ДЗ условия (1175500), страница 3
Текст из файла (страница 3)
= 16 , 0 < < 4, > 0,⎧ 2⎨ , 0 6 6 2,(, 0) = 2⎩4 − , 2 < 6 4,(0, ) = (4, ) = 0.12.5. = 4 , 0 < < 5, > 0,⎧ 2⎨ 2 , 0 6 6 5 ,52(, 0) =⎩5 − , 5 < 6 5,2(0, ) = (5, ) = 0.12.6. = , 0 < < 3, > 0,⎧ 2⎨ 2 , 0 6 6 3 ,32(, 0) =⎩3 − , 3 < 6 3,2(0, ) = (3, ) = 0.12.7. = 25 , 0 < < 8, > 0,⎧ 2⎨ , 0 6 6 4,(, 0) = 4⎩8 − , 4 < 6 8,(0, ) = (8, ) = 0.12.8. = 9 , 0 < < 2, > 0,⎧⎨2 , 0 6 6 1,(, 0) =⎩2 − , 1 < 6 2,(0, ) = (2, ) = 0.12.9.
= 16 , 0 < < 1, > 0,⎧⎨22 , 0 6 6 1 ,2(, 0) =⎩1 − , 1 < 6 1,2(0, ) = (1, ) = 0.12.10. = 4 , 0 < < 4, > 0,⎧ 2⎨ , 0 6 6 2,(, 0) = 2⎩4 − , 2 < 6 4,(0, ) = (4, ) = 0.12.11. = 9 , 0 < < 10, > 0,⎧ 2⎨ , 0 6 6 5,(, 0) = 5⎩10 − , 5 < 6 10,(0, ) = (10, ) = 0.12.12. = 25 , 0 < < 9, > 0,⎧ 2⎨ 2 , 0 6 6 9 ,92(, 0) =⎩9 − , 9 < 6 9,2(0, ) = (9, ) = 0.12.13. = 9 , 0 < < 3, > 0,⎧ 2⎨ 2 , 0 6 6 3 ,32(, 0) =⎩3 − , 3 < 6 3,2(0, ) = (3, ) = 0.12.14. = , 0 < < 5, > 0,⎧ 2⎨ 2 , 0 6 6 5 ,52(, 0) =⎩5 − , 5 < 6 5,2(0, ) = (5, ) = 0.12.15. = 4 , 0 < < 7, > 0,⎧ 2⎨ 2 , 0 6 6 7 ,72(, 0) =⎩7 − , 7 < 6 7,2(0, ) = (7, ) = 0.12.16.
= 25 , 0 < < 1, > 0,⎧⎨22 , 0 6 6 1 ,2(, 0) =⎩1 − , 1 < 6 1,212.23. = 16 , 0 < < 6, > 0,⎧ 2⎨ , 0 6 6 3,(, 0) = 3⎩6 − , 3 < 6 6,(0, ) = (6, ) = 0.12.24. = 4 , 0 < < 1, > 0,⎧⎨22 , 0 6 6 1 ,2(, 0) =⎩1 − , 1 < 6 1,2(0, ) = (1, ) = 0.(0, ) = (1, ) = 0.12.17. = 9 , 0 < < 4, > 0,⎧ 2⎨ , 0 6 6 2,(, 0) = 2⎩4 − , 2 < 6 4,12.25. = 9 , 0 < < 5, > 0,⎧ 2⎨ 2 , 0 6 6 5 ,52(, 0) =⎩5 − , 5 < 6 5,(0, ) = (4, ) = 0.(0, ) = (5, ) = 0.12.18. = , 0 < < 10, > 0,⎧ 2⎨ , 0 6 6 5,(, 0) = 5⎩10 − , 5 < 6 10,(0, ) = (10, ) = 0.12.19. = 4 , 0 < < 2, > 0,⎧⎨2 , 0 6 6 1,(, 0) =⎩2 − , 1 < 6 2,(0, ) = (2, ) = 0.12.20.
