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Консультация № 174518

24.11.2009, 20:01

35.00 руб.

0 2 1

Образование

Доброе время суток. У меня задача по дисциплине "Уравнения математической физики".
Задание: Привести к каноническому виду 2u_xx+2u_xy+u_yy+3xu_x-2u_y-xy+13=0.
Я уже начал решать:
L₀u(x,y)=2u_xx+2u_xy+u_yy
L₁u(x,y)=3xu_x-2u_y-xy+13
1) Определим тип уравнения: d=b²-ac=1-2=-1 < 0 [$8658$] уравнение элиптического типа
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2) Определим семейство характеристик
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Помогите пожалуйста найти [$958$] и [$627$] и решить до нахождения частных производных.

Обсуждение

давно

Специалист
246813
155

24.11.2009, 23:19

общий

Всем большое спасибо. Уже не надо, сделал сам.

Неизвестный

24.11.2009, 23:31

общий

это ответ

Здравствуйте, LfiN.
Для уравнений эллиптического типа характеристическое уравнение распадается на два уравнения с комплексно сопряженными правыми частями.
y’ = [b±i(b^2p-ac)^1/2]/a = (1 ± i)/2.
Поэтому общие интегралы этих уравнений будут также комплексно сопряженными
y - x/2 - ix/2 = C1,
y – x/2 + ix/2 = C2.
Чтобы не иметь дела с комплексными переменными и функциями, введем новые, уже веществен-
ные, переменные ξ и ɳ
ξ = re(y - x/2 - ix/2) = y – x/2,
ɳ = im(y – x/2 + ix/2) = x/2.
Подставляя выражения
u_x = u_ξ ξ_x + u_ɳ ɳ _x = -u_ξ/2 + u_ɳ/2,
u_y = u_ξ ξ_y + u_ɳ ɳ _y = u_ξ,
u_xx = 1/4( u_{ξ ξ} - 2 u_{ξ ɳ} + u_{ɳ ɳ}),
u_xy = 1/2( u_{ξ ɳ} - u_{ξ ξ}),
u_yy = u_{ξ ξ},
x = 2ɳ,
y = ξ + ɳ
в исходное уравнение, получаем канонический вид уравнения
1/2( u_{ξ ξ} + u_{ɳ ɳ} +3 ɳ(u_ɳ - u_ξ) -2 u_ξ - 2 ɳ(ξ + ɳ)+13=0.
Удачи.

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