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روش تفاضل محدود(متناهی) و درون¬يابی چند جمله¬ای جبری برای حل عددی معادله پواسون در دامنه دلخواه

تعداد 103 صفحه فایل word

چکیده:

این پایان­نامه به روش تفاضل محدود ودرون­یابی چند جمله­ای جبری برای حل عددی معادله پواسون در دامنه­های دلخواه می­پردازد.

روش تفاضل محدود (FDM) بر مبنای دستگاه مختصات دکارتی را می توان با تحليل عددی در هر دامنه مركب بكار برد. دامنه مركب معمولاً به معنای هندسه جسم فرو رفته در سیال می­باشد. در اينجا به معنای ساده­تر يك دامنه تحليلی از وضعيت دلخواه است. در روش تفاضل محدود به بررسی رفتار مرزهای بيرونی و درونی متفاوت در محاسبات عددی نيازی نیست، چرا كه هر دو در يك روش مشابه يكسان رفتار می­كنند. با استفاده ازروش درون­یابی چند­جمله­ای جبری درمحاسبه عناصر نزدیک دیواره همه محاسبات در دامنه نامنظم به محاسبه در دامنه منظم تبدیل می­شود. گسسته­سازی تفاضلات فضا معمولاً با ­استفاده از سری تیلور به­دست می­آید، با این حال اگر از درون­یابی چند­جمله­ای به صورت سیستماتیک استفاده شود، در به­دست آمدن تفاضلات مرتبه بالا مزیت­های ویژه­ای حاصل می­شود. با ­استفاده از درون­یابی چند­جمله­ای می­توان، آزادانه معادله پواسون را به روش عددی در هر دامنه مرکب حل کرد.

در فصل اول این پایان نامه به تعاریف و مفاهیم اصلی پرداخته شده، در فصل دوم برخی از روش­های درون­یابی همراه با خطا آورده شده ودر فصل سوم روش FD درجه دو وچهار همراه با برون­یابی های مختلف مورد بررسی قرار گرفته و میزان دقت آن­ها با هم مقایسه شده ­است. سر انجام در فصل چهارم جداول ومثال­های عددی آورده شده است وحل معادله پواسون در حالت دو بعدی و سه بعدی مورد بررسی قرار گرفته و به نتیجه­گیری نهایی پرداخته شده­ است.

فهرست مطالب

فهرست مطالب

چکیده: 1

مقدمه: 2

فصل اول: مفاهیم اولیه

1-1- مقدمه: 4

1-2- شبکه منظم وشبکه پیچیده: 4

1-3- تقسیم بندی  : 5

1-4- تعریف شرط اولیه: 7

 1-13- روشهای تکراری: 12

1-16- تعریف تنش برشی: 14

1-17- لزجت یا گرانوی: 15

1-19- جریان تراکم ناپذیر وتراکم پذیر: 16

1-20- معادله تراکم ناپذیر ناویر-استوکس: 16

1-21-1- تبدیل فوریه: 17

1-21-2- سری فوریه: 17

فصل دوم: روش های تک گامی و چند گامی برای حل عددی معادلات دیفرانسیل معمولی

2- مقدمه : 20

2-1- روش های تک گامی : 21

2-2- روش سری تیلور 22

2-3- روش اویلر. 24

2-4- روش های رونگه- کوتا: 25

2-5- دستگاه معادلات دیفرانسیل مرتبه اول: 31

2-5-1- روش رونگه- کوتای مرتبه چهار برای حل دستگاه معادلات دیفرانسیل 2×2 مرتبه اول: 31

2-6- روش های چند گامی: 32

2-6-3- روش آدامز بشفورث: 34

2-6-4- روش آدامز مولتون: 37

2-6-5- روش پیشگو-اصلاحگر : 39

 نتیجه گیری: 41

فصل سوم: روش تفاضل متناهی و درونیابی چند جمله­ای جبری  برای حل عددی معادله پواسون در دامنه اختیاری

3-1- مقدمه: 43

3-2-1- حل عددی معادله پواسون: 45

3-2- روش FD دقیق با درجه ی بیشتر: 53

3-3- رابطه بین روش FD و درونیابی چندجمله­ای: 55

نتیجه گیری: 61

فصل چهارم: جداول، مثال­های عددی و نتیجه گیری

4-1- مقدمه: 63

4-2- بررسی روش  FDمرتبه دو وچهار همراه با برونیابی­های گوناگون: 63

4-3- حل معادله پواسون در حالت 2بعدی: 75

4-4- حل معادله پواسون درحالت 3بعدی: 82

نتیجه گیری: 86

منابع فارسی: 89

منابع انگلیسی: 90

 

 

فهرست جداول:

جدول 4-1 ……………………………………………………………………………………………………… 62

جدول4-2………………………………………………………………………………………………………… 64

جدول4-3………………………………………………………………………………………………………… 66

جدول4-4………………………………………………………………………………………………………… 68

 

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