= 16 , 0 < < 8, > 0,⎧ 2⎨ , 0 6 6 4,(, 0) = 4⎩8 − , 4 < 6 8,212.26. = 25 , 0 < < 6, > 0,⎧ 2⎨ , 0 6 6 3,(, 0) = 3⎩6 − , 3 < 6 6,(0, ) = (6, ) = 0.12.27. = , 0 < < 12, > 0,⎧ 2⎨ , 0 6 6 6,(, 0) = 6⎩12 − , 6 < 6 12,(0, ) = (12, ) = 0.12.28. = 16 , 0 < < 2, > 0,⎧⎨2 , 0 6 6 1,(, 0) =⎩2 − , 1 < 6 2,(0, ) = (8, ) = 0.(0, ) = (2, ) = 0.12.21. = , 0 < < 1, > 0,⎧⎨22 , 0 6 6 1 ,2(, 0) =⎩1 − , 1 < 6 1,12.29. = 4 , 0 < < 6, > 0,⎧ 2⎨ , 0 6 6 3,(, 0) = 3⎩6 − , 3 < 6 6,(0, ) = (1, ) = 0.(0, ) = (6, ) = 0.212.22.
= 25 , 0 < < 4, > 0,⎧ 2⎨ , 0 6 6 2,(, 0) = 2⎩4 − , 2 < 6 4,(0, ) = (4, ) = 0.12.30. = 36 , 0 < < 3, > 0,⎧ 2⎨ , 0 6 6 3,2(, 0) = 3⎩3 − , 3 < 6 3,2(0, ) = (3, ) = 0.12.31. = 9 , 0 < < 8, > 0,⎧ 2⎨ , 0 6 6 4,(, 0) = 4⎩8 − , 4 < 6 8,(0, ) = (8, ) = 0.Задача 13. Найти решение первой смешанной краевой задачи для уравнения теплопроводностив круге.13.1. = 16D, 0 6 < 5, > 0,13.11. = 10D, 0 6 < 2, > 0,(, 0) = 25 − 2 ,(, 0) = 4 − 2 ,(5, ) = 0.(2, ) = 0.13.2.
= 25D, 0 6 < 3, > 0,13.12. = 25D, 0 6 < 4, > 0,(, 0) = 9 − 2 ,(, 0) = 16 − 2 ,(3, ) = 0.(4, ) = 0.13.3. = D, 0 6 < 7, > 0,13.13. = 6D, 0 6 < 6, > 0,(, 0) = 49 − 2 ,(, 0) = 36 − 2 ,(7, ) = 0.(6, ) = 0.13.4. = 10D, 0 6 < 1, > 0,13.14. = 4D, 0 6 < 1, > 0,(, 0) = 1 − 2 ,(, 0) = 1 − 2 ,(1, ) = 0.(1, ) = 0.13.5. = 9D, 0 6 < 8, > 0,(, 0) = 64 −2 ,(8, ) = 0.2 ,(2, ) = 0.2 ,(1, ) = 0.2 ,(3, ) = 0.13.17. = D, 0 6 < 1, > 0,(, 0) = 1 − 2 ,(, 0) = 16 −2 ,(4, ) = 0.(, 0) = 4 − 2 ,13.19. = 9D, 0 6 < 3, > 0,(, 0) = 9 − 2 ,(3, ) = 0.13.10.
= D, 0 6 < 5, > 0,(, 0) = 25 −13.18. = D, 0 6 < 2, > 0,(2, ) = 0.13.9. = 5D, 0 6 < 4, > 0,(5, ) = 0.(, 0) = 49 − 2 ,(1, ) = 0.13.8. = 3D, 0 6 < 3, > 0,(, 0) = 9 −13.16. = 10D, 0 6 < 7, > 0,(7, ) = 0.13.7. = 16D, 0 6 < 1, > 0,(, 0) = 1 −(, 0) = 25 − 2 ,(5, ) = 0.13.6. = 25D, 0 6 < 2, > 0,(, 0) = 4 −13.15. = 7D, 0 6 < 5, > 0,2 ,13.20.
= 2D, 0 6 < 4, > 0,(, 0) = 16 − 2 ,(4, ) = 0.13.21. = 36D, 0 6 < 7, > 0,(, 0) = 49 −2 ,(7, ) = 0.13.27. = 5D, 0 6 < 1, > 0,(, 0) = 1 − 2 ,(1, ) = 0.13.22. = 4D, 0 6 < 2, > 0,(, 0) = 4 −2 ,(2, ) = 0.13.28. = 9D, 0 6 < 2, > 0,(, 0) = 4 − 2 ,(2, ) = 0.13.23. = D, 0 6 < 4, > 0,(, 0) = 16 −2 ,(4, ) = 0.13.29. = 6D, 0 6 < 7, > 0,(, 0) = 49 − 2 ,(7, ) = 0.13.24.
