right ) \right ]}_{A_{mn}} \sinh \frac{m\pi x}{L} \sin \frac{n\pi y}{M} \end{align} $$

$$ \begin{align} A_{mn}&=\frac{\int\limits_0^M \int \limits_0^L F(x,y) \sinh \frac{m\pi x}{L} \sin \frac{n\pi y}{M} dx dy}{\int\limits_0^M \sin^2 \frac{n\pi y}{M} dy \int\limits_0^L \sinh^2 \frac{m\pi x}{L} dx } \\ &=\frac{8\pi}{LM}\frac{m}{\left(\sinh(2\pi m) - 2\pi m\right)}\int\limits_0^M \int \limits_0^L F(x,y) \sinh \frac{m\pi x}{L} \sin \frac{n\pi y}{M} dx dy \end{align} $$

$$ a_{mn}=\frac{8\pi}{LM}\frac{m}{\left(\sinh(2\pi m) - 2\pi m\right)\left [ \frac{(m+1)^2\pi^2}{L^2} - \frac{(n+1)^2\pi^2}{M^2} \right ]}\int\limits_0^M \int \limits_0^L F(x,y) \sinh \frac{m\pi x}{L} \sin \frac{n\pi y}{M} dx dy; m,n=1,2,3,\cdots $$

Solution to General Case with 4 Non-homogeneous Boundary Conditions
Let's consider the following example, where $$u_{xx}+u_{yy}=F(x,y), (x,y) \in \lbrack 0,L \rbrack \times \lbrack 0,M \rbrack~.$$ and the boundary conditions are as follows: $$ \begin{align} u(x,0)&=f_1 \\ u(x,M)&=f_2 \\ u(0,y)&=f_3 \\ u(L,y)&=f_4 \end{align} $$ The boundary conditions can be Dirichlet, Neumann or Robin type.

Step 1: Decompose Problem
For the Poisson equation, we must decompose the problem into 2 sub-problems and use superposition to combine the separate solutions into one complete solution. \begin{cases} u_{xx}+u_{yy}=0 \\ u(x,0)=f_1 \\ u(x,M)=f_2 \\ u(0,y)=f_3 \\ u(L,y)=f_4 \end{cases} $$ \begin{cases} u_{xx}+u_{yy}=F(x,y) \\ u(x,0)=0 \\ u(x,M)=0 \\ u(0,y)=0 \\ u(L,y)=0 \end{cases} $$
 * 1) The first sub-problem is the homogeneous Laplace equation with the non-homogeneous boundary conditions. The individual conditions must retain their type (Dirichlet, Neumann or Robin type) in the sub-problem: $$
 * 1) The second sub-problem is the non-homogeneous Poisson equation with all homogeneous boundary conditions. The individual conditions must retain their type (Dirichlet, Neumann or Robin type) in the sub-problem: $$

Step 2: Solve Subproblems
Depending on how many boundary conditions are non-homogeneous, the Laplace equation problem will have to be subdivided into as many sub-problems. The Poisson sub-problem can be solved just as described above. f(x,y)=x+3*y-2

Step 3: Combine Solutions
The complete solution to the Poisson equation is the sum of the solution from the Laplace sub-problem $$u_1(x,y)$$ and the homogeneous Poisson sub-problem $$u_2(x,y)$$: $$ u(x,y)=u_1(x,y)+u_2(x,y) $$


text: File:Learning cycle.svg

title: File:Learning cycle.svg


text:
Heat equation/Solution to the 1-D Heat Equation

Definition
For the 1-dimensional case, the solution takes the form $$u(x,t)$$ since we are only concerned with one spatial direction and time. Thus the heat equation takes the form: $$u_t=ku_{xx}+h(x,t)$$ where k is our diffusivity constant and h(x,t) is the representation of internal heat sources. For our example, we impose the Robin boundary conditions, the initial condition, and the following bounds on our variables: $$(x,t)\in[0,L]\times[0,\infty)$$ $$u(x,0)=f(x)$$ $$\alpha_1u(0,t)-\beta_1u_x(0,t)=b_1(t)$$ $$\alpha_2u(L,t)+\beta_2u_x(L,t)=b_2(t)$$ $$\alpha_j,\beta_j \ge 0, \alpha_j^2+\beta_j^2 \ne 0$$ Notice that the boundary conditions are non-homogeneous and time-dependent. The next few steps will illustrate one way to achieve a solution.

