Типовые
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МИНИСТЕРСТВО ВНУТРЕННИХ ДЕЛ РОССИЙСКОЙ ФЕДЕРАЦИИ
ВОРОНЕЖСКИЙ ИНСТИТУТ
Кафедра теоретических и прикладных
математических дисциплин
Думачев В.Н.
Уравнения математической физики
ВОРОНЕЖ - 2002
2
УДК 517.53
Думачев В.Н.
Типовой расчет. Уравнения математической физики – Воронеж:
Воронежский институт МВД России, 2002. - 96 с.
Методические указания для выполнения типового расчета, проведения практических занятий и самоподготовки по курсу «математика» для курсантов Радиотехнического факультета обучающихся по специальности 075600 – информационная безопасность телекоммуникационных систем.
Воронежский институт МВД России, 2002.
3
РАСЧЕТНЫЕ ЗАДАНИЯ
Задача 1. Найти в указанной области отличные от тождественного нуля решения y=y(x) дифференциального уравнения, удовлетворяющие заданным краевым условиям. (Задача Штурма-Лиувилля)
1.1. |
y"+λy = 0, 1 ≤ x ≤ 2 |
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= 0 |
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y(1) = y'(2) |
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1.3. |
y"+λy = 0, π / 2 ≤ x ≤ π |
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y(π / 2) = y'(π ) = 0 |
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1.5. |
y"+λy = 0, 1/ 2 ≤ x ≤ 1 |
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y(1/ 2) = y'(1) = 0 |
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1.7. |
y"+λy = 0, π ≤ x ≤ 2π |
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y(π ) = y'(2π ) = 0 |
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1.9. |
y"+λy = 0, 1 ≤ x ≤ 3 / 2 |
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y(1) = y'(3 / 2) = 0 |
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1.11. |
y"+λy = 0, |
π / 2 ≤ x ≤ 3π / 2 |
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y(π / 2) = y'(3π / 2) = 0 |
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1.13. |
y"+λy = 0, 1/ 2 ≤ x ≤ 3 / 2 |
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y(1/ 2) = y'(3 / 2) = 0 |
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y"+λy = 0, π ≤ x ≤ 3π / 2
y(π ) = y'(3π / 2) = 0
1.17. |
y"+λy = 0, 1 ≤ x ≤ 2 |
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= 0 |
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y'(1) = y(2) |
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1.19. |
y"+λy = 0, |
π / 2 ≤ x ≤ π |
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y'(π / 2) = y(π ) = 0 |
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1.21. |
y"+λy = 0, 1/ 2 ≤ x ≤ 1 |
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y'(1/ 2) = y(1) = 0 |
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y"+λy = 0, 3 / 2 ≤ x ≤ 2 |
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1.2. |
