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,...

Ngày tải lên: 10/09/2015, 15:54

... 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 24...

Ngày tải lên: 13/08/2014, 05:21

... (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 9...

Ngày tải lên: 13/08/2014, 05:21

... −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 23...

Ngày tải lên: 13/08/2014, 05:21

... ) + 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...

Ngày tải lên: 13/08/2014, 05:21

... 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...

Ngày tải lên: 13/08/2014, 05:21

... )(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, deri...

Ngày tải lên: 13/08/2014, 05:21

... 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 5...

Ngày tải lên: 13/08/2014, 05:21

... 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 bec...

Ngày tải lên: 13/08/2014, 05:21

... 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 1.74576...

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: 978-0-470-72335-7 318 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 tha...

Ngày tải lên: 13/08/2014, 05:21

... 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) Stabili...

Ngày tải lên: 13/08/2014, 05:21

... 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 DI...

Ngày tải lên: 13/08/2014, 05:21

... 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 inco...

Ngày tải lên: 13/08/2014, 05:21