1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 7 491

5. Coordinates and values of the correspondingui are:( 1 / 7 , 1 / 7 ),5α;
   ( 2 / 7 , 1 / 7 ),10α;( 3 / 7 , 1 / 7 ),14α;( 1 / 7 , 2 / 7 ),21α;( 2 / 7 , 2 / 7 ),32α.Here
   α= 19 /1159.
   7.u 1 = 0 .670,u 2 = 0 .721,u 3 = 0 .961,u 4 = 1 .212,u 5 = 0 .954,u 6 = 0 .651.
   The remaining values are found by symmetry.
   9.u 1 = 0 .386,u 2 = 0 .542,u 3 = 0 .784,u 4 = 0 .595. The remaining values are
   found by symmetry.

Section 7.5

1. Use Eq. (8) withr= 1 /4.

i

m 1234 5 6

0 0000 0 0

10 0 0 1 / 41 / 41 / 4

21 / 16 1 / 16 1 / 16 5 / 16 3 / 85 / 16

33 / 32 1 / 83 / 32 23 / 64 27 / 64 23 / 64

3. Note thatu 1 =u 2 =u 4 =u 5 ; replacement equations become
   u 1 (m+ 1 )=u 3 (m)/4,u 3 (m+ 1 )=u 1 (m).

i

m 13

0 11

11 / 41

21 / 41 / 4

31 / 16 1 / 4

41 / 16 1 / 16

5. Use Eq. (8) withr= 1 /4. Note thatu 4 =u 2 ,u 7 =u 3 ,u 8 =u 6.

i

m 12356 9

0 00000 0

10 0 1 /40 1/ 41 / 2

20 1 / 16 5 / 16 1 / 87 / 16 5 / 8

31 / 32 7 / 64 3 / 81 / 417 / 64 23 / 32

7. Use the same numbering as for Exercise 5. Note thatu 1 =u 3 =u 7 =u 9
   andu 2 =u 4 =u 6 =u 8. The running equations become

u 1 (m+ 1 )=^1

2

u 2 (m)−u 1 (m− 1 ),