Mathematical theory and numerical methods for gross pitaevskii equations and applications

... physics, and motivates numerous mathematical and numerical studies on GPE 1.1 The Gross- Pitaevskii equation Many diﬀerent physical applications lead to the Gross- Pitaevskii equation (GPE) For example, ... uniform convergence rates for CNFD and SIFD, independent of the perturbation We overcome the diﬃculties and obtain uniform error bounds for both CNFD and SIFD, in one, two and three dimensions Numerical ... short-range and a long-range interaction (see (2.17) for details) and thus we can reformulate the GPE (2.5) into a GrossPitaevskii-Poisson type system In addition, based on the new mathematical formulation,...

Numerical Methods for Ordinary Dierential Equations Episode 1 docx

... 10 5 10 5 10 5 10 7 10 7 10 9 11 1 11 2 11 3 11 4 11 4 11 5 11 6 11 9 12 0 12 1 12 2 12 2 12 3 12 4 12 4 12 7 12 8 12 8 13 0 13 1 13 2 13 3 Runge–Kutta Methods 13 7 30 31 Preliminaries ... 16 2 16 3 16 5 16 6 17 0 17 0 17 1 17 5 18 1 18 7 19 0 19 2 19 5 19 8 19 8 19 8 2 01 202 208 210 211 213 213 214 215 219 222 230 230 230 232 238 240 243 245 248 252 254 259 259 2 61 262 266 2 71 273 CONTENTS ... (k1 + k2 y2 ) (10 4i) 16 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 1. 0 0.5 y −0.0 10 4 −0.2 10 4 10 1 y3 10 −2 10 −3 10 −4 10 −5 −0.6 10 4 λ y2 0.2 0.5 −0.4 10 4 y1 10 20 50 10 2 10 3 λ −0.8 10 4...

Numerical Methods for Ordinary Dierential Equations Episode 2 docx

... (y2 + y3 + (y1 + µ − µy3 (y2 3 /2 1 )2 ) + (y1 + µ − 3 /2 1 )2 ) − 3 /2 1 )2 ) (1 − µ)(y1 + µ) (y2 − + y3 + (y1 + µ )2 ) + y3 , (1 − µ)y2 (y2 + y3 + (y1 + µ )2 ) (1 − µ)y3 (y2 3 /2 + (y1 + µ )2 ) 3 /2 3 /2 ... −3.0000000000 1 .25 00000000 −6.5000000000 1.6875000000 2. 6 625 00×10 0.8398437500 −3.800703×1 02 2.13450 622 56 −7 .26 0579×104 −0. 421 6106015 2. 635873×109 1.40063389 92 √ √ y0 ∈ {− 2, + 2} : √ √ y0 ∈ (− 2, + 2) : ... given by 32 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 1 02 Figure 121 (ii) 1+ √ 2+ √ 2 √ √ 1+ √ 10 Solution to neutral delay diﬀerential equation ( 121 c) the formula for y (x) for x positive...

Numerical Methods for Ordinary Dierential Equations Episode 3 pot

... −0.1 632 03 −0.087 530 −0.045 430 −0.0 231 58 −0.0116 93 −0.005875 −0.002945 y4 −0.454109 −0.517588 −0.548 433 −0.5 632 27 −0.57 038 7 −0.5 738 95 −0.575 630 −0.576491 Error 0.569602 0 .30 7510 0.160 531 0.082 134 ... Error 0. 231 124 0.121426 0.06 233 3 0. 031 5 93 0.015906 0.007981 0.0 039 98 0.002001 Euler method: problem (201d) with e = y2 0 .35 1029 0.181229 0.091986 0.04 631 9 0.0 232 38 0.011 638 0.005824 0.0029 13 y3 −0.288049 ... 0.62 236 7 0 .32 2011 0.1 632 35 0.082042 0.041102 0.020567 0.010287 y3 −0. 739 430 −0.47 832 9 −0.284524 −0.158055 −0.0 838 29 −0.0 432 52 −0.021980 −0.011081 y4 0.029212 −0.168796 −0.276187 −0 .32 9290 −0 .35 4 536 ...