= y'(2) = 0 |
y(3 / 2) |
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y"+λy = 0, π / 4 ≤ x ≤ π / 2
y(π / 4) = y'(π / 2) = 0
1.6. |
y"+λy = 0, 3 / 4 ≤ x ≤ 1 |
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= y'(1) = 0 |
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y(3 / 4) |
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1.8. |
y"+λy = 0, π / 2 ≤ x ≤ 3π / 4 |
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= y'(3π / 4) = 0 |
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y(π / 2) |
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1.10. |
y"+λy = 0, 1/ 4 ≤ x ≤ 1/ 2 |
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y(1/ 4) = y'(1/ 2) = 0 |
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1.12. |
y"+λy = 0, 3 / 4 ≤ x ≤ 5 / 4 |
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= y'(5 / 4) = 0 |
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y(3 / 4) |
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1.14. |
y"+λy = 0, π / 2 ≤ x ≤ 5π / 4 |
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= y'(5π / 4) = 0 |
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y(π / 2) |
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y"+λy = 0, 3π / 4 ≤ x ≤ 5π / 2
y(3π / 4) = y'(5π / 2) = 0
y"+λy = 0, 3 / 2 ≤ x ≤ 2 |
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1.18. |
= y(2) = 0 |
y'(3 / 2) |
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y"+λy = 0, π / 4 ≤ x ≤ π / 2
y'(π / 4) = y(π / 2) = 0
y"+λy = 0, 3 / 4 ≤ x ≤ 1 |
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1.22. |
= y(1) = 0 |
y'(3 / 4) |
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1.23. |
y"+λy = 0, π ≤ x ≤ 2π |
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y'(π ) = y(2π ) = 0 |
1.25. |
y"+λy = 0, 1 ≤ x ≤ 3 / 2 |
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y'(1) = y(3 / 2) = 0 |
1.27. |
y"+λy = 0, π / 2 ≤ x ≤ 3π / 2 |
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y'(π / 2) = y(3π / 2) = 0 |
1.29. |
y"+λy = 0, π ≤ x ≤ 3π / 2 |
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y'(π ) = y(3π / 2) = 0 |
y"+λy = 0, 1/ 2 ≤ x ≤ 3 / 2
y'(1/ 2) = y(3 / 2) = 0
4
1.24. |
y"+λy = 0, π / 2 ≤ x ≤ 3π / 4 |
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y'(π / 2) = y(3π / 4) = 0 |
1.26. |
y"+λy = 0, 1/ 4 ≤ x ≤ 1/ 2 |
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y'(1/ 4) = y(1/ 2) = 0 |
1.28. |
y"+λy = 0, 3 / 4 ≤ x ≤ 5 / 4 |
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y'(3 / 4) = y(52π ) = 0 |
1.30. |
y"+λy = 0, π / 2 ≤ x ≤ 5π / 2 |
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y'(π / 2) = y(5π / 2) = 0 |
Задача 2. Найти общее решение уравнения, приведя его к каноническому виду.
2.1. |
ux x+2uxy−uyy+ux+uy=0 |
2.2. |
ux x+4uxy+4uyy−ux−2uy=0 |
