1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

490 Answers to Odd-Numbered Exercises

5.

i

tm m 01 2 3 4

0001 /21 1/ 20

0.177 1 0 1 / 23 / 41 / 2 0

0.354 2 0 3 / 81 / 43 / 8 0

0.530 3 0 0 − 1 / 800

0.707 4 0 − 7 / 16 − 3 / 8 − 7 / 16 0

0.884 5 0 − 5 / 8 − 11 / 16 − 5 / 8 0

7.

i

m 01 2 3 4

0 00000

1 0 0001

2 0 0011

3 0 0111

4 0 1110

5 0 110 − 1

6 0 00 − 1 − 1

7 0 − 1 − 2 − 1 − 1

8 0 − 2 − 2 − 2 0

9. Run:ui(m+ 1 )=( 2 − 2 ρ^2 − 16 t^2 )ui(m)+ρ^2 ui− 1 (m)+ρ^2 ui+ 1 (m)−
   ui(m− 1 ).Start:ui( 1 )=^12 (( 2 − 2 ρ^2 − 16 t^2 )ui( 0 )+ρ^2 ui− 1 ( 0 )+
   ρ^2 ui+ 1 ( 0 )). Longest stable time step: t= 1 /

√

24 (ρ^2 = 2 / 3 ).

i

m 01 2 34

0 00 .50 1.00 0.50 0

1 0 0. 33 0. 33 0. 33 0

2 0 − 0. 28 − 0. 56 − 0. 28 0

3 0 − 0. 70 − 0. 70 − 0. 70 0

4 0 − 0. 19 − 0. 38 − 0. 19 0

5 0 0. 45 0. 45 0. 45 0

6 0 0. 49 0. 98 0. 49 0

7 0 0. 21 0. 21 0. 21 0

8 0 − 0. 35 − 0. 71 − 0. 35 0

Section 7.4

1. At( 1 / 4 , 1 / 4 ),11/256; at( 1 / 2 , 1 / 4 ),14/256; at( 1 / 2 , 1 / 2 ),18/256.


2. In both this exercise and Exercise 4, the exact solution isu(x,y)=xy,and
   the numerical solutions are exact.