Step 1: Partition Solution
In order to get a solution, we can partition the function into a "transient" or "variable" solution and a "steady-state" solution: $$u(x,t)=\underbrace{s(x,t)}_{\text{steady-state}}+\underbrace{v(x,t)}_{\text{variable}}$$ Substitute this relation into our original heat equation, boundary conditions and initial condition: $$s_t+v_t=ks_{xx}+kv_{xx}+h(x,t)$$ $$\alpha_1s(0,t)+\alpha_1v(0,t)-\beta_1s_x(0,t)-\beta_1v_x(0,t)=b_1(t)$$ $$\alpha_2s(L,t)+\alpha_2v(L,t)+\beta_2s_x(L,t)+\beta_2v_x(L,t)=b_2(t)$$ $$s(x,0)+v(x,0)=f(x)$$ We will also impose conditions on our partitioned solutions. Part of the beauty of partitioning the solution is being able to divide the boundary conditions among both parts. The following conditions will be imposed: Thus, we have constructed two separate initial boundary value problems (IBVPs): $$ \begin{cases} s_{xx} = 0 \\ \alpha_1s(0,t)-\beta_1s_x(0,t)=b_1(t) \\ \alpha_2s(L,t)+\beta_2s_x(L,t)=b_2(t) \end{cases} $$ $$ \begin{cases} v_t=kv_{xx}-s_t+h(x,t) \\ \alpha_1v(0,t)-\beta_1v_x(0,t)=0\\ \alpha_2v(L,t)+\beta_2v_x(L,t)=0\\ v(x,0)=f(x)-s(x,0) \end{cases} $$ Note that the sum $$s(x,t)+v(x,t)$$ satisfies all conditions of the original IBVP.
 * 1) We will choose the steady-state solution to be linear in x, which means that $$s_{xx}=0~.$$
 * 2) We will let the steady-state solution handle the non-homogeneous boundary conditions.
 * 3) We will let the variable portion satisfy the non-homogeneous equation and homogeneous boundary conditions.

Assumption 1
We are assuming that S is linear in x, so our equation take the form: $$s(x,t)=m(t)x+b(t)$$ We will see why this is only an assumption because doesn't cover all the different boundary condition types. Applying the boundary conditions, we get: $$ \begin{align} \alpha_1 & b(t)-\beta_1 & m(t) = & b_1(t) \\ \alpha_2 & b(t)+(\alpha_2 L+\beta_2) & m(t) = & b_2(t) \end{align} $$ Solving for $$b(t)$$ and $$m(t)$$ : $$ b(t)=\frac{\begin{vmatrix} b_1(t) & -\beta_1 \\ b_2(t) & \alpha_2 L+\beta_2 \end{vmatrix} }{\begin{vmatrix} \alpha_1 & -\beta_1 \\ \alpha_2 & \alpha_2 L+\beta_2 \end{vmatrix} }=\frac{(\alpha_2 L+\beta_2)b_1(t)+\beta_1b_2(t)}{\alpha_1\alpha_2L+\alpha_1\beta_2+\alpha_2\beta_1} $$ $$ m(t)=\frac{\begin{vmatrix} \alpha_1 & b_1(t) \\ \alpha_2 & b_2(t) \end{vmatrix} }{\begin{vmatrix} \alpha_1 & -\beta_1 \\ \alpha_2 & \alpha_2 L+\beta_2 \end{vmatrix} }=\frac{\alpha_1b_2(t)-\alpha_2b_1(t)}{\alpha_1\alpha_2L+\alpha_1\beta_2+\alpha_2\beta_1} $$ $$ s(x,t)=\frac{\alpha_1b_2(t)-\alpha_2b_1(t)}{\alpha_1\alpha_2L+\alpha_1\beta_2+\alpha_2\beta_1} x + \frac{(\alpha_2 L+\beta_2)b_1(t)+\beta_1b_2(t)}{\alpha_1\alpha_2L+\alpha_1\beta_2+\alpha_2\beta_1} $$ That means that the coefficients are only defined as long as $$\alpha_1\alpha_2L+\alpha_1\beta_2+\alpha_2\beta_1 \ne 0~.$$ This means that the linear assumption of s(x,t) only holds if the boundary conditions are not both Neumann type (i.e. $$\alpha_1,\alpha_2 \ne 0$$ ). Knowing the coefficients $$\alpha_1,\alpha_2,\beta_1,\beta_2$$ from the original boundary conditions will yield the coefficients for s(x,t).