2.3. |
ux x−2uxy+uyy+2ux−2uy=0 |
2.4. |
ux x+6uxy+9uyy+ux+3uy=0 |
2.5. |
ux x−6uxy+9uyy−2ux+6uy=0 |
2.6. |
ux x+2uxy+uyy−3ux−3uy=0 |
2.7. |
ux x−4uxy+4uyy+3ux−6uy=0 |
2.8. |
9ux x+6uxy+uyy−9ux−3uy=0 |
2.9. |
ux x+8uxy+16uyy−ux−4uy=0 |
2.10. |
ux x−2uxy+uyy+4ux−4uy=0 |
2.11. |
16ux x+8uxy+uyy−8ux−2uy=0 |
2.12. |
4ux x+4uxy+uyy+8ux+4uy=0 |
2.13. |
ux x−8uxy+16uyy+3ux−12uy=0 |
2.14. |
9ux x+6uxy+uyy−12ux−4uy=0 |
2.15. |
16ux x+8uxy+uyy−16ux+4uy=0 |
2.16. |
ux x+10uxy+25uyy+ux+5uy=0 |
2.17. |
ux x+2uxy+uyy+5ux+5uy=0 |
2.18. |
ux x−10uxy+25uyy+2ux−10uy=0 |
2.19. |
4ux x−4uxy+uyy−10ux+5uy=0 |
2.20. |
25ux x−10uxy+uyy−15ux+3uy=0 |
2.21. |
ux x+6uxy+9uyy+5ux+15uy=0 |
2.22. |
25ux x+10uxy+uyy+20ux+4uy=0 |
2.23. |
ux x+8uxy+16uyy+5ux+20uy=0 |
2.24. |
ux x−10uxy+25uyy+5ux−25uy=0 |
2.25. |
ux x+12uxy+36uyy+ux+6uy=0 |
2.26. |
ux x−2uxy+uyy+6ux−6uy=0 |
2.27. |
ux x−12uxy+36uyy+2ux−12uy=0 |
2.28. |
36ux x+12uxy+uyy+18ux+3uy=0 |
2.29. |
ux x+14uxy+49uyy+2ux+14uy=0 |
2.30. |
36ux x−12uxy+uyy+18ux−3uy=0 |
2.31. |
49ux x−14uxy+uyy+14ux−2uy=0 |
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Задача3. Найти общее решение уравнения, приведя его к каноническому виду. |
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3.1. |
4uxx+8uxy+3uyy=0 |
3.2. |
3uxx+8uxy+4uyy=0 |
3.3. |
3uxx+4uxy+uyy=0 |
3.4. |
uxx+4uxy+3uyy=0 |
3.5. |
16uxx+16uxy+3uyy=0 |
3.6. |
3uxx+16uxy+16uyy=0 |
3.7. |
25uxx+20uxy+3uyy=0 |
3.8. |
uxx+8uxy+12uyy=0 |
3.9. |
12uxx+8uxy+uyy=0 |
3.10. |
49uxx+28uxy+3uyy=0 |
3.11. |
64uxx+32uxy+3uyy=0 |
3.12. |
3uxx+20uxy+25uyy=0 |
3.13. |
uxx+3uxy+2uyy=0 |
3.14. |
2uxx+3uxy+uyy=0 |
3.15. |
uxx+12uxy+27uyy=0 |
3.16. |
uxx+16uxy+48uyy=0 |
3.17. |
uxx+20uxy+75uyy=0 |
3.18. |
uxx+24uxy+108uyy=0 |
3.19. |
uxx+28uxy+147uyy=0 |
3.20. |
uxx+32uxy+192uyy=0 |
3.21. |
uxx+36uxy+243uyy=0 |
3.22. |
3uxx+28uxy+49uyy=0 |
3.23. |
3uxx+32uxy+64uyy=0 |
3.24. |
27uxx+12uxy+uyy=0 |
3.25. |
48uxx+16uxy+uyy=0 |
3.26. |
75uxx+20uxy+uyy=0 |
3.27. |
108uxx+24uxy+uyy=0 |
3.28. |
147uxx+28uxy+uyy=0 |
3.29. |
192uxx+32uxy+uyy=0 |
3.30. |
4uxx+3uxy−uyy=0 |
3.31. |
2uxx+5uxy−3uyy=0 |
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Задача 4. Решить смешанную задачу. |
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4.1 |
ut=2uxx; u(x,0)= 1sin(3πx); |
u(0,t)=u(8,t)=0 |
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4.2 |
ut=9uxx; u(x,0)= 2sin(2πx) + 3sin(3πx); |
u(0,t)=u(1,t)=0 |
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4.3 |
ut=3uxx; u(x,0)= 3sin(2πx) ; |
u(0,t)=u(7,t)=0 |
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4.4 |
ut=2uxx; u(x,0)= 4sin(3πx) + 5sin(4πx); |
u(0,t)=u(2,t)=0 |
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4.5 |
ut=4uxx; u(x,0)= 5sin(3πx) ; |
u(0,t)=u(6,t)=0 |