Numerical Methods for Ordinary Dierential Equations Episode 4 doc

... ) + O(h4 ), 2 110 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 244 (I) k β1 23 12 55 24 1901 720 42 77 144 0 198721 6 048 0 16083 44 80 β2 β0 β3 −1 4 − 59 24 − 1387 360 − 2 641 48 0 − ... 244 (II) k Coeﬃcients and error constants for Adams–Bashforth methods 12 37 24 109 30 49 91 720 235183 20160 242 653 1 344 0 4 β5 −3 − 637 360 − 3 649 720 − 107 54 945 − 296053 1 344 0 251 720 959 48 0 ... a32 c2 + b4 a42 c2 + b4 a43 c3 = b2 c3 + b3 c3 + b4 c3 = b3 c3 a32 c2 + b4 c4 a42 c2 + b4 c4 a43 c3 = b3 a32 c2 + b4 a42 c2 + b4 a43 c2 = 2 b4 a43 a32 c2 = , , , , , , 12 24 (235a) (235b) (235c)...

Numerical Methods for Ordinary Dierential Equations Episode 5 docx

... 0 189 92 33 75 5 152 33 75 5 152 25 168 25 168 0 0 0 52 9 − 33 75 152 25 1 1 0 3904 33 75 − 127 25 0 0 4232 33 75 − 419 100 19 96 19 96 1472 33 75 1118 − 57 5 − 55 2 − 55 2 0 0 [n] For this method, the output ... quantities are 4232 52 9 3904 1472 yn−1 + yn−2 + h fn−1 + fn−2 , 33 75 33 75 33 75 33 75 189 152 127 419 1118 yn−1 − yn−2 + h fn−1 − fn−2 , fn−7/ 15 − yn = 25 25 92 100 57 5 25 33 75 19 fn + fn−7/ 15 + fn−1 − ... 4.842 85 1.22674 3.30401 × 10−1 8.28328 × 10−2 2.33986 × 10−2 4. 952 05 × 10−3 1.04 655 × 10−3 2.24684 × 10−4 4.89663 × 10 5 1.023 65 × 10 5 2. 151 23 × 10−6 4 .53 436 × 10−7 9 .57 567 × 10−8 2.011 65 × 10−8...

Numerical Methods for Ordinary Dierential Equations Episode 6 ppsx

... )(ci − c6 )ci = i=1 c6 − , 12 (326b) (326c) (326d) RUNGE–KUTTA METHODS 193 , (326e) , (326f) bi = 1, (326g) bi (1 − ci )ci = i=1 bi ci = i=1 i=1 c3 − , 60 24 (326h) bi (1 − ci )(ci − c6 )aij (cj ... a method derived in this manner: 4 1 8 0 16 −3 7 90 −3 8 16 45 16 − 12 15 16 45 90 192 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The ﬁrst methods of this order, derived by Kutta (1901), ... of the matrix A For i corresponding to a member of row k for k = 1, 2, , m, the only non-zero 190 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS aij are for j = and for j corresponding...

Numerical Methods for Ordinary Dierential Equations Episode 7 potx

... 865 473 13055 19 10295610642 − 28 670 62 571 978 869 8 171 12 67 − 245133801 4 679 8 271 1 98 1 071 64 390963 0 − 83 − 11 76 5 1600 75 225 22 576 00 188 1593 2943 1 97 576 2 476 099 0 3315 75 53 20800 975 2 275 29348800 ... 548 74 75 37 312 41 520 31 − 78 0 −1 299 − 1000 688 2 875 0 78 125 572 2 875 33 58 165 38 165 13 50 88 − 575 16 135 − 104 135 132 299 − 100 1 17 50 351 350 351 575 79 2 575 2 376 575 − 1188 15 15 For ... 208 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 1 9 1 18 18 1 24 −1 15 − 63 −1 8 24921 73 5 73 5 − 93 − 10059 22 1408 70 4 1408 70 4 9 670 70 67 15526951598 279 49088 452648800 270 189568 17 865 473 13055...