Assumption 2
For the case where both boundary conditions are Neumann type, a different assumption must be made. The linear assumption didn't yield any solution, so let's assume s has a quadratic relationship with x: $$s(x,t)=a(t)x^2+b(t)x+c(t)~.$$ For convenience, we choose $$c(t)=0~.$$ In this Neumann-Neumann condition, the only boundary conditions we have are: $$ \begin{cases} s_x(0,t)=b_1(t), \beta_1=1 \text{ for simplicity}\\ s_x(L,t)=b_2(t), \beta_2=1 \text{ for simplicity} \end{cases} $$ Solving the assumption for the boundary conditions: $$ \begin{align} s_x(0,t)&=b_1(t)=b(t) \\ s_x(L,t)&=b_2(t)=2a(t)L+b(t) \end{align} \quad \Rightarrow \quad \left \{ \begin{align} b(t)&=b_1(t)\\ a(t)&=\frac{b_2(t)-b_1(t)}{2L} \end{align} \right. $$ Thus we arrive at the solution to the "steady-state" solution: $$s(x,t)=\frac{b_2(t)-b_1(t)}{2L} x^2 + b_1(t) x~.$$

Step 3: Solve Variable Partition
The equation for the variable portion is: $$v_t=kv_{xx}-s_t+h(x,t)+ks_{xx}$$

For Assumption 1
In order to simplify the equation, we can define a new function: $$q(x,t):=h(x,t)-s_t(x,t)$$

For Assumption 2
In order to simplify the equation, we can define a new function: $$q(x,t):=h(x,t)-s_t(x,t)+ks_{xx}(x,t)$$

Step 3.1: Solve the Associated Homogeneous IBVP
The associated homogeneous IBVP is as follows: $$ \begin{cases} v_t=kv_{xx} \\ \alpha_1v(0,t)-\beta_1v_x(0,t)=0 \\ \alpha_2v(L,t)+\beta_2v_x(L,t)=0 \end{cases} $$

Separate Variables
$$v(x,t)=X(x)T(t) \Rightarrow XT'=kXT \Rightarrow \frac{T'}{kT} = \frac{X}{X} = \mu$$

Translate Boundary Conditions
$$ \begin{align} 0 & = \alpha_1X(0)T(t)-\beta_1X'(0)T(t) & \\ & = \left [ \alpha_1X(0)-\beta_1X'(0) \right ]T(t) & \forall t \in [0,\infty) \\ \end{align} \quad \Rightarrow \quad \alpha_1X(0)-\beta_1X'(0)=0 $$ $$ \begin{align} 0 & = \alpha_2X(L)T(t)+\beta_2X'(L)T(t) & \\ & = \left [ \alpha_2X(L)+\beta_2X'(L) \right ]T(t) & \forall t \in [0,\infty) \\ \end{align} \quad \Rightarrow \quad \alpha_2X(L)+\beta_2X'(L)=0 $$

