Numerical Methods for Ordinary Dierential Equations Episode 5 docx

... is              000 − 52 9 33 75 3904 33 75 4232 33 75 1472 33 75 189 92 00 152 25 − 127 25 − 419 100 − 1118 57 5 33 75 5 152 25 168 0 10 19 96 − 1 55 2 33 75 5 152 25 168 0 10 19 96 − 1 55 2 000 1000 001 0000 000 0010              . For ... 10 −2 3 .54 007 13 8 5 4. 952 05 × 10 −3 4.7 250 4 19 8 −6 1.04 655 × 10 −3 4.73180 30 8 −7 2.24684...

Ngày tải lên: 13/08/2014, 05:21

... radius 440 55 3 Derivation of methods with IRK stability 442 55 4 Methods with property F 4 45 555 Some non-stiff methods 446 55 6 Some stiff methods 447 55 7 Scale and modify for stability 448 55 8 Scale ... method 433 54 7 ARK methods for stiff problems 434 55 Methods with Inherent Runge–Kutta Stability 436 55 0 Doubly companion matrices 436 55 1 Inherent Runge–Kutta...

Ngày tải lên: 13/08/2014, 05:21

... were (y 1 ,y 2 ,y 3 ,y 4 ,y 5 ,y 6 )=(0.994, 0, 0, 0, −2.00 158 510637908 252 24, 0) and the period was 17.0 652 1 656 0 157 96. 34 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS y 1 y 2 Figure 122(i) Solution to ... k<1, such that, for any η, ζ ∈ M, ρ(φ(η),φ(ζ)) ≤ kρ(η, ζ). Then there exists a unique ξ ∈ M such that φ(ξ)=ξ. 46 NUMERICAL METHODS FOR ORDINARY DI...

Ngày tải lên: 13/08/2014, 05:21

... −0.000039 45 n y 3 error Ratio y 4 error Ratio 32 0.04038636 −0.0 154 8 159 3.9497 4. 155 2 64 0.0102 252 5 −0.0037 258 5 4.0132 4.0 659 128 0.00 254 793 −0.00091636 4.0163 4.0294 256 0.00063440 −0.00022742 4.01 05 ... stepsizes we have used, the errors are respectively 1.7247 45 − 1.68 755 5 = 0.037190, 1.7247 45 − 1.70 657 0 = 0.0181 75 and 1.7247 45 −1.7 157 60 = 0.0089 85. These...

Ngày tải lên: 13/08/2014, 05:21

... −0.002 153 39 −0.00019419 3.2 451 8.1221 128 −0.00066 358 −0.00002391 3. 655 1 8.1017 256 −0.00018 155 −0.000002 95 3.8 351 8.0620 51 2 −0.00004734 −0.00000037 3.9194 8.0339 1024 −0.00001208 −0.000000 05 experiments ... − 1 2 2 3 2 − 1 2 5 12 3 23 12 − 4 3 5 12 − 3 8 4 55 24 − 59 24 37 24 − 3 8 251 720 5 1901 720 − 1387 360 109 30 − 637 360 251 720 − 95 288 6 4277 1440 − 2...

Ngày tải lên: 13/08/2014, 05:21

... equations are used in the choice of particular coefficients of the method: 0 (325b) (325f) (325b) (325f) (325e) (325b) (325f) (325d) (325e) (325b) (325f) (325g) (325e) (325h) (325a) (325i) (325i) ... is 0 1 3 1 3 2 5 4 25 6 25 1 1 4 −3 15 4 2 3 2 27 10 9 − 50 81 8 81 4 5 2 25 12 25 2 15 8 75 0 23 192 0 1 25 192 0 − 27 64 1 25 192 . As we have pointed out in Subsection 31...

Ngày tải lên: 13/08/2014, 05:21

... − 1 2 1 13 20 13 200 − 299 1000 78 1 25 13 50 4 5 548 74 75 688 28 75 572 28 75 − 88 57 5 132 299 1 37 312 0 4 33 8 9 − 100 117 57 5 792 41 52 0 0 58 1 65 16 1 35 50 351 57 5 2376 1 15 − 31 780 0 38 1 65 − 104 1 35 350 351 − 57 5 1188 1 15 . For ... =2,p=4, 1 2 − √ 3 6 1 4 1 4 − √ 3 6 1 2 + √ 3 6 1 4 + √ 3 6 1 4 1 2 1 2 ; s =3,p=6, 1 2 − √ 15 10 5 36 2 9 − √...

Ngày tải lên: 13/08/2014, 05:21

... + 3 22 z 2 − 2 99 z 3 + 1 52 8 z 4 − 1 9240 z 5 + 1 332640 z 6 7 1+ 6 13 z + 5 52 z 2 + 5 429 z 3 + 1 1144 z 4 + 1 257 40 z 5 + 1 12 355 20 z 6 1 − 7 13 z + 7 52 z 2 − 35 1716 z 3 + 7 3432 z 4 − 7 51 480 z 5 + 7 12 355 20 z 6 − 1 8648640 z 7 For ... + 1 7 z 2 − 2 1 05 z 3 + 1 840 z 4 5 1+ 4 9 z + 1 12 z 2 + 1 126 z 3 + 1 3024 z 4 1 − 5 9 z + 5 36 z 2 − 5 252 z 3 +...

Ngày tải lên: 13/08/2014, 05:21

... +3ξ 2 − 2 3 ξ 3 + 1 24 ξ 4 51 − 5 +5 2 − 5 3 ξ 3 + 5 24 ξ 4 − 1 120 ξ 5 61− 6ξ + 15 2 ξ 2 − 10 3 ξ 3 + 5 8 ξ 4 − 1 20 ξ 5 + 1 720 ξ 6 71− 7ξ + 21 2 ξ 2 − 35 6 ξ 3 + 35 24 ξ 4 − 7 40 ξ 5 + 7 720 ξ 6 − 1 50 40 ξ 7 81− ... acceptable for many problems. In each of the values of ξ for which there is a single underline, the method is A(α)-stable with α ≥ 1 .55 ≈ 89 ◦ . 272...

Ngày tải lên: 13/08/2014, 05:21

... requirements. Numerical Methods for Ordinary Differential Equations, Second Edition. J. C. Butcher © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-723 35- 7 332 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... β 4 + 1 2 β 2 + 1 24 95 9 + β 9 β 9 +4β 5 +6β 3 +4β 2 + 1 5 10 5 α 10 + β 2 α 3 + β 10 β 10 +2β 6 +β 5 +β 4 + 5 2 β 3 +2β 2 + 1 10 11 5 α 11 + β 3 α 2 +...

Ngày tải lên: 13/08/2014, 05:21