Numerical Methods for Ordinary Dierential Equations Episode 8 ppsx

... =5,p =8) , 0 1 20 −7− √ 21 120 1 15 −7+ √ 21 120 0 7− √ 21 14 1 20 343+9 √ 21 2520 56−15 √ 21 315 343−69 √ 21 2520 0 1 2 1 20 49+12 √ 12 360 8 45 49−12 √ 12 360 0 7+ √ 21 14 1 20 343+69 √ 21 2520 56+15 √ 21 315 343−9 √ 21 2520 0 1 1 20 119−3 √ 21 360 13 45 119+3 √ 21 360 0 1 20 49 180 16 45 49 180 1 20 Lobatto IIIC (s =5,p =8) , 0 1 20 − 7 60 2 15 − 7 60 1 20 7− √ 21 14 1 20 29 180 47...

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... manner: 0 1 4 1 4 1 4 1 8 1 8 1 2 00 1 2 3 4 3 16 − 3 8 3 8 9 16 1 − 3 7 8 7 6 7 − 12 7 8 7 7 90 0 16 45 2 15 16 45 7 90 . 166 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Another approach ... Φ(t)=1/γ(t)is satisfied for trees t =[τ k−1 [τ l−1 ]] for k ≤ η and l ≤ ζ.Thisisanecessary condition for orders at least η + ζ. 188 NUMERICAL METHODS FOR ORDI...

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... −b s 280 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 38 Algebraic Properties of Runge–Kutta Methods 380 Motivation For any specific N-dimensional initial value problem, Runge–Kutta methods can ... polynomial 185 976λ 12 − 1490400λ 11 + 46014 48 10 − 72571 68 9 + 684 285 3λ 8 −4 181 760λ 7 +1724256λ 6 − 487 296λ 5 +94176λ 4 −12192λ 3 +10 08 2 − 48 +1. 37 Symple...

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... DIFFERENCE EQUATIONS 13 Table 103(I) Period of simple pendulum for various amplitudes Θ T 0 ◦ 6. 283 185 3072 3 ◦ 6. 284 262 083 1 6 ◦ 6. 287 4944421 9 ◦ 6.29 288 8 488 0 12 ◦ 6.3004544311 15 ◦ 6.3102066431 18 ◦ 6.3221637356 21 ◦ 6.3363 486 630 24 ◦ 6.352 788 8501 27 ◦ 6.3715163462 30 ◦ 6.3925 680 085 where ... Fourth order methods 98 236 Higher orders 99 237 Implicit Runge–Kutta m...

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... these assumptions, the equations of motion become 38 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 13 Difference Equation Problems 130 Introduction to difference equations While differential equations ... 2a 1 (a 0 − a 1 ) −4a 2 (a 1 − a 2 ) −8a 3 (a 2 − a 3 ) −···  . 18 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 0246 u 024 68 v Figure 106(i) Ph...

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... −0.03396724 28 10 0.0 181 7 489 −0.017370 78 47 20 0.0 089 8 483 −0.0 087 8393 80 40 0.00446704 −0.00441 680 149 80 0.00222721 −0.00221462 240 160 0.00111203 −0.0011 088 9 480 320 0.00055562 −0.00055 484 960 640 0.00027771 ... −0.0401 685 8 3.9676 3.9 688 64 0.00372 781 −0.010120 98 4.0417 4.0001 1 28 0.00092233 −0.00253020 4.0361 4.0041 256 0.0002 285 2 −0.00063190 4.0219 4.0031 512...

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... − 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 − 1 387 360 109 30 − 637 360 251 720 − 95 288 6 4277 1440 − 2641 480 4991 720 − 3649 720 959 480 − 95 288 19 087 60 480 7 1 987 21 60 480 − 186 37 2520 235 183 20160 − 10754 945 135713 20160 − 5603 2520 19 087 60 480 − 5257 17 280 8 16 083 4 480 − 1152169 120960 242653 13440...

Ngày tải lên: 13/08/2014, 05:21

... 7 8 −1 1.22674 3.94773 8 8 −2 3.30401 × 10 −1 3.71 289 8 8 −3 8. 283 28 × 10 −2 3. 988 76 10 8 −4 2.33 986 × 10 −2 3.54007 13 8 −5 4.95205 × 10 −3 4.72504 19 8 −6 1.04655 × 10 −3 4.73 180 30 8 −7 2.24 684 × 10 −4 4.65 786 50 8 8 4 .89 663 ... 10 −4 4.65 786 50 8 8 4 .89 663 × 10 −5 4. 588 54 82 8 −9 1.02365 × 10 −5 4. 783 50 137 8 −10 2.15123 × 10 −6 4.7 584 5 2...

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... − 1 2 3 4 − 93 22 0 24921 14 08 − 10059 704 735 14 08 735 704 17 19 86 547313055 10295610642 0 − 96707067 286 7062 155269515 98 5719 788 69 27949 088 81 711267 − 4526 488 00 24513 380 1 270 189 5 68 467 982 711 1 98 765 00 − 9 83 1071 1600 − 11 75 64 225 390963 2257600 188 3315 00 1593 7553 2943 2 080 0 197 975 576 2275 2476099 293 488 00 2 39 − 142 1 989 00 2412 7553 − 549 1040...

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... 1) 2 − 3 8 (z − 1) 3 + 251 720 (z − 1) 4 − 95 288 (z − 1) 5 + 19 087 60 480 (z − 1) 6 − 5257 17 280 (z −1) 7 + 1070017 36 288 00 (z − 1) 8 − 25713 89 600 (z − 1) 9 + 2 684 2253 9 580 0320 (z − 1) 10 −··· . For ... requirements. Numerical Methods for Ordinary Differential Equations, Second Edition. J. C. Butcher © 20 08 John Wiley & Sons, Ltd. ISBN: 9 78- 0-470-72335-...

Ngày tải lên: 13/08/2014, 05:21