OpenLLM-France/wikimedia huggingface dataset:
# English-Chinese/Outside the mood disorders

Outside the mood disorders: borderline personality disorder often features an extremely intense depressive mood; adjustment disorder with depressed mood is a mood disturbance appearing as a psychological response to an identifiable event or stressor, in which the resulting emotional or behavioral symptoms are significant but do not meet the criteria for a major depressive episode; and posttraumatic stress disorder, a mental disorder that sometimes follows trauma, is commonly accompanied by depressed mood.
情绪障碍之外：边缘型人格障碍往往具有极度强烈的抑郁情绪; 情绪低落的情绪障碍是一种情绪障碍，表现为对可识别事件或压力源的心理反应，其中由此产生的情绪或行为症状是显着的，但不符合重度抑郁发作的标准; 和创伤后应激障碍，一种有时伴随创伤的精神障碍，通常伴有抑郁情绪。
# English-Chinese/Other symptoms of psychosis may include incoherent speech and behavior that is inappropriate for the situation

Other symptoms of psychosis may include incoherent speech and behavior that is inappropriate for the situation.
精神病的其他症状可能包括不连贯的言语和不适合这种情况的行为。
# English-Chinese/Outer space

Space, also known as outer space, is the near-vacuum between celestial bodies. It is where everything (all of the planets, stars, galaxies and other objects) is found.
太空，也 被称为 外太空，是 天体 之间的 近真空。它是 所有东西（所有 行星，恒星，星系 和 其他 物体）被发现的 地方。
# English-Chinese/Over the centuries, better observations and better ideas of gravity led to Copernicus's Sun-centred model

Over the centuries, better observations and better ideas of gravity led to Copernicus's Sun-centred model.
几个世纪以来，更好的观察和更好的重力思想导致了哥白尼以太阳为中心的模型。
# English-Chinese/Other side effects of olanzapine include hypotension when standing, allergic reactions, antipsychotic malignant syndrome, hyperglycemia, seizures, gynecomastia and tardive dyskinesia

Other side effects of olanzapine include hypotension when standing, allergic reactions, antipsychotic malignant syndrome, hyperglycemia, seizures, gynecomastia and tardive dyskinesia.
奥氮平的其他副作用包括站立时的低血压，过敏反应，抗精神病药恶性综合征，高血糖，癫痫发作，男子女性型乳房和迟发性运动障碍。
# English-Chinese/Other signs and symptoms of diabetes include blurred vision, headache, fatigue, slow healing of cuts, and itchy skin

Other signs and symptoms of diabetes include blurred vision, headache, fatigue, slow healing of cuts, and itchy skin.
糖尿病的其他症状和体征包括视力模糊，头痛，疲劳，伤口愈合缓慢和皮肤瘙痒。
# Bloom Clock/Keys/Southeastern Pennsylvania/June/Black Flowers/DPL

This is a DPL-generated list, used to create the key for Black-flowered plants Southeastern Pennsylvania in June
* BCP/Arisaema ringens
* BCP/Iris
* BCP/Aquilegia x hybrida
* BCP/Viola tricolor
* BCP/Hemerocallis
* BCP/Rosa
* BCP/Viola
* BCP/Arisaema ringens
* BCP/Iris
* BCP/Aquilegia x hybrida
* BCP/Viola tricolor
* BCP/Hemerocallis
* BCP/Rosa
* BCP/Viola
# Bloom Clock/Keys/Southeastern Pennsylvania/June/Green Flowers/DPL

This is a DPL-generated list, used to create the key for Green-flowered plants Southeastern Pennsylvania in June
* BCP/Amaranthus blitoides
* BCP/Arisaema ringens
* BCP/Dactylis glomerata
* BCP/Schizachne purpurascens
* BCP/Vitis aestivalis
* BCP/Alchemilla mollis
* BCP/Vitis × labruscana
* BCP/Asparagus officinalis
* BCP/Cotinus coggygria
* BCP/Smilax herbacea
* BCP/Toxicodendron radicans
* BCP/Phalaris arundinacea
* BCP/Arisaema triphyllum
* BCP/Plantago major
* BCP/Matricaria discoidea
* BCP/Spigelia marilandica
* BCP/Rhus typhina
* BCP/Amaranthus blitoides
* BCP/Arisaema ringens
* BCP/Dactylis glomerata
* BCP/Schizachne purpurascens
* BCP/Vitis aestivalis
* BCP/Alchemilla mollis
* BCP/Vitis × labruscana
* BCP/Asparagus officinalis
* BCP/Cotinus coggygria
* BCP/Smilax herbacea
* BCP/Toxicodendron radicans
* BCP/Phalaris arundinacea
* BCP/Arisaema triphyllum
* BCP/Plantago major
* BCP/Matricaria discoidea
* BCP/Spigelia marilandica
* BCP/Rhus typhina
# Bloom Clock/Keys/Southeastern Pennsylvania/June
## June