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4.6 |
ut=7uxx; u(x,0)= 6sin(2πx) + 7sin(3πx); |
u(0,t)=u(3,t)=0 |
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4.7 |
ut=5uxx; u(x,0)= 7sin(2πx); |
u(0,t)=u(5,t)=0 |
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4.8 |
ut=6uxx; u(x,0)= 8sin(3πx) + 9sin(4πx); |
u(0,t)=u(4,t)=0 |
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4.9 |
ut=6uxx; u(x,0)= 9sin(3πx); |
u(0,t)=u(4,t)=0 |
4.10 |
ut=5uxx; u(x,0)=10sin(2πx) + 3sin(3πx); |
u(0,t)=u(5,t)=0 |
4.11 |
ut=7uxx; u(x,0)=11sin(2πx); |
u(0,t)=u(3,t)=0 |
4.12 |
ut=4uxx; u(x,0)=12sin(3πx) + 5sin(4πx); |
u(0,t)=u(6,t)=0 |
4.13 |
ut=8uxx; u(x,0)=13sin(3πx); |
u(0,t)=u(2,t)=0 |
4.14 |
ut=3uxx; u(x,0)=14sin(2πx) + 7sin(3πx); |
u(0,t)=u(7,t)=0 |
4.15 |
ut=9uxx; u(x,0)=15sin(2πx); |
u(0,t)=u(1,t)=0 |
4.16 |
ut=2uxx; u(x,0)=16sin(3πx) + 9sin(4πx); |
u(0,t)=u(8,t)=0 |
4.17 |
ut=2uxx; u(x,0)=17sin(2πx); |
u(0,t)=u(2,t)=0 |
4.18 |
ut=3uxx; u(x,0)=18sin(3πx) + 3sin(4πx); |
u(0,t)=u(7,t)=0 |
4.19 |
ut=3uxx; u(x,0)=19sin(3πx); |
u(0,t)=u(3,t)=0 |
4.20 |
ut=8uxx; u(x,0)=20sin(2πx) + 7sin(3πx); |
u(0,t)=u(6,t)=0 |
4.21 |
ut=4uxx; u(x,0)=21sin(2πx); |
u(0,t)=u(4,t)=0 |
4.22 |
ut=4uxx; u(x,0)=22sin(3πx) + 5sin(4πx); |
u(0,t)=u(5,t)=0 |
4.23 |
ut=5uxx; u(x,0)=23sin(3πx); |
u(0,t)=u(5,t)=0 |
4.24 |
ut=6uxx; u(x,0)=24sin(2πx) + 9sin(3πx); |
u(0,t)=u(4,t)=0 |
4.25 |
ut=6uxx; u(x,0)=25sin(2πx); |
u(0,t)=u(6,t)=0 |
4.26 |
ut=5uxx; u(x,0)=26sin(3πx) + 3sin(4πx); |
u(0,t)=u(3,t)=0 |
4.27 |
ut=7uxx; u(x,0)=27sin(3πx); |
u(0,t)=u(7,t)=0 |
4.28 |
ut=4uxx; u(x,0)=28sin(2πx) + 5sin(3πx); |
u(0,t)=u(2,t)=0 |
4.29 |
ut=8uxx; u(x,0)=29sin(2πx); |
u(0,t)=u(8,t)=0 |
4.30 |
ut=3uxx; u(x,0)=30sin(3πx) + 7sin(4πx); |
u(0,t)=u(1,t)=0 |
4.31 |
ut=9uxx; u(x,0)=31sin(3πx); |
u(0,t)=u(9,t)=0 |
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7 |
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Задача 5. Решить смешанную задачу. |
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5.1 |
ut=2uxx; u(x,0)=1cos(3πx) + 2cos(4πx); |
u(0,t)=ux(8,t)=0 |
5.2 |
ut=2uxx; u(x,0)=2cos(2πx) ; |
u(0,t)=ux(2,t)=0 |
5.3 |
ut=3uxx; u(x,0)=3cos(3πx) + 4cos(4πx); |
u(0,t)=ux(7,t)=0 |
5.4 |
ut=3uxx; u(x,0)=4cos(3πx) ; |
u(0,t)=ux(3,t)=0 |
5.5 |
ut=8uxx; u(x,0)=5cos(2πx) + 6cos(3πx); |
u(0,t)=ux(6,t)=0 |
5.6 |
ut=4uxx; u(x,0)=6cos(2πx); |
u(0,t)=ux(4,t)=0 |
5.7 |
ut=4uxx; u(x,0)=7cos(3πx) + 8cos(4πx); |
u(0,t)=ux(5,t)=0 |
5.8 |
ut=5uxx; u(x,0)=8cos(3πx); |
u(0,t)=ux(5,t)=0 |
5.9 |
ut=6uxx; u(x,0)=9cos(2πx)+ 10cos(3πx); |
u(0,t)=ux(4,t)=0 |
5.10 |
ut=6uxx; u(x,0)=10cos(2πx); |
u(0,t)=ux(6,t)=0 |
5.11ut=5uxx; u(x,0)=11cos(3πx) + 12cos(4πx); u(0,t)=ux(3,t)=0
5.12 ut=7uxx; u(x,0)=12cos(3πx); |
u(0,t)=ux(7,t)=0 |
5.13ut=4uxx; u(x,0)=13cos(2πx) + 14cos(3πx); u(0,t)=ux(2,t)=0