Numerical Methods for Ordinary Dierential Equations Episode 8 ppsx

... omitted 2 38 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 352(V) m Error constants for diagonal and ﬁrst two sub-diagonals Cm−2,m −1 − 480 75600 − 20321 280 83 825 280 00 − 493 180 0473600 ... 665 280 z 1 5+ 44 z − 66 z + 792 z − 1 584 0 z 665 280 z 5 1 312 z + 3432 z + 11440 z + 3 088 80 z + 17297 280 z 5 1 312 z + 3432 z − 11440 z + 3 088 80 z − 17297 280 z because the expression on the left-hand ... 120 z 3 1 + z + 28 z + 84 z + 1 680 z 1 − z + 28 z − 84 z + 1 680 z 1 1 z + z + 72 z + 10 08 z + 30240 z 1 1 z + z − 72 z + 10 08 z − 30240 z 1 44 z + 66 z + 792 z + 1 584 0 z + 665 280 z 1 5+ 44 z −...

Numerical Methods for Ordinary Dierential Equations Episode 9 ppsx

... 1.4134030 591 3. 596 4257710 7.08581000 59 0.2228466042 9. 8374674184 1.18 893 21017 15 .98 287 398 06 2 .99 27363261 5.7751435 691 0. 193 0436766 8.1821534446 1.026664 895 3 2.5678767450 12.734180 291 8 19. 395 7278623 4 .90 03530845 ... polynomials for degrees s = 1, 2, , s ξ , , ξs 1.0000000000 0.5857864376 3.4142135624 0.4157745568 2. 294 2803603 6.2 899 4508 29 0.3225476 896 1.7457611012 4.536620 296 9 9. 395 07 091 23 0.2635603 197 12.6408008443 ... 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 ≈ 89 268 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS...

Numerical Methods for Ordinary Dierential Equations Episode 10 pot

... requirements Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-72335-7 318 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ... therefore over several steps 322 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 405 Necessity of conditions for convergence We formally prove that stability and consistency are necessary for ... regarded as excessive, then this gives information about the correct value of h to use in a second attempt 310 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS For robustness, a stepsize controller...

Numerical Methods for Ordinary Dierential Equations Episode 11 pptx

... of this test in Subsection 433 346 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Algorithm 432α Boundary locus method for low order Adams–Bashforth methods % Second order % -w = ... 348 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 2i −6 −4 −2 −2i Figure 432(iii) Stability region for the third order Adams–Moulton method 2i −2i Figure 432(iv) Stability region for ... following equations for the predicted and corrected values: ∗ ∗ ∗ yn = yn−1 + hfn−1 − hfn−2 , (434a) 2 ∗ ∗ (434b) yn = yn−1 + hfn + hfn−1 2 350 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS...

Numerical Methods for Ordinary Dierential Equations Episode 12 pptx

... yr Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-72335-7 374 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ... one of these formulations, can be transformed into the sequence that would have been generated using the alternative formulation 376 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS It ... now   0 23 − 12 12     12 − 12       12 − 12 (503a)  0 0       0 0  0 0 The one-leg methods, introduced by Dahlquist (1976) as counterparts of linear multistep methods, have...

Numerical Methods for Ordinary Dierential Equations Episode 13 pps

... form given by Exercise 53.1 420 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 54 Methods with Runge–Kutta stability 540 Design criteria for general linear methods We consider some of the ... NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS stability properties that are usually superior to those of alternative methods For example, A-stability is inconsistent with high order for ... V as a simple matrix, for example a matrix with rank 422 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS If p = q, it is a simple matter to write down conditions for this order and stage...

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