* White flowers (110)
* Pink flowers (53)
* Red flowers (24)
* Orange flowers (18)
* Yellow flowers (60)
* Green flowers (14)
* Blue flowers (20)
* Purple flowers (41)
* Brown flowers (8)
* Black flowers (7)
# Bloom Clock/Keys/Southeastern Pennsylvania/June/Blue Flowers/DPL

This is a DPL-generated list, used to create the key for Blue-flowered plants Southeastern Pennsylvania in June
* BCP/Cichorium intybus
* BCP/Clematis
* BCP/Salvia nemorosa
* BCP/Brunnera macrophylla
* BCP/Nepeta x faassenii
* BCP/Iris sibirica
* BCP/Torenia fournieri
* BCP/Vinca sardoa
* BCP/Platycodon grandiflorus
* BCP/Geranium
* BCP/Geranium maculatum
* BCP/Geranium pratense
* BCP/Commelina communis
* BCP/Veronica americana
* BCP/Iris
* BCP/Ageratum houstonianum
* BCP/Glechoma hederacea
* BCP/Prunella vulgaris
* BCP/Myosotis
* BCP/Hydrangea macrophylla
* BCP/Sisyrinchium montanum
* BCP/Tradescantia
* BCP/Tradescantia virginiana
* BCP/Salvia guaranitica
* BCP/Viola
* BCP/Centaurea montana
* BCP/Nepeta subsessilis
* BCP/Cichorium intybus
* BCP/Clematis
* BCP/Salvia nemorosa
* BCP/Brunnera macrophylla
* BCP/Nepeta x faassenii
* BCP/Iris sibirica
* BCP/Torenia fournieri
* BCP/Vinca sardoa
* BCP/Platycodon grandiflorus
* BCP/Geranium
* BCP/Geranium maculatum
* BCP/Geranium pratense
* BCP/Commelina communis
* BCP/Veronica americana
* BCP/Iris
* BCP/Ageratum houstonianum
* BCP/Glechoma hederacea
* BCP/Prunella vulgaris
* BCP/Myosotis
* BCP/Hydrangea macrophylla
* BCP/Sisyrinchium montanum
* BCP/Tradescantia
* BCP/Tradescantia virginiana
* BCP/Salvia guaranitica
* BCP/Viola
* BCP/Centaurea montana
* BCP/Nepeta subsessilis
# Bloom Clock/Keys/Southeastern Pennsylvania/June/Brown Flowers/DPL

This is a DPL-generated list, used to create the key for Brown-flowered plants Southeastern Pennsylvania in June
* BCP/Asarum europaeum
* BCP/Arisaema triphyllum
* BCP/Arisaema ringens
* BCP/Iris
* BCP/Viola tricolor
* BCP/Hemerocallis
* BCP/Rosa
* BCP/Viola
* BCP/Asarum europaeum
* BCP/Arisaema triphyllum
* BCP/Arisaema ringens
* BCP/Iris
* BCP/Viola tricolor
* BCP/Hemerocallis
* BCP/Rosa
* BCP/Viola
# Data Networking/Spring 2015/GROUP4/VPN
## CONFIGURATION AT THE VPN SERVER

1.Install pptpd and ufw
2.open the ports needed.The guide suggest 22 ssh and 1723 for pptpvpn
3.edit /etc/ppp/pptpd.conf
4. edit /etc/ppp/chap-secrets
5.Restart pptpd,To run this command
6.edit /etc/sysctl.conf uncomment the following for enabling IP forwarding
To reload configuration:

## CONFIGURATION AT VPN CLIENT

1.Install PPTP in client machine
2.Add required Kernel module
3.Now open the peers file and add the following commands.
4.call the pptpd server using this command .Establish connection between pptpd server and client
5.Now open the syslog file in which we can see successful connection from PPTP server
# Data Networking/Spring 2015/GROUP4/Web Server Backup
## configuration of back up server

Backup server must be in a rot mode before the execution.
After entering rootmode
# Data Networking/Spring 2015/GROUP4/WEB-SERVER

o configure and implement an Apache web server, the software of Apache server has to be installed.
But before installing the software, all the package updates available and supported by host Linux machine have to be installed.
To update all the available packages.
To install the apache web server on host machine
When the web server is installed, a default html page is created in the local host which can be accessed by other hosts in the network.
To configure the web pages of the organization, html pages are created and placed in the default fetch directory /var/www. The pages in this path are fetched by the server when a http request is generated to the server.
To implement the web server browse the localhost/IP address of the host from web browser of the server machine. Browser should be able to load the content of the configured web page of the server.
If the html pages which are to be accessed are located in /Directory path/, change the path from which the server fetches the contents.
In this file, replace the default path from /var/www to /Directory path/, save the file.
Once the configuration is done, the webserver has to be started.
In case, if the server needs to be stopped
The web server has to be restarted whenever configurations changes are made.
# Data Networking/Spring 2015/GROUP4/Virtual hosting

1.create new file in sites-available folder that will be used by server to access the new site
2.Now change the directory to sites-available using below command
3.Now open the new site2.conf file and modify the directory path and server name using below command
4.save the file and restart the apache server
# English-Chinese/Sino-Tibetan languages

The Sino-Tibetan languages form a language family composed of, at least, the Chinese and the Tibeto-Burman languages, including some 250 languages of East Asia. They are second only to the Indo-European languages in terms of their number of speakers.
汉藏语言 构成了 一个 语言家族，至少 包括 中文 和 藏缅语言，包括 约 250种 东亚语言。就 使用人数 而言，它们 仅次于 印欧语系。
# English-Chinese/Sinusitis

Sinusitis is inflammation of the sinuses resulting in symptoms. Common symptoms include thick nasal mucus, a plugged nose, and pain in the face. Other symptoms may include fever, headaches, poor sense of smell, sore throat, and cough. The cough is often worse at night. Serious complications are rare. It is defined as acute rhinosinusitis (ARS) if it lasts less than 4 weeks, and as chronic rhinosinusitis (CRS) if it lasts for more than 12 weeks.
鼻窦炎 是 导致 多种 症状的 鼻窦 发炎。常见 症状 包括 鼻粘稠，鼻塞，脸部 疼痛。其他 症状 可能 包括 发烧，头痛，嗅觉不佳，喉咙痛 和 咳嗽。夜间 咳嗽 往往 更糟。严重的 并发症 很少见。如果 持续 时间 少于 4周，则 定义为 急性鼻窦炎（ARS）；如果 持续 超过 12周，则 定义为 慢性鼻窦炎（CRS）。
# English-Chinese/Skin

Skin is the outside covering of animals. The skin of different kinds of animals is very different. Many kinds of animals have hair or fur on their skin. Birds have feathers on their skin. Most fish, and reptiles, like snakes and lizards, have scales on their skin.
皮肤 是 动物的 外部 覆盖物。不同 种类 动物的 皮肤 差别 很大。许多 动物的 皮肤上 都有 毛发。鸟 在 它们的 皮肤上 有 羽毛。大多数 鱼类 和 爬行动物，如 蛇 和 蜥蜴，皮肤上 都有 鳞屑。
# English-Chinese/Sleep increases the sensory threshold. In other words, sleeping persons perceive fewer stimuli, but can generally still respond to loud noises and other salient sensory events

Sleep increases the sensory threshold. In other words, sleeping persons perceive fewer stimuli, but can generally still respond to loud noises and other salient sensory events.
睡眠会增加感觉阈值。 换句话说，睡着的人感觉到较少的刺激，但通常仍能对大声的声音和其他突出的感觉有反应。
# English-Chinese/Skin allergies frequently cause rashes, or swelling and inflammation within the skin

