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Консультация № 185425
12.02.2012, 15:35
70.47 руб.
0
7
1
Здравствуйте! Прошу помощи в следующем вопросе:
необходимо решить дифференциальное уравнение для объекта с запаздыванием методом конечных разностей и построить график:
Tx'(t)+x(t)=k*Ф(t-tau), x(0)=4, где Ф - единичная ступенчатая функция, к=5, Т=10, tau=2
необходимо решить дифференциальное уравнение для объекта с запаздыванием методом конечных разностей и построить график:
Tx'(t)+x(t)=k*Ф(t-tau), x(0)=4, где Ф - единичная ступенчатая функция, к=5, Т=10, tau=2
Обсуждение
13.02.2012, 02:32
общий
это ответ
Здравствуйте, Невалёный В.!
Задаём сетку по переменной t так, чтобы один из узлов попадал в [$964$] (тау).
ti=i*dt, i=0..n
[$916$]t=0.1
Соответственно этим точкам будем искать значения функции
x(ti)=xi
x'(ti)=x'i
Разностную аппроксимацию в данном случае можно задать такую
x'i = (xi+1-xi)/[$916$]tлюдаетс\я
Тогда из исходного уравнения получаем разностную схему
T[$149$](xi+1-xi)/[$916$]t + xi = k*Ф(ti-[$964$])
Отсюда выражаем рассчётную формулу
xi+1 = xi + ( -xi + k*Ф(ti-[$964$]) )[$149$][$916$]t/T
Рассчёты приведены прикреплённом файле (Excel).
Общий интервал по t в задаче не задан, поэтому взял произвольно n=100. Потом увеличил до n=200. Наблюдается сходимость к стационарному значению. Если надо точнее -- можно продолжить таблицу дальше.
Задаём сетку по переменной t так, чтобы один из узлов попадал в [$964$] (тау).
ti=i*dt, i=0..n
[$916$]t=0.1
Соответственно этим точкам будем искать значения функции
x(ti)=xi
x'(ti)=x'i
Разностную аппроксимацию в данном случае можно задать такую
x'i = (xi+1-xi)/[$916$]tлюдаетс\я
Тогда из исходного уравнения получаем разностную схему
T[$149$](xi+1-xi)/[$916$]t + xi = k*Ф(ti-[$964$])
Отсюда выражаем рассчётную формулу
xi+1 = xi + ( -xi + k*Ф(ti-[$964$]) )[$149$][$916$]t/T
Рассчёты приведены прикреплённом файле (Excel).
Общий интервал по t в задаче не задан, поэтому взял произвольно n=100. Потом увеличил до n=200. Наблюдается сходимость к стационарному значению. Если надо точнее -- можно продолжить таблицу дальше.
Прикрепленные файлы:
13.02.2012, 03:15
общий
Действительно невозможно. Аналитическое решение имеет вид(не пропадать же :))
13.02.2012, 18:30
общий
Адресаты:
Первый - да. А второй, если автор вопроса заинтересован в аналитическом решении, лучше оставить.
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