# chapter9 - i book 2005/4/7 22:31 page S-76#97 i i i Answers...

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± ± “book” — 2005/4/7 — 22:31 — page S-76 — #97 ± ± ± ± Answers to Selected Exercises for Chapter 9 Section 9.1 (page 739) 3. (c) Set w ( x, y )= e A ( x )+ B ( y ) u ( x, y ) and then ˜ f ( x, y e A ( x )+ B ( y ) f ( x, y ), where A ( x ) is an antiderivative of a and B ( y ) is an antiderivative of b . 5. (a) ³ 1 h 2 a j,k ´ w j 1 ,k + ³ 1+ h 2 a j,k ´ w j +1 ,k + ³ 1 h 2 b j,k ´ w j,k 1 + ³ h 2 b j,k ´ w j,k +1 4 w j,k = h 2 f j,k (c) h< 2 /L , where L = max( | a ( x, y ) | , | b ( x, y ) | ). 7. (a) 4 1 1 4 / 17 14 1 1 0 1 4 / 65 1 1 2 / 5 1 1 1 0 1 1 2 / 17 1 892 / 1025 1 1 4 / 13 1 1948 / 4745 (b) h rms error rms error / h 2 1/4 0.001360757 0.02177 1/6 0.000566060 0.02038 1/8 0.000307143 0.01966 1/10 0.000192197 0.01922 1/12 0.000131438 0.01893 S-76

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± ± “book” — 2005/4/7 — 22:31 — page S-77 — #98 ± ± ± ± Selected answers for Section 9.2 S-77 9. (a) 4 1 1 (13 / 4) cos(5 / 2) + cos 1 + cos(3 / 2) 14 1 1 (13 / 4) cos(7 / 2) + cos 2 1 (13 / 4) cos(9 / 2) + cos 3 + cos(11 / 2) 1 1 (13 / 4) cos 4 + cos 3 1 1 1 (13 / 4) cos 5 1 1 (13 / 4) cos 6 + cos 7 1 (13 / 4) cos(11 / 2) + cos(9 / 2) + cos 7 1 1 (13 / 4) cos(13 / 2) + cos 8 1 (13 / 4) cos(15 / 2) + cos(17 / 2) + cos 9 (b) h rms error rms error / h 2 1/4 0.1254981 2.00797 1/6 0.0477298 1.71827 1/8 0.0250989 1.60633 1/10 0.0154798 1.54798 1/12 0.0105030 1.51244 11. (a) With λ =(∆ x ) 2 / (∆ y ) 2 , w j 1 ,k + w j +1 ,k + λw j,k 1 + λw j,k +1 2(1 + λ ) w j,k x ) 2 f j,k . (c) The coeﬃcient matrix will always be nonsingular. 13. (a) The coeﬃcient matrix has the form 4 1 1 4 1 1 4 1 1 1 1 4 1 1 4 1 1 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 Section 9.2 (page 752) 1. q ( x ) /p ( x ) 0 3. 4 2 1 h 2 f 0 , 1 + g 0 , 0 +2 1 1 1 h 2 f 1 , 1 + g 1 , 0 1 1 h 2 f 2 , 1 + g 2 , 0 24+ 2 h q 1 p 1 1 h 2 f 3 , 1 + g 3 , 0 h r 1 p 1 2 h 2 f 0 , 2 + g 0 , 3 2 1 1 h 2 f 1 , 2 + g 1 , 3 1 1 h 2 f 2 , 2 + g 2 , 3 1 2 h q 2 p 2 h 2 f 3 , 2 + g 3 , 3
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chapter9 - i book 2005/4/7 22:31 page S-76#97 i i i Answers...

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