Numerical Methods for Ordinary Dierential Equations Episode 12 pptx

... representation is                      000 − 225 128 200 128 153 128 225 128 300 128 45 128 384 155 00 540 128 − 297 31 − 212 31 − 1395 155 − 2130 155 − 309 155 2304 3085 465 3085 0 783 617 − 135 617 − 31 617 − 135 3085 − 495 3085 − 39 3085 2304 3085 465 3085 0 783 617 − 135 617 − 31 617 − 135 3085 − 495 3085 − 39 3085 000100 ... general linear formulation...

Ngày tải lên: 13/08/2014, 05:21

... equation for the error in step n, which will be written as  n = y(x n ) −y n ,intheform  n − k  i=1 α i  n−i = Kh p+1 , 362 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS For U ∈ R kN ,writeU i , ... y n ). (452c) 350 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 1 2 3 1 2 i −i −2 −1 0 Figure 434(i) Stability regions for Adams–Moulton methods (s...

Ngày tải lên: 13/08/2014, 05:21

... derivatives 120 255 Further numerical examples 121 26 Hybrid Methods 122 260 Historical introduction 122 261 Pseudo Runge–Kutta methods 123 262 Generalized linear multistep methods 124 263 General ... k 2 y 3 +2k 3 y 2 )λ +2k 3 y 2 (k 1 + k 2 y 2 ). (104i) xiv NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS problems. For implicit methods for stiff prob...

Ngày tải lên: 13/08/2014, 05:21

... comprising (124 d), (124 e), (124 f) has been reduced, first to an index 1 formulation (124 d), (124 e), (124 h), and then to an ordinary differential equation system (124 d), (124 e), (124 i). Exercises 12 12.1 ... + Cθ 2 (θ−1)(θ−2) θ n ,θ=1,θ=2. 32 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 0 1 2 3 4 √ 2 1+ √ 2 2 √ 2 2+ √ 2 1+2 √ 2 10 10 2 1 Figure 121 (ii) Solutio...

Ngày tải lên: 13/08/2014, 05:21

... value. 62 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 0123 0.05 0.10 0.15 x h Figure 203(iii) Stepsize h against x for the ‘mildly stiff’ problem (203a) with variable stepsize for T =0.02 The ... This is done by writing Φ(ξ, x) for the solution at x to the differential equation w  (x)= ∂f ∂y w(x),w(ξ)=I, 66 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIO...

Ngày tải lên: 13/08/2014, 05:21

... (252a) 110 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 244(I) Coefficients and error constants for Adams–Bashforth methods k β 1 β 2 β 3 β 4 β 5 β 6 β 7 β 8 C 1 1 − 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 − 2641 480 4991 720 − 3649 720 959 480 − 95 28...

Ngày tải lên: 13/08/2014, 05:21

... labelled by Numerical Methods for Ordinary Differential Equations, Second Edition. J. C. Butcher © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-72335-7 158 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... 126 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The second order Adams–Bashforth and Adams–Moulton and PECE methods based on these are, respectivel...

Ngày tải lên: 13/08/2014, 05:21

... 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 ORDINARY DIFFERENTIAL EQUATIONS Again we see ... condition is 178 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS b 1 = 6c 3 c 2 − 2c 3 − 2c 2 +1 12c 3 c 2 , b 2 = − (2c 3 − 1) 12c 2 (c 2 − 1)(c 3 − c 2 ) , b 3...

Ngày tải lên: 13/08/2014, 05:21

... 360. 226 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Radau II (s =3,p =5), 4− √ 6 10 24− √ 6 120 24−11 √ 6 120 0 4+ √ 6 10 24+11 √ 6 120 24+ √ 6 120 0 1 6− √ 6 12 6+ √ 6 12 0 16− √ 6 36 16+ √ 6 36 1 9 Radau ... 342(i). To turn this result into a recipe for constructing methods of order 2s we have: 224 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQU...

Ngày tải lên: 13/08/2014, 05:21

... P k−1 (v) is √ 2k −1  2k−2 k−1  , with a similar formula for the leading coefficient of P k (u). 240 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The fact that D s (z) cannot vanish now ... Vandermonde matrix V =  1 cc 2 ··· c s−1  , 256 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Lemma 359A Let X G = W BAW, where A and B =diag(b) are as for the Ga...

Ngày tải lên: 13/08/2014, 05:21