Solve the SLP
$$ \left. \begin{align} X''-\mu X = 0\\ \alpha_1X(0)-\beta_1X'(0)=0 \\ \alpha_2X(L)+\beta_2X'(L)=0 \end{align} \right \} \begin{align} -\mu & = \lambda^2 \\ \lambda_n & =\text{ solutions to equation } (\alpha_1\alpha_2-\beta_1\beta_2\lambda^2)\sin (\lambda L)+(\alpha_1\beta_2+\alpha_2\beta_1)\lambda \cos (\lambda L) = 0 \\ X_n(x) & =\beta_1\lambda_n \cos (\lambda_n x) + \alpha_1 \sin(\lambda_n x) \end{align} $$

Step 3.2: Solve the Non-homogeneous BVP
The BVP is as follows: $$ \begin{cases} v_t=kv_{xx}+q(x,t) \\ \alpha_1v(0,t)-\beta_1v_x(0,t)=0 \\ \alpha_2v(L,t)+\beta_2v_x(L,t)=0 \\ v(x,0)=f(x)-s(x,0) \end{cases} $$ In order to solve the problem, let: $$v(x,t):=\sum_{n=0}^\infty T_n(t)X_n(x)$$ where the time-dependent Fourier coefficients will be determined later. Also, let: $$q(x,t):=\sum_{n=0}^\infty Q_n(t)X_n(x)$$ where the Fourier coefficients are given by the inner product: $$Q_n(t)=\frac{\int\limits_0^L q(x,t)X_n(x) dx}{\int\limits_0^L X_n(x) \cdot X_n(x) dx}~.$$ In order to solve for the coefficients $$T_n(t)$$, we have to substitute the relations above into the original PDE. The term-by-term differentiation of v is legitimate since the partial derivatives of v are continuous. v satisfies the same spatial boundary conditions as the eigenfunction: $$\frac{\partial}{\partial t} \left [ \sum_{n=0}^\infty T_n(t)X_n(x) \right ] = k \frac{\partial^2}{\partial x^2} \left [ \sum_{n=0}^\infty T_n(t)X_n(x) \right ] + \sum_{n=0}^\infty Q_n(t)X_n(x)$$ $$\sum_{n=0}^\infty T_n'(t)X_n(x) = k \sum_{n=0}^\infty T_n(t)X_n''(x) + \sum_{n=0}^\infty Q_n(t)X_n(x)$$ Since the eigenfunction $$X_n(x)$$ satisfies the ODE $$X+\lambda_n^2 X=0 \Rightarrow X_n(x)=-\lambda_n^2 X_n(x)$$ , $$\sum_{n=0}^\infty T_n'(t)X_n(x) = \sum_{n=0}^\infty -k\lambda_n^2T_n(t)X_n(x) + \sum_{n=0}^\infty Q_n(t)X_n(x)$$ $$\sum_{n=0}^\infty \left [ T_n'(t)+k\lambda_n^2T_n(t) \right ] X_n(x) = \sum_{n=0}^\infty Q_n(t)X_n(x)$$ Since $$X_n(x)$$ forms a basis and is therefore linearly independent: $$T_n'(t)+k\lambda_n^2T_n(t)=Q_n(t)$$ which is a first order non-homogeneous linear equation. The integrating factor $$\mu(t)=e^{\int k\lambda_n^2 dt}=e^{k\lambda_n^2t}$$. Solving the equation yields: $$T_n(t)=e^{-k\lambda_n^2t} \int\limits_0^t e^{k\lambda_n^2s}Q_n(s) ds + C_ne^{-k\lambda_n^2t}$$ The constant $$C_n$$ allows us to satisfy the initial condition: $$v(x,0)=f(x)-s(x,0)=\sum_{n=0}^\infty T_n(0)X_n(x)=\sum_{n=0}^\infty C_nX_n(x)~.$$ $$C_n=\frac{\int\limits_0^L \left [ f(x)-s(x,0) \right ] X_n(x) dx}{\int\limits_0^L X_n(x) \cdot X_n(x) dx}~.$$ We can now substitute all the expressions to get the variable portion $$v(x,t)$$:
 * 1) Compute and substitute $$C_n$$ and $$Q_n(t)$$ into $$T_n(t)~.$$
 * 2) Compute and substitute $$T_n(t)$$ and $$X_n(x)$$ into $$v(x,t):=\sum_{n=0}^\infty T_n(t)X_n(x)~.$$

Step 4: Combine Solutions
The complete solution for $$u(x,t)$$ can be found by adding the "steady-state" solution and the "variable" solutions. The result can easily be checked by graphing in a symbolic solver like Mathematica or Maple.