Skin allergies frequently cause rashes, or swelling and inflammation within the skin, in what is known as a "weal and flare" reaction characteristic of hives and angioedema.
皮肤过敏经常引起皮疹，皮肤的肿胀和炎症，这被称为荨麻疹和血管水肿的“红肿和灼热”反应特征。
# English-Chinese/Sleep is a highly conserved behavior across animal evolution

Sleep is a highly conserved behavior across animal evolution.
睡眠是动物进化过程中高度保守的行为。
# English-Chinese/Skeleton

A skeleton is the hard structure that protects the internal organs of a living thing. Skeletons can be inside the body or outside the body. In mammals, which include humans, the skeleton is made of bones. All the bones, when they are joined together, make the "skeletal system" of a body. The skeletal system or "skeleton" is under the skin, the muscle and the tissue of the body. The skeleton supports the skin, muscle and tissue, and all the organs that are inside the body. The skeleton protects important internal organs like the brain, heart and lungs. If humans did not have a skeleton then the body would be flat as the skeleton gives the body its frame.
骨骼 是 保护 生物 内脏的 坚硬 结构。骨骼 可以 在 体内 或 体外。在 包括 人类 在内的 哺乳动物中，骨骼 由 骨头 制成。所有的 骨头，当 它们 连在一起 时，都会 形成 一个 身体的 “骨架系统”。骨骼系统 或 “骨骼” 位于 皮肤，肌肉 和 身体组织 之下。骨骼 支撑 身体 内部的 皮肤，肌肉 和 组织 以及 所有 器官。骨骼 保护 重要的 内部器官，如 大脑，心脏 和 肺部。如果 人类 没有 骨骼，那么 身体 就会 变得 平坦，因为 骨骼 会 给 身体 提供 框架。
# English-Chinese/Skipping school is a bad thing to do

Skipping school is a bad thing to do.
逃学是一件坏事。
# English-Chinese/Sleep disorders such as sleep disordered breathing and insomnia, as well as particularly short duration of sleep or particularly long duration of sleep, have been found to be associated with a higher cardiometabolic risk

Sleep disorders such as sleep disordered breathing and insomnia, as well as particularly short duration of sleep or particularly long duration of sleep, have been found to be associated with a higher cardiometabolic risk.
已经发现睡眠障碍如睡眠呼吸紊乱和失眠，以及特别是短暂的睡眠持续时间或特别长的睡眠持续时间与更高的心脏代谢风险相关。
# English-Chinese/Skin rashes that can occur in diabetes are known as diabetic dermadromes

A number of skin rashes that can occur in diabetes are collectively known as diabetic dermadromes.
可能在糖尿病中发生的许多皮疹统称为糖尿病性皮肤病。
# Boundary Value Problems/Numerical Methods

Return to Main Page *Boundary Value Problems
These lessons are to introduce you to Numerical Methods used to calculate numerical solutions to the 2-D BVPs discussed earlier. Two resources that would be useful for the exercises are
* MATLAB MathWorks
* MAPLE MAPLE
However any computational system that can solve a system of equations will meet the needs of many of the elementary exercises. Three Linux based packages (free) are:
* Scilab Scilab (similar to Matlab)
* Maxima Computer Algebra System Maxima
* Octave

## Finite Difference Method

The Finite Difference method is a numerical method used for approximating the solution to a differential equation. The fundamentals of FD methods with application to an ODE and the 1-D heat equation can be found at [Finite Difference: http://en.wikipedia.org/wiki/Finite_difference_method]
A particular application is the Finite Difference Time Domain method to wave propagation problems

### Assignment: FD-1

* Read the material at [Finite Difference: http://en.wikipedia.org/wiki/Finite_difference_method].
* Also read Chapter 7 in Power's
* Attempt the following problems FD-1 problems
* Check your results against the FD-1 posted solutions posted solutions.
* If your solution provides additional insight (you think) to the solving of the problem, please create a solution using the solution template solution template

## Finite Elements Method

The Finite Elements Method constitutes a numerical approach to approximating the solution of an ordinary differential equation over a two-dimensional grid that is not rectangular, or one in which the data points, or nodes, are not evenly spaced. This gives it an advantage over the Finite Differences Method.
The first step of the method consists of triangulating the region to be integrated by dividing it into triangles, with the nodes as vertices. The next step is to determine the values of the unknown function at the nodes. Here is where the "finite elements" come in. A finite element is a function $\phi _{j}$ whose graph is a pyramid with its peak over a node $v_{j}$. A finite element solution to an ODE is a linear combination of finite elements: $U(v_{j})=\sum _{j=1}^{n}C_{j}\phi _{j}(v_{j})$ where $\phi _{j}(v_{j})={\begin{cases}1{\text{ at j-node}}\\0{\text{ at other nodes}}\end{cases}}$. The constants $C_{j}$ are then the values of the function at the nodes. Once these values are found, each triangle is integrated using the "Three Corners" method (the triangular version of the Trapezoid method) and summed.