5.14 ut=8uxx; u(x,0)=14cos(2πx); |
u(0,t)=ux(8,t)=0 |
5.15ut=3uxx; u(x,0)=15cos(3πx) + 16cos(4πx); u(0,t)=ux(1,t)=0
5.16 ut=9uxx; u(x,0)=16cos(3πx); |
u(0,t)=ux(9,t)=0 |
5.17ut=2uxx; u(x,0)=17cos(3πx) + 18cos(4πx); u(0,t)=ux(2,t)=0
5.18 ut=4uxx; u(x,0)=18cos(3πx); |
u(0,t)=ux(6,t)=0 |
5.19ut=7uxx; u(x,0)=19cos(2πx) + 20cos(3πx); u(0,t)=ux(3,t)=0
5.20 ut=5uxx; u(x,0)=20cos(2πx); |
u(0,t)=ux(5,t)=0 |
5.21ut=6uxx; u(x,0)=21cos(3πx) + 22cos(4πx); u(0,t)=ux(4,t)=0
5.22 ut=6uxx; u(x,0)=22cos(3πx); |
u(0,t)=ux(4,t)=0 |
5.23ut=5uxx; u(x,0)=23cos(2πx) + 24cos(3πx); u(0,t)=ux(5,t)=0
5.24 ut=7uxx; u(x,0)=24cos(2πx); |
u(0,t)=ux(3,t)=0 |
5.25ut=4uxx; u(x,0)=25cos(3πx) + 26cos(4πx); u(0,t)=ux(6,t)=0
5.26 ut=8uxx; u(x,0)=26cos(3πx); |
u(0,t)=ux(2,t)=0 |
5.27ut=3uxx; u(x,0)=27cos(2πx) + 28cos(3πx); u(0,t)=ux(7,t)=0
5.28 ut=9uxx; u(x,0)=28cos(2πx); |
u(0,t)=ux(1,t)=0 |
5.29ut=9uxx; u(x,0)=29сos(2πx) + 30cos(3πx); u(0,t)=ux(1,t)=0
5.30 |
ut=3uxx; u(x,0)=30cos(2πx); |
u(0,t)=ux(7,t)=0 |
5.31 |
ut=2uxx; u(x,0)=31cos(3πx) + cos(4πx); |
u(0,t)=ux(6,t)=0 |
8
Задача 6. Решить смешанную задачу.
6.1ut=2uxx; u(x,0)=19sin(5πx);
6.2ut=5uxx; u(x,0)=8cos(πx);
6.3ut=8uxx; u(x,0)=17sin(3πx);
6.4ut=uxx; u(x,0)=6cos(9πx) ;
6.5ut=4uxx; u(x,0)=15sin(3πx);
6.6ut=2uxx; u(x,0)=4cos(5πx) ;
6.7ut=3uxx; u(x,0)=13sin(5πx);
6.8ut=uxx; u(x,0)=2cos(7πx) ;
6.9ut=5uxx; u(x,0)=19sin(3πx);
6.10ut=uxx; u(x,0)=8cos(5πx) ;
6.11ut=6uxx; u(x,0)=17sin(3πx);
6.12ut=2uxx; u(x,0)=6cos(7πx);
6.13ut=uxx; u(x,0)=15sin(9πx);
6.14ut=3uxx; u(x,0)=4cos(5πx) ;
6.15ut=7uxx; u(x,0)=13sin(3πx);
6.16ut=9uxx; u(x,0)=12cos(3πx);
6.17ut=2uxx; u(x,0)=9sin(7πx);
6.18ut=5uxx; u(x,0)=18cos(πx) ;
6.19ut=8uxx; u(x,0)=7sin(3πx);
6.20ut=uxx; u(x,0)=16cos(9πx) ;
6.21ut=4uxx; u(x,0)=5sin(3πx);
6.22ut=2uxx; u(x,0)=14cos(5πx);
6.23ut=3uxx; u(x,0)=3sin(5πx);
6.24ut=uxx; u(x,0)=12cos(7πx) ;
6.25ut=5uxx; u(x,0)=9sin(3πx);
6.26ut=uxx; u(x,0)=18cos(5πx) ;
6.27ut=6uxx; u(x,0)=7sin(3πx);
6.28ut=2uxx; u(x,0)=16cos(7πx);
6.29ut=uxx; u(x,0)=5sin(9πx);
6.30ut=3uxx; u(x,0)=14cos(5πx);
6.31ut=7uxx; u(x,0)=3sin(3πx);
u(0,t)=0, ux(0,5,t)=0
ux(0,t)=0,u(1,5,t)=0
u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(4,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(1,5,t)=0 u(0,t)=0, ux(4,5,t)=0 ux(0,t)=0, u(1,5,t)=0 u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(4,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(4,5,t)=0 ux(0,t)=0, u(1,5,t)=0 u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(4,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(1,5,t)=0 u(0,t)=0, ux(0,5,t)=0 ux(0,t)=0, u(1,5,t)=0 u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(4,5,t)=0 ux(0,t)=0, u(3,5,t)=0 u(0,t)=0, ux(2,5,t)=0 ux(0,t)=0, u(1,5,t)=0 u(0,t)=0, ux(0,5,t)=0
9
Задача 7. Решить смешанную задачу.
7.1. ut=9uxx; |
u(x,0)=5sin(2πx) – 1 + x; |
u(0,t)= –1 , |
u(2,t)= 5 |
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7.2. ut=8uxx; |
u(x,0)=6sin(3πx) + 2 – x; |
u(0,t)=2, |
u(3,t)= –7 |
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7.3. ut=7uxx; |
u(x,0)=7sin(2πx) – 3 + x; |
u(0,t)= –3 , |
u(1,t)= 1 |
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7.4. ut=6uxx; |
u(x,0)=8sin(4πx) + 4 – x; |
u(0,t)=4 , |
u(2,t)= –6 |
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7.5. ut=5uxx; |
u(x,0)=9sin(3πx) – 5 + x; |
u(0,t)= –5 , |
u(3,t)= 1 |
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7.6. ut=9uxx; |
u(x,0)=8sin(3πx) + 6 – x; |
u(0,t)=6, |
u(4,t)= –2 |
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7.7. ut=8uxx; |
u(x,0)=7sin(2πx) – 7 + x; |
u(0,t)= –7 , |
u(3,t)= 2 |
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7.8. ut=7uxx; |
u(x,0)=6sin(3πx) + 8 – x; |
u(0,t)=8 , |
u(4,t)= – 4 |
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7.9. |
ut=4uxx; |
u(x,0)=5sin(4πx) – 9 + x; |
u(0,t)= –9 , |
u(2,t)= 1 |
7.10. ut=3uxx; |
u(x,0)=4sin(5πx) + 9 – x; |
u(0,t)=9 , |
u(3,t)= –3 |
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7.11. ut=2uxx; |
u(x,0)=3sin(6πx) – 8 + x; |
u(0,t)= –8 , |
u(2,t)= 2 |
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7.12. ut=3uxx; |
u(x,0)=2sin(4πx) + 7 – x; |
u(0,t)=7 , |
u(1,t)= 2 |
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7.13. ut=5uxx; |
u(x,0)=3sin(3πx) – 6 + x; |
u(0,t)= –6 , |
u(3,t)= 6 |
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7.14. ut=6uxx; |
u(x,0)=4sin(4πx) + 5 – x; |
u(0,t)=5 , |
u(2,t)= –3 |
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7.15. ut=8uxx; |
u(x,0)=5sin(2πx) – 4 + x; |
u(0,t)= –4 , |
u(1,t)= –1 |
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7.16. ut=7uxx; |
u(x,0)=6sin(3πx) + 3 + x; |
u(0,t)=3 , |
u(2,t)= 7 |
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7.17. ut=6uxx; |
u(x,0)=7sin(4πx) – 2 + x; |
u(0,t)= –2 , |
u(3,t)= 1 |
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7.18. ut=2uxx; |
u(x,0)=8sin(7πx) + 1 – x; |
u(0,t)=1 , |
u(2,t)= –1 |
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7.19. ut=4uxx; |
u(x,0)=9sin(3πx) – 1 – x; |
u(0,t)= –1 , |
u(1,t)= –3 |
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7.20. ut=6uxx; |
u(x,0)=8sin(4πx) + 3 – x; |
u(0,t)=3 , |
u(2,t)= – 5 |
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7.21. ut=7uxx; |
u(x,0)=7sin(3πx) – 5 + x; |
u(0,t)= –5 , |
u(1,t)= 1 |
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7.22. ut=8uxx; |
u(x,0)=6sin(2πx) + 7 – x; |
u(0,t)=7 , |
u(2,t)= –3 |
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7.23. ut=9uxx; |
u(x,0)=5sin(3πx) – 9 + x; |
u(0,t)= –9 , |
u(3,t)= 3 |
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7.24. ut=8uxx; |
u(x,0)=4sin(3πx) + 8 – x; |
u(0,t)= 8, |
u(2,t)=2 |
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7.25. ut=7uxx; |
u(x,0)=3sin(2πx) – 6 + x; |
u(0,t)= –6 , |
u(3,t)= 0 |
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7.26. ut=6uxx; |
u(x,0)=2sin(4πx) + 4 + x; |
u(0,t)=4 , |
u(4,t)= 8 |
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7.27. ut=5uxx; |
u(x,0)=3sin(3πx) – 2 – x; |
u(0,t)= –2 , |
u(3,t)= –5 |
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7.28. ut=3uxx; |
u(x,0)=4sin(5πx) + 3 – x; |
u(0,t)=3 , |
u(2,t)= – 1 |
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7.29. ut=2uxx; |
u(x,0)=5sin(7πx) – 1 – x; |
u(0,t)= –1 , |
u(1,t)= –4 |
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7.30. ut=4uxx; |
u(x,0)=6sin(4πx) + 2 – x; |
u(0,t)=2 , |
u(2,t)= –6 |
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7.31. |
ut=5uxx; |
u(x,0)=7sin(3πx) – 4 – x; |
u(0,t)= –4 , |
u(1,t)= –9 |
10
Задача 8. Решить смешанную задачу для данного неоднородного уравнения теплопроводности с нулевыми начальными и граничными данными.
u(x,0)=0; u(0,t)=0; u(π,t)=0.
8.1. |
ut= |
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uxx + 5sin2t sin3x |
8.2. |
ut= |
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uxx + e−2t sin4x |
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8.3. |
ut= |
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uxx + 10cos3t sin2x |
8.4. |
ut=2uxx + e−10t sin3x |
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8.5. |
ut= |
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uxx + 10sin3t sin4x |
8.6. |
ut= |
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uxx + e−3t sin2x |
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8.7. |
ut= |
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uxx + 2cost sin3x |
8.8. |
ut=3uxx + e−43t sin4x |
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8.9. |
ut= |
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uxx + 5sin2t sin2x |
8.10. |
ut= |
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uxx + e−4t sin3x |
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8.11. |
ut= |
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uxx + 10cos3t sin4x |
8.12. |
ut=5uxx + e−46t sin3x |
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8.13. |
ut= |
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uxx + 2sint sin3x |
8.14. |
ut= |
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uxx + e−5t sin4x |
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8.15. |
ut= |
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uxx + 10cos3t sin3x |
8.16. |
ut=4uxx + e−64t sin4x |
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8.17. |
ut= |
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uxx + 2sint sin4x |
8.18. |
ut= |
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uxx + e−2t sin2x |
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8.19. |
ut= |
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uxx + 5cos2t sin3x |
8.20. |
ut=7uxx + e−63t sin3x |
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uxx + e−3t sin3x |
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8.21. |
ut= |
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uxx + 10sin3t sin2x |
8.22. |
ut= |
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8.23. |
ut= |
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uxx + 5cos2t sin4x |
8.24. |
ut=5uxx + e−20t sin2x |
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uxx + e−4t sin4x |
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8.25. |
ut= |
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uxx + 10sin3t sin3x |
8.26. |
ut= |
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8.27. |
ut= |
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uxx + 5cos2t sin2x |
8.28. |
ut=6uxx + e−24t sin2x |
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8.29. |
ut= |
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uxx + 5sin2t sin4x |
8.30. |
ut= |
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8.31. |
ut= |
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uxx + 2cost sin4x |
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