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Heat equation/Solution to the 2-D Heat Equation

Definition
The solution to the 2-dimensional heat equation (in rectangular coordinates) deals with two spatial and a time dimension, $$u(x,y,t)$$. The heat equation, the variable limits, the Robin boundary conditions, and the initial condition are defined as: $$ \begin{align} u_t=k \left [ u_{xx} + u_{yy} \right ] + h(x,y,t) \\ (x,y,t) \in [0,L] \times [0,M] \times [0,\infty) \\ \alpha_1u(0,y,t)-\beta_1u_x(0,y,t)=b_1(y,t) \\ \alpha_2u(L,y,t)+\beta_2u_x(L,y,t)=b_2(y,t) \\ \alpha_3u(x,0,t)-\beta_3u_y(x,0,t)=b_3(x,t) \\ \alpha_4u(x,M,t)+\beta_4u_y(x,M,t)=b_4(x,t) \\ u(x,y,0)=f(x,y) \end{align} $$

Solution
The solution is just an advanced version of the solution in 1 dimension. If you have questions about the steps shown here, review the 1-D solution.

Step 1: Partition Solution
Just as in the 1-D solution, we partition the solution into a "steady-state" and a "variable" portion: $$ u(x,y,t)=\underbrace{s(x,y,t)}_{ \text{steady-state} }+\underbrace{v(x,y,t)}_{ \text{variable} } $$ We substitute this equation into the initial boundary value problem (IBVP): $$ \begin{cases} s_t+v_t=k \left [ s_{xx}+v_{xx}+s_{yy}+v_{yy} \right ] + h(x,y,t) \\ \alpha_1s(0,y,t)+\alpha_1v(0,y,t)-\beta_1s_x(0,y,t)-\beta_1v_x(0,y,t)=b_1(y,t) \\ \alpha_2s(0,y,t)+\alpha_2v(L,y,t)+\beta_2s_x(L,y,t)+\beta_2v_x(L,y,t)=b_2(y,t) \\ \alpha_3s(x,0,t)+\alpha_3v(x,0,t)-\beta_3s_y(x,0,t)-\beta_3v_y(x,0,t)=b_3(x,t) \\ \alpha_4s(x,M,t)+\alpha_4v(x,M,t)+\beta_4s_y(x,M,t)+\beta_4v_y(x,M,t)=b_4(x,t) \\ s(x,y,0)+v(x,y,0)=f(x,y) \end{cases} $$ We want to set some conditions on s and v: We end up with 2 separate IBVPs: $$ \begin{cases} s_{xx}+s_{yy}=0 \\ \alpha_1s(0,y,t)-\beta_1s_x(0,y,t)=b_1(y,t) \\ \alpha_2s(0,y,t)+\beta_2s_x(L,y,t)=b_2(y,t) \\ \alpha_3s(x,0,t)-\beta_3s_y(x,0,t)=b_3(x,t) \\ \alpha_4s(x,M,t)+\beta_4s_y(x,M,t)=b_4(x,t) \\ \end{cases} $$ $$ \begin{cases} v_t=k \left [ v_{xx}+v_{yy} \right ] + h(x,y,t) - s_t(x,y,t) \\ \alpha_1v(0,y,t)-\beta_1v_x(0,y,t)=0 \\ \alpha_2v(L,y,t)+\beta_2v_x(L,y,t)=0 \\ \alpha_3v(x,0,t)-\beta_3v_y(x,0,t)=0 \\ \alpha_4v(x,M,t)+\beta_4v_y(x,M,t)=0 \\ v(x,y,0)=f(x,y)-s(x,y,0) \end{cases} $$
 * 1) Let $$s$$ satisfy the Laplace equation: $$s_{xx}+s_{yy}=0~.$$
 * 2) Let $$s$$ satisfy the non-homogeneous boundary conditions.
 * 3) Let v satisfy the non-homogeneous equation and homogeneous boundary conditions.