## Plot of Finite Difference Example

Example of $u_{rr}+{\frac {1}{r}}u_{r}+{\frac {1}{r^{2}}}u_{\theta \theta }=0$ on a half annulus geometry with:
* $u(1,\theta )=sin(\theta )$
* $u(2,\theta )=0$
* $u(r,0)=0$ for $1<r<2$
* $u(r,\pi )=0$ for $1<r<2$
* ↑ Boundary Value Problems,Powers 5th edition,Elsevier Academic Press
* ↑ T. Young and M. Mohlenkamp, Numerical Methods and Matlab Programming For Engineers
# Boundary Value Problems/Problem Lookup

A directory/database of BVP problems.
| PDE | BCs | ICs | Click for link to solution |
| 1 | 2 | 3 | 4 |
| $u_{xx}={\frac {1}{k}}u_{t}$ | $u(0,t)=0{\mbox{ }}u(L,t)=0$ | $u(x,0)=f(x)$ | BVP_1D_heat_hom_dirichlet_hom |
| $u_{xx}={\frac {1}{k}}u_{t}$ | $u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}$ | $u(x,0)=f(x)$ | BVP_1D_heat_hom_dirichlet_non_hom |
| $u_{xx}={\frac {1}{k}}u_{t}$ | $u_{x}(0,t)=0{\mbox{ }}u(L,t)=T_{1}$ | $u(x,0)=f(x)$ | BVP_1D_heat_hom_neumann_dirichlet |
| $u_{xx}={\frac {1}{k}}u_{t}$ | $u_{x}(0,t)=-k(u(0,t)-A){\mbox{ }}u(L,t)=T_{2}$ | $u(x,0)=f(x)$ | BVP_1D_heat_hom_dirichlet_non_hom |
| $u_{xx}+g(x)={\frac {1}{k}}u_{t}$ | $u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}$ | $u(x,0)=f(x)$ | BVP_1D_heat_hom_dirichlet_non_hom |
| $u_{xx}={\frac {1}{k}}u_{t}+k(u(x,t)-A)$ | $u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}$ | $u(x,0)=f(x)$ | BVP_1D_heat_nonhom_dirichlet_non_hom |
| $u_{xx}={\frac {1}{k}}u_{t}$ | $u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}$ | $u(x,0)=f(x)$ | BVP_2D_Laplaces_hom_BCs_non_hom |
| $\displaystyle \nabla ^{2}u=0$ | $u(0,y)=\pi -y$ $u(1,x)=0$ $u_{y}(x,0)=x$ $u_{y}(x,\pi )=x$ | none | click here to see pdf of Maple file that has solution. |
| $\displaystyle u_{xx}=u_{tt}+10$ | $u(0,t)=0$ $u(1,t)=0$ | $u(x,0)=0$ $u_{t}(x,0)=x(1-x)$ | Click here to see pdf of a Maple file that has solution. |
| $\displaystyle \nabla ^{2}u={\frac {1}{4}}u_{t}$ | $u(x,0,t)=0$ $u(x,5,t)=0$ $u(0,y,t)=0$ $u(3,y,t)=0$ | $u(x,y,t)=sin({\frac {\pi x}{3}})sin({\frac {\pi y}{5}})$ | click here to see pdf of Maple file that has solution. |
| $\displaystyle \nabla ^{2}u={\frac {1}{4}}u_{t}$+ F(x,y,t) | $u(x,0,t)=0$ $u(x,M,t)=0$ $u(0,y,t)=0$ $u(L,y,t)=0$ | $u(x,y,t)=sin({\frac {\pi x}{L}})sin({\frac {\pi y}{M}})$ | |
# Boundary Value Problems/Lesson 8