= 25D, 0 6 < 5, > 0,(, 0) = 25 −2 ,(5, ) = 0.13.30. = 4D, 0 6 < 8, > 0,(, 0) = 64 − 2 ,(8, ) = 0.13.25. = 2D, 0 6 < 3, > 0,(, 0) = 9 −2 ,(3, ) = 0.13.31. = 7D, 0 6 < 4, > 0,(, 0) = 16 − 2 ,(4, ) = 0.13.26. = 3D, 0 6 < 6, > 0,(, 0) = 36 − 2 ,(6, ) = 0.Задача 14. Используя формулу Кирхгофа, найти решение задачи Коши для волнового уравненияна плоскости.14.1.
= 2 ( + ),|=0 = 0, |=0 = ( + )2 .14.2. = 2 ( + ),|=0 = 0, |=0 = ( − )2 .14.3. = 2 ( + ),|=0 = 0, |=0 = (2 + )2 .14.4. = 2 ( + ),|=0 = 0, |=0 = ( + 2)2 .14.5. = 2 ( + ),|=0 = 0, |=0 = (2 − )2 .14.6. = 2 ( + ),|=0 = 0, |=0 = ( − 2)2 .14.7. = 2 ( + ),|=0 = 0, |=0 = (3 + )2 .14.8. = 2 ( + ),|=0 = 0, |=0 = ( − 3)2 .14.9. = 2 ( + ),|=0 = 0, |=0 = (2 + 3)2 .14.10. = 2 ( + ),|=0 = 0, |=0 = (2 − 3)2 .14.11.
= 2 ( + ),|=0 = 0, |=0 = (3 + 4)2 .14.12. = 2 ( + ),|=0 = 0, |=0 = (3 − 4)2 .14.13. = 2 ( + ),|=0 = 0, |=0 = (5 + 6)2 .14.14. = 2 ( + ),|=0 = 0, |=0 = (5 − 6)2 .14.15. = 2 ( + ),|=0 = 0,14.24. = 2 ( + ), |=0 = (6 + 7)2 .14.16. = 2 ( + ),|=0 = 0,14.17.
= 2 ( + ), |=0 = 0. |=0 = 0. |=0 = 0. |=0 = 0.|=0 = 22 + 2 , |=0 = 0.|=0 = 2 + 2 2 , |=0 = 0.|=0 = 2 − 2 , |=0 = 0.14.30. = 2 ( + ), |=0 = 0.14.22. = 2 ( + ),|=0 = 52 − 6 2 , |=0 = 0.14.29. = 2 ( + ),14.21. = 2 ( + ),|=0 = 52 + 6 2 ,|=0 = 2 − 2 ,14.28. = 2 ( + ),14.20. = 2 ( + ),|=0 = 42 − 5 2 , |=0 = 0.14.27. = 2 ( + ),14.19. = 2 ( + ),|=0 = 42 + 5 2 ,|=0 = 2 + 2 ,14.26. = 2 ( + ),14.18. = 2 ( + ),|=0 = 32 − 4 2 , |=0 = 0.14.25.
= 2 ( + ), |=0 = (6 − 7)2 .|=0 = 32 + 4 2 ,|=0 = 62 − 7 2 ,|=0 = 32 + 2 2 , |=0 = 0.14.31. = 2 ( + ), |=0 = 0.|=0 = 32 + 2 , |=0 = 0.14.23. = 2 ( + ),|=0 = 62 + 7 2 , |=0 = 0.Задача 15. Используя формулу Кирхгофа, найти решение задачи Коши для волнового уравненияв пространстве.15.1. = + + ,|=0 = 0, |=0 = ( − +15.7.