Step 2: Solve Steady-State Portion
Solving for the steady-state portion is exactly like solving the Laplace equation with 4 non-homogeneous boundary conditions. Using that technique, a solution can be found for all types of boundary conditions.

Step 3.1: Solve Associated Homogeneous BVP
The associated homogeneous BVP equation is: $$v_t=k \left [ v_{xx}+v_{yy} \right ]$$ The boundary conditions for v are the ones in the IBVP above.

Separate Variables
$$v(x,y,t)=X(x)Y(y)T(t)$$ $$\Rightarrow XYT'=k \lbrack XYT+XYT \rbrack$$ $$\Rightarrow \frac{T'}{kT} = \frac{X}{X} + \frac{Y}{Y} = \mu$$ By similar methods, you obtain the following ODEs: $$ \begin{cases} T'-\mu kT=0 \\ X''-\rho X=0 \\ Y''-\delta Y=0 \\ \mu=\rho + \delta \quad \text{ (coupling equation) } \end{cases} $$

Translate Boundary Conditions
$$ \left. \begin{align} \left [ \alpha_1X(0)-\beta_1X'(0) \right ] Y(y)T(t) & = 0 \\ \left [ \alpha_2X(L)+\beta_2X'(L) \right ] Y(y)T(t) & = 0 \\ \left [ \alpha_3Y(0)-\beta_3Y'(0) \right ] X(x)T(t) & = 0 \\ \left [ \alpha_4Y(M)+\beta_4Y'(M) \right ] X(x)T(t) & = 0 \end{align} \right \} \Rightarrow \begin{align} \alpha_1X(0)-\beta_1X'(0) = 0 \\ \alpha_2X(L)+\beta_2X'(L) = 0 \\ \alpha_3Y(0)-\beta_3Y'(0) = 0 \\ \alpha_4Y(M)+\beta_4Y'(M) = 0 \end{align} $$

Solve SLPs
$$ \left. \begin{align} X''-\rho X=0\\ \alpha_1X(0)-\beta_1X'(0) = 0 \\ \alpha_2X(L)+\beta_2X'(L) = 0 \end{align} \right \} \begin{align} & -\rho =\lambda^2 \\ & \text{Eigenvalues } \lambda_n \text{: solutions to equation } (\alpha_1\alpha_2-\beta_1\beta_2\lambda^2) \sin (\lambda L) + (\alpha_1\beta_2+\alpha_2\beta_1)\lambda \cos (\lambda L)=0 \\ & X_n(x)=\beta_1\lambda_n \cos (\lambda_n x) + \alpha_1 \sin (\lambda_n x), n=0,1,2,\cdots \end{align} $$ $$ \left. \begin{align} Y''-\delta Y=0 \\ \alpha_3Y(0)-\beta_3Y'(0) = 0 \\ \alpha_4Y(M)+\beta_4Y'(M) = 0 \end{align} \right \} \begin{align} & -\delta =\hat \lambda^2 \ \\ & \text{Eigenvalues } \hat{\lambda}_m \text{: solutions to equation } (\alpha_3\alpha_4-\beta_3\beta_4 \hat{\lambda}^2) \sin (\hat{\lambda}M ) + (\alpha_3\beta_4+\alpha_4\beta_3) \hat{\lambda} \cos ( \hat{\lambda} M)=0 \\ & Y_m(x)=\beta_3 (\hat{\lambda}_m ) \cos ( \hat{\lambda}_m y) + \alpha_3 \sin ( \hat{\lambda}_m y), m=0,1,2,\cdots \end{align} $$ We have obtained eigenfunctions that we can use to solve the nonhomogeneous IBVP.