Return to Boundary Value Problems
Bessel's equation $\displaystyle x^{2}y''+xy'+(x^{2}-n^{2})y=0$
Series Solutions-Fourier series - Power Series
Return to main Boundary Value Problems page.
# Boundary Value Problems/Main PDE's to be addressed in course

Return Boundary Value Problems

## Heat Equation

$\scriptstyle u_{t}-\alpha ^{2}\nabla ^{2}u=F$

## Poisson's Equation

$\scriptstyle \nabla ^{2}u=F$

## The Wave Equation

$\scriptstyle u_{tt}-c^{2}\nabla ^{2}u=F$

## Maxwell's Equations

$div(E)=4\pi \rho$
$\scriptstyle curl(E)+{\frac {1}{c}}{\frac {\partial B}{\partial t}}=0$
$\scriptstyle curl(B)-{\frac {1}{c}}{\frac {\partial E}{\partial t}}={\frac {4\pi }{c}}J$
$\scriptstyle div(B)=0$
# Boundary Value Problems/ODE BVPs
## Two point BVPs for an ODE

Begin with second order DEs, $x''=f(t,x,x')$, with conditions on the solution at $t=a$ and $t=b$.
$\displaystyle {\frac {d^{2}x}{dt^{2}}}+p(t){\frac {dx}{dt}}+q(t)x(t)=f(t)$ with $\displaystyle a_{0}x(a)+a_{1}x'(a)=g$ and $\displaystyle b_{0}x(b)+b_{1}x'(b)=h$ on the interval $\displaystyle I_{ab}=\{x|a\leq t\leq b\}$

## Example

$\displaystyle {\frac {d^{2}x}{dt^{2}}}+4{\frac {dx}{dt}}+2x(t)=f(t)$ with $\displaystyle x(0)=0$ and $\displaystyle x(1)=0$ on the interval $\displaystyle I_{ab}=\{x|0\leq t\leq 1\}$
# Boundary Value Problems/Lesson 7

Boundary Value Problems

## Rectangular Domain (R$^2$)

$\displaystyle u_{xx}+u_{yy}=0$

## Disk Domain (Polar)

For a disk with a radius of "c", let the polar coordinates be $0<r<c$, and $-\pi <\theta <\pi$
$r^{2}u_{rr}+ru_{r}+u_{\theta \theta }=0$
$u(c,\theta )=f(\theta )$ , boundary condition.
$u(r,\pi )=u(r,-\pi )$ continuity of potential.
$u_{\theta }(r,\pi )=u_{\theta }(r,-\pi )$ continuity of derivative.
$\displaystyle u(r,\theta )=R(r)\Theta (\theta )$ The solution as a product of two independent functions. By substitution into the above PDE we have:
$\displaystyle r^{2}R''\Theta +rR'\Theta +R\Theta ''=0$
Separate,
$\displaystyle {\frac {r^{2}R''+rR'}{R}}+{\frac {\Theta ''}{\Theta }}=0$
$\displaystyle {\frac {r^{2}R''+rR'}{R}}=-{\frac {\Theta ''}{\Theta }}={\mbox{Constant}}$
The constant may be greater than , equal to or less than zero.
* $\displaystyle \lambda ^{2}>0$
$\displaystyle -{\frac {\Theta ''}{\Theta }}=\lambda ^{2}$
$\displaystyle \Theta ''+\lambda ^{2}\Theta =0$
$\displaystyle \Theta (\theta )=Acos(\lambda \theta )+Bsin(\lambda \theta )$
Use the continuity conditions and try to determine something more about A, B and λ.
$\displaystyle u(r,\pi )=u(r,-\pi )$ thus $\displaystyle \Theta (\pi )=\Theta (-\pi )$ and $\displaystyle Acos(\lambda \pi )+Bsin(\lambda \pi )=Acos(\lambda -\pi )+Bsin(\lambda -\pi )$
$\displaystyle Bsin(\lambda \pi )=-Bsin(\lambda \pi )$
$\displaystyle 2Bsin(\lambda \pi )=0$
Either $\displaystyle B=0$ or $\displaystyle sin(\lambda \pi )=0$
Before choosing, apply the second boundary condition:
The continuity of the derivative provides a second condition:
$\displaystyle u_{\theta }(r,\pi )=u_{\theta }(r,-\pi )$ thus $\displaystyle \Theta _{\theta }(\pi )=\Theta _{\theta }(-\pi )$
$\displaystyle -A\lambda sin(\lambda \pi )+\lambda Bcos(\lambda \pi )=-A\lambda sin(\lambda -\pi )+B\lambda cos(\lambda -\pi )$
$\displaystyle -A\lambda sin(\lambda \pi )=A\lambda sin(\lambda \pi )$
$\displaystyle 2A\lambda sin(\lambda \pi )=0$
Either $\displaystyle A=0$ or $\displaystyle sin(\lambda \pi )=0$
If either A or B are zero then $\displaystyle sin(\lambda \pi )=0$ also must hold. So all we need is $\displaystyle sin(\lambda \pi )=0$ which implies $\displaystyle \lambda =n$. Remember $\displaystyle sin(n\pi )=0,{\mbox{ }}n=1,2,...$