= 7( + + ),)2 .15.2. = 2( + + ),|=0 = 0, |=0 = ( − 2 + |=0 = ( + − )2 . |=0 = ( + −|=0 = 0, |=0 = (2 + 3 + 4)2 . |=0 = (2 + + )2 . |=0 = (2 + +|=0 = 0, |=0 = (3 + 2 + 4)2 .15.11. = 11( + + ),15.6. = 6( + + ),|=0 = 0, |=0 = ( + 2 + 3)2 .15.10. = 10( + + ),2)2 .15.5. = 5( + + ),|=0 = 0,|=0 = 0,15.9. = 9( + + ),15.4. = 4( + + ),|=0 = 0, |=0 = ( + + 3)2 .15.8. = 8( + + ),)2 .15.3.
= 3( + + ),|=0 = 0,|=0 = 0,|=0 = 0, |=0 = (2 + 4 + 3)2 .15.12. = 12( + + ),2)2 .|=0 = 0, |=0 = (4 + 2 + 3)2 .15.13. = 13( + + ), |=0 = (4 + 3 +|=0 = 0,15.23. = 9( + + ),2)2 .15.14. = 14( + + ),15.15. = 15( + + ), |=0 = (5 + 4 +|=0 = 0, |=0 = (3 + 4 + 5)2 .15.17. = 15( + + ),|=0 =2−2+2, |=0 = 0.15.18. = 14( + + ),|=0 =2−2 2+2, |=0 = 0.15.19. = 13( + + ),|=0 =2+2−2, |=0 = 0.15.20.
= 12( + + ),|=0 =2+2−2 2 , |=0 = 0.15.21. = 11( + + ),|=0 = 22 + 2 + 2 2 ,|=0 = 2 + 2 2 + 3 2 , |=0 = 0.15.25. = 7( + + ),3)2 .15.16. = 16( + + ),|=0 = 0, |=0 = 0.15.24. = 8( + + ), |=0 = (3 + 4 + 2)2 .|=0 = 0,|=0 = 2 + 2 + 3 2 , |=0 = 0.|=0 = 32 + 2 2 + 2 , |=0 = 0.15.26. = 6( + + ),|=0 = 32 + 2 + 2 2 , |=0 = 0.15.27.
= 5( + + ),|=0 = 22 + 2 + 3 2 , |=0 = 0.15.28. = 4( + + ),|=0 = 22 + 3 2 + 2 , |=0 = 0.15.29. = 3( + + ),|=0 = 32 + 4 2 + 5 2 , |=0 = 0.15.30. = 2( + + ),|=0 = 42 + 5 2 + 3 2 , |=0 = 0.15.31. = 15( + + ),|=0 = 52 + 3 2 + 4 2 , |=0 = 0.15.22. = 10( + + ),|=0 = 22 + 2 + 2 , |=0 = 0.Задача 16. Используя формулу Пуассона, найти решение задачи Коши для уравнения теплопроводности.16.1. = ,|=0 =16.6. = 6 ,2− + .|=0 = −16.2. = 2 ,|=0 = −2|=0 = −2.216.9.
= 9 ,.16.5. = 5 ,|=0 =.|=0 = −3 .16.4. = 4 ,−22 +2 +216.8. = 8 ,2−2 .|=0 = .16.7. = 7 ,16.3. = 3 ,|=0 =2 −2−2 − .|=0 = −32 +.16.10. = 10 ,|=0 = −22 −2.16.11. = 11 ,|=0 =16.22. = 10 ,2−3 − .|=0 = −216.12. = 12 ,|=0 = −.16.13. = 13 ,|=0 =16.14. = 14 ,|=0 = −4|=0 = −3|=0 = −3.2 −.2 −6.2 +6.16.29. = 3 ,2−3 +3 .16.19. = 13 ,|=0 = −42 −6.16.30. = 2 ,−22 +4|=0 = .16.20. = 12 ,|=0 = −42 +6.16.31. = ,2− −2 .|=0 = −416.21. = 11 ,|=0 = −32 +216.28. = 4 ,2− +2 .16.18. = 14 ,|=0 =.16.27. = 5 ,2−4 −2 .16.17. = 15 ,|=0 =2 −416.26.
= 6 ,2−3 −2 .16.16. = 16 ,|=0 =|=0 = −|=0 = −4.16.15. = 15 ,|=0 =.16.25. = 7 ,−42 +|=0 =2 +416.24. = 8 ,2−3 +2 .|=0 = .16.23. = 9 ,−42|=0 = 2 −42 −3.2 −4..