Setup Problem
Just like in the 1-D case, we define v(x,y,t) and q(x,y,t) as infinite sums: $$ v(x,y,t):=\sum_{m,n=0}^\infty T_{mn}(t)X_n(x)Y_m(y) $$ $$ q(x,y,t):=\sum_{m,n=0}^\infty Q_{mn}(t)X_n(x)Y_m(y)~,~Q_{mn}(t)=\frac{\int\limits_0^L \int\limits_0^M q(x,y,t)X_n(x)Y_m(y) dy dx}{\int\limits_0^L X_n^2(x) dx \int\limits_0^M Y_m^2(y) dy} $$

Determine Coefficients
We then substitute expansion into the PDE: $$\frac{\partial}{\partial t} \left [ \sum T_{mn}(t)X_n(x)Y_m(y) \right ] = k \left \{ \frac{\partial}{\partial x^2} \left [ \sum T_{mn}(t)X_n(x)Y_m(y) \right ] + \frac{\partial}{\partial y^2} \left [ \sum T_{mn}(t)X_n(x)Y_m(y) \right ] \right \} + \sum Q_{mn}(t)X_n(x)Y_m(y)$$

$$\Rightarrow \sum T_{mn}'(t)X_n(x)Y_m(y) = k \left \{ \sum T_{mn}(t)X_n(x)Y_m(y) + \sum T_{mn}(t)X_n(x)Y_m(y) \right \} + \sum Q_{mn}(t)X_n(x)Y_m(y)$$

$$\Rightarrow \sum T_{mn}'(t)X_n(x)Y_m(y) = \sum k \left \{ T_{mn}(t)[-\lambda_n^2X_n(x)]Y_m(y) + \sum T_{mn}(t)X_n(x)[-\hat{\lambda}_m^2Y_m(y)] \right \} + \sum Q_{mn}(t)X_n(x)Y_m(y)$$

$$\Rightarrow \sum \left [ T_{mn}'(t)+k(\lambda_n^2+\hat{\lambda}_m^2) \right ] X_n(x)Y_m(y) = \sum Q_{mn}(t)X_n(x)Y_m(y)$$

This implies that $$X_n(x) \otimes Y_m(y)~$$ forms an orthogonal basis. This means that we can write the following:

$$\Rightarrow T_{mn}'(t)+k(\lambda_n^2+\hat{\lambda}_m^2)=Q_{mn}(t)$$

This is a first-order ODE which can be solved using the integration factor:

$$\mu(t)=e^{\int k(\lambda_n^2+\hat{\lambda}_m^2) dt}=e^{k(\lambda_n^2+\hat{\lambda}_m^2)t}$$

Solving for our coefficient we get:

$$T_{mn}(t)=e^{-k(\lambda_n^2+\hat{\lambda}_m^2)t} \int\limits_0^t e^{k(\lambda_n^2+\hat{\lambda}_m^2)s} Q_{mn}(s) ds + C_{mn}e^{-k(\lambda_n^2+\hat{\lambda}_m^2)t}$$

Satisfy Initial Condition
We apply the initial condition to our equation above: $$ \begin{align} v(x,y,0)&=f(x,y)-s(x,y,0) \\ &=\sum T_{mn}(0)X_n(x)Y_m(y) \\ &=\sum C_{mn}X_n(x)Y_m(y) \end{align} $$ The Fourier coefficients can be solved using the inner product definition: $$ C_{mn}=\frac{\int\limits_0^L \int\limits_0^M \left [ f(x,y)-s(x,y,0) \right ] X_n(x)Y_m(y) dy dx}{\int\limits_0^L X_n^2(x) dx \int\limits_0^M Y_m^2(y) dy} $$ We have all the necessary information about the variable portion of the function.

Step 4: Combine Solutions
We now have solved for the "steady-state" and "variable" portions, so we just add them together to get the complete solution to the 2-D heat equation.


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