## Example of Potential equation on semi-annulus.

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Return to Main Page Boundary Value Problems
# Boundary Value Problems/Main Examples of PDE's
## Linear Elasticity Equations

Return Boundary Value Problems
# Boundary Value Problems/Lesson 8: 2-D Potential Equation
## Rectangular Domain (R$^2$)

$\displaystyle u_{xx}+u_{yy}=0$

## Disk Domain (Polar)

$\displaystyle u_{rr}+{\frac {1}{r}}u_{r}+{\frac {1}{r^{2}}}u_{\theta \theta }=0$
Example of Potential equation on semi-annulus.
Return to Main Page Boundary Value Problems
# Boundary Value Problems/Lesson 6

Return to BVP main page Boundary Value Problems

### 1D Wave Equation

Derivation of the wave equation using string model.

### General form for boundary conditions.

$\alpha _{11}u(a,t)+\alpha _{12}u_{x}(a,t)=\gamma _{1}$ $\alpha _{21}u(b,t)+\alpha _{22}u_{x}(b,t)=\gamma _{2}$

### Wave equation with Dirichlet Homogeneous Boundary conditions.

$\displaystyle \alpha _{11}u(a,t)=0$
$\displaystyle \alpha _{21}u(b,t)=0$
In the homogeneous problem $\displaystyle u_{xx}-{\frac {1}{c^{2}}}u_{tt}=0$ with $\displaystyle u(0,t)=0$ , $\displaystyle u(L,t)=0$

### Finding a solution: u(x,t)

Let $u(x,t)=X(x)T(t)$
then substitute this into the PDE.
$X''T={\frac {1}{c^{2}}}XT''$
${\frac {X''}{X}}={\frac {1}{c^{2}}}{\frac {T''}{T}}=\mu$
Where $\mu$ is a constant that can be positive, zero or negative. We need to check each case for a solution.

## Wave Equation with nonhomogeneous Dirichlet Boundary Conditions

In the homogeneous problem $u_{xx}-{\frac {1}{c^{2}}}u_{tt}=0$
$\displaystyle \alpha _{11}u(x,t)=\gamma _{1}(t)$ $\displaystyle \alpha _{21}u(x,t)=\gamma _{2}(t)$

## Wave Equation with resistive damping

In the homogeneous problem $u_{xx}={\frac {1}{c^{2}}}u_{tt}+ku_{t}$
# Automation (disambiguation)

If you would like to learn about automation here at Wikiversity, here are a few links to get you started.
* Library automation software
* Automation engineering
* Business process management and automation
* Electronic design automation
* Automate processes
# Automaton

An automaton or sequential machine is a discrete and usually finite system that proceeds sequentially from an initial state to a final discrete state upon changing its inputs, and that also produces an output at each step determined by the input and the internal state of the automaton. An automaton thus defined has a state space which is a finite semigroup or monoid, and that can be decomposed into certain types of modules or parts in a canonical manner. Two functions or maps usually determine mathematically the functioning of the automaton : the transition function from one state to the next, and the ouput function that maps an input and a specific state to a specific output of the automaton. Examples of automata are abundant as machines, computers and robots. Their underlying operational logic is Boolean. Exceptions are fuzzy systems based on 'fuzzy logic'.
Automata are very useful automation tools.
# Automotive Technology/Accumulator

Hydraulic assembly