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Консультация № 186242

29.05.2012, 19:15

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30.05.2012, 23:02

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Обсуждение

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Киселев Виталий Сергеевич

Академик
324866
619

30.05.2012, 05:46

общий

это ответ

Здравствуйте, sereggg!
Решение 1 задачи в формате docx Вы можете скачать по ссылке: решение.
Будут вопросы обращайтесь в минифорум.
Удачи

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Орловский Дмитрий

Мастер-Эксперт
319965
1463

30.05.2012, 19:11

общий

В задаче Дирихле (задача 4) краевое условие должно ставиться при r=2, либо круг должен быть радиуса R=1. Выберите что-нибудь.

Неизвестный

30.05.2012, 21:48

общий

r=2

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Орловский Дмитрий

Мастер-Эксперт
319965
1463

31.05.2012, 23:19

общий

это ответ

Здравствуйте, sereggg!
4. Решение задачи Дирихле в круге радиуса R определяется формулой
U=(a₀/2)+[$8721$]_n=1^[$8734$](r/R)ⁿ(a_ncos n[$966$]+b_nsin n[$966$]),
где a_n,b_n - коэффициенты Фурье граничной функции ( в данном случае это 3[$966$]²+[$966$]+2).
Вычисляем эти коэффициенты:
a₀=(1/Pi)[$8747$]₀^2Pi(3[$966$]²+[$966$]+2)d[$966$]=(1/Pi)[[$966$]³+([$966$]²/2)+2[$966$]]|₀^2Pi=8Pi²+2Pi+4
При n>0
a_n=(1/Pi)[$8747$]₀^2Pi(3[$966$]²+[$966$]+2)cos n[$966$]d[$966$]=
=(1/Pi)[3{([$966$]²sin n[$966$]/n)+(2[$966$]cos n[$966$]/n²)-(2sin n[$966$]/n³)+{([$966$]sin n[$966$]/n)+(cos n[$966$]/n²)}+2(sin n[$966$]/n)]|₀^2Pi=12/n²

b_n=(1/Pi)[$8747$]₀^2Pi(3[$966$]²+[$966$]+2)sin n[$966$]d[$966$]=
=(1/Pi)[3{(-[$966$]²cos n[$966$]/n)+(2[$966$]sin n[$966$]/n²)+(2cos n[$966$]/n³)+{(-[$966$]cos n[$966$]/n)+(sin n[$966$]/n²)}+2(-cos n[$966$]/n)]|₀^2Pi=-2(6n+1)/n

Ответ:
U=4Pi²+Pi+2+[$8721$]_n=1^[$8734$](r/2)ⁿ[(12/n²)cos n[$966$]-(2(6n+1)/n)sin n[